Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures: Foundations and Applications (STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health) [1st ed. 2023] 3031393333, 9783031393334

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STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health

Mahouton Norbert Hounkonnou Melanija Mitrović Mujahid Abbas Madad Khan   Editors

Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures Foundations and Applications

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health Series Editor Bourama Toni, Department of Mathematics, Howard University, Washington, DC, USA

This interdisciplinary series highlights the wealth of recent advances in the pure and applied sciences made by researchers collaborating between fields where mathematics is a core focus. As we continue to make fundamental advances in various scientific disciplines, the most powerful applications will increasingly be revealed by an interdisciplinary approach. This series serves as a catalyst for these researchers to develop novel applications of, and approaches to, the mathematical sciences. As such, we expect this series to become a national and international reference in STEAM-H education and research. Interdisciplinary by design, the series focuses largely on scientists and mathematicians developing novel methodologies and research techniques that have benefits beyond a single community. This approach seeks to connect researchers from across the globe, united in the common language of the mathematical sciences. Thus, volumes in this series are suitable for both students and researchers in a variety of interdisciplinary fields, such as: mathematics as it applies to engineering; physical chemistry and material sciences; environmental, health, behavioral and life sciences; nanotechnology and robotics; computational and data sciences; signal/image processing and machine learning; finance, economics, operations research, and game theory. The series originated from the weekly yearlong STEAM-H Lecture series at Virginia State University featuring world-class experts in a dynamic forum. Contributions reflected the most recent advances in scientific knowledge and were delivered in a standardized, self-contained and pedagogically-oriented manner to a multidisciplinary audience of faculty and students with the objective of fostering student interest and participation in the STEAM-H disciplines as well as fostering interdisciplinary collaborative research. The series strongly advocates multidisciplinary collaboration with the goal to generate new interdisciplinary holistic approaches, instruments and models, including new knowledge, and to transcend scientific boundaries. Peer reviewing All monographs and works selected for contributed volumes within the STEAMH series undergo peer review. The STEAM-H series follows a single-blind review process. A minimum of two reports are asked for each submitted manuscript. The Volume Editors act in cooperation with the Series Editor for a final decision. The Series Editors agrees with and follows the guidelines published by the Committee on Publication Ethics. Titles from this series are indexed by Scopus, Mathematical Reviews, and zbMATH.

Mahouton Norbert Hounkonnou • Melanija Mitrovi´c • Mujahid Abbas • Madad Khan Editors

Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures Foundations and Applications

Editors Mahouton Norbert Hounkonnou International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair) University of Abomey-Calavi Cotonou, Benin Mujahid Abbas Government College University Lahore, Pakistan

Melanija Mitrovi´c CAM-FMEN University of Niš Niš, Serbia Madad Khan Abbottabad Campus COMSATS University Islamabad Abbottabad, Pakistan

ISSN 2520-193X ISSN 2520-1948 (electronic) STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health ISBN 978-3-031-39333-4 ISBN 978-3-031-39334-1 (eBook) https://doi.org/10.1007/978-3-031-39334-1 Mathematics Subject Classification: 03F65, 17-XX, 18-XX, 22-XX, 35-XX © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To Jelena and Nikola, children of Nebojša Stevanovi´c

Foreword

Mathematics is in essence a universal language but needs to further break down the borders between fields and researchers. This volume is a valuable contribution presenting some recent developments in the area of algebra, in particular nonassociative algebras, their generalizations, and interactions with other domains like physics, knot theory, and cryptography. The subjects of algebra have grown spectacularly since several decades; algebra reasoning and combinatorial aspects turn to be very efficient in solving various problems in different domains. The objective of this volume is to report on the new trends of research in algebra and related topics, likewise to provide an insight into the fast development of new concepts and theories. The first chapter is dedicated to quandles which are non-associative algebraic structures whose axioms are modeled on the three Reidemeister moves in knot theory. They aim to obtain invariants of knots and links, by considering them from many different points of views like Yang-Baxter equation and in relation with Lie algebras, Frobenius algebras, Hopf algebras, representation theory, quasigroups, and Moufang loops. The second chapter aims to introduce and interpret new splittings of operations of Poisson algebras and transposed Poisson algebras in terms of their representations. These types of constructions were first introduced by Loday who used dendriform algebras to split associative operations. Such structures turn out to be useful in algebraic K-theory and many other fields in mathematics and physics, such as arithmetic, combinatorics, Hopf algebras, operads, and quantum field theory. The third chapter deals with some loops of Bol-Moufang types (right Bol loop and central loops), a particular category of loops (called Osborn loops) of nonBol-Moufang type (among which are Moufang loops, conjugacy closed loops (CC-loops), universal weak inverse property loops). These structures appeared in algebra, geometry, topology, and combinatorics. An importance of some algebraic tools in physics is shown when dealing with symmetries and generalizing Dirac equation for quarks in which one needs to consider the cyclic group .Z3 . Chapter 4 proposes the construction of .Z3 -

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graded extension of the Poincaré algebra and discusses representations in terms of differential operators, generalized Casimir operators, and symmetry properties. Ternary operations and more generally n-ary operations are natural generalization of binary operations. They appear in various domains of physics, mathematics, and data processing. Relevant applications are considered in string theory and in Nambu mechanics, generalizing usual hamiltonian mechanics, in which Lie algebra is replaced by a 3-Lie algebra. Chapter 5 provides a study of ternary Leibniz color algebras, which are G-graded ternary Leibniz algebras. Various related structures are considered as bimodule over ternary Leibniz color algebra, ternary LeibnizPoisson color algebras, color Lie triple systems, and Comstrans color algebras, while Chap. deals with dual structures of ternary algebras. Partially and totally (co)associative ternary (co)algebras, and infinitesimal bialgebras are studied in classical case and twisted case (Hom-case). They show that there is a duality between algebra structures and coalgebra structures. The concepts of trimodules and matched pairs are introduced and discussed as well as some relevant constructions. Chapter 7 explores Elduque-Myung type mutations of Hom-algebras, which are twisted version through a homomorphism of classical-type algebra. They appeared first in physics literature where some q-deformations of some Lie algebras of vector fields were considered and led to structures that no longer satisfy Jacobi condition but a modified version by a homomorphism. Such structures were called Homalgebras, and various classical structures were extended to this context. In this work, scalar and non-scalar mutations of associative algebras to mutations of nonassociative algebras and of several kinds of Hom-algebras of Hom-associative type are studied. Moreover, characterizations are given in terms of Hom-associators, commutators, and general mutation parameters. In similar spirit, generalizations of Witt algebra, which plays an important role in physics and may be expressed as an operator algebra, are presented in Chap. 8. They are called f -generalized Witt algebras and obtained by modifying the commutator using two parameters. Their cohomology and central extensions are investigated and f -generalized Virasoro algebras naturally derived. Chapter 8 deals with commutation relation for linear operators or, in general, in an associative algebra, mainly of the form .AB = BF (A) where F is a polynomial. Such relations are important objects of investigation because of their interest in quantum mechanics, non-commutative geometry, and non-commutative analysis. Algebraic properties of representations are studied, and additivity property of representations of polynomial covariance commutation relations is derived for operator algebras. The last part shows in Chap. 9 applications of quasigroups in cryptography and coding theory. A complete survey is presented with many algorithms and applications. The development of quantum computers questioned security based on associative structures (number theory, group and finite field theory, Boolean algebras, etc.). So, nowadays, the use of quasigroups for building cryptographic products is becoming important and promising for future research. The quasigroups are also suitable algebraic structures for building error-detecting and error-correcting codes. Chapter 10 is dedicated to generalized quadratic quasigroup functional equations.

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Notice that quasigroups have already many applications in geometry, combinatorics, statistics, economy, and engineering. The volume ends with an interplay between theoretical physics and mathematics, where a survey on dynamics of the motion of particles with non-commutative Poisson structure is presented. Souriau’s method of orbit is used to study this exotic mechanics on the tangent bundle of the configuration space or velocity phase space. Poisson manifolds and Schouten-Nijenhuis bracket turn out to be key tools. The contributions in this volume highlight the interaction between many mathematical topics and applications. They encompass surveys of basic theories and also more new and recent algebraic structures with relevant results and applications. Exchanges between the various fields allow for mutual enrichment and considerable progress. Mulhouse, France February 2023

Abdenacer Makhlouf

Preface

This first volume of the series Algebras without Borders–Classical and Constructive Nonassociative Algebraic Structures: Foundations and Applications is inspired by the conference on Classical and Constructive Nonassociative Algebraic Structures: Foundations and Applications—CaCNAS:FA 2021, held in 2021, and dedicated to the memory of Nebojša Stevanovi´c (1962–2009). Introduction to nonassociative algebra OR Playing havoc with the product rule? For modern mathematics, thick intertwining of very many directions and subdisciplines is typical. So the algebraic structures, nonassociative algebras among them, are percolating other branches of mathematics accommodating special demands and purposes, and acquiring new features and properties to serve ‘for the simplification of theoretical constructions, wrote Bernard Russo in 2012. The 2021 edition of the conference on Classical and Constructive Nonassociative Algebraic Structures: Foundations and Applications—CaCNAS: FA 2021 is the first of its series among the set of conferences entitled, “Algebra Without Borders”. The organizers will strive to hold it every two years in a country, to truly live to its name. As a specificity, the CaCNAS: FA 2021, (see http://cacnas.masfak.ni. ac.rs/), is offered in the memory of professor Nebojša Stevanovi´c (1962–2009), for his significant contribution to this field. He was a member of the Department of Mathematics, Physics, and Informatics at the Faculty of Civil Engineering and Architecture, University of Niš, Serbia. He was one of the founders of the first paragliding club in Niš, “Albatross”, one of the oldest paragliding clubs in Serbia. He was the first President of the Paragliding Association of Serbia. Professor Nebojša Stevanovi´c’s life was marked by a wish to conquer both algebraic and heavenly spaces—he sailed the structures of groupoids, especially those of AbelGrassmann’s law, and semigroups, just as he sailed the skies as a true sportsman and paraglider. He published his works in numerous renowned journals, participated in international and domestic conferences, reviewed the papers from his area of interest, was quoted many times, and so forth. CaCNAS:FA 2021 is organized by higher training and research institutions from three continents: Europe (Center of Applied xi

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Mathematics of the Faculty of Mechanical Engineering Niš, CAM-FMEN, Serbia), Africa (International Chair in Mathematical Physics and Applications (ICMPAUNESCO Chair), University of Abomey-Calavi, Benin), and Asia (Government College University, Lahore, and Department of Mathematics, Faculty of Sciences, COMSATS University Islamabad, Abbottabad Campus, Pakistan). Organizers of this event are Academician Mahouton Norbert Hounkonnou (the President of the Network of African Science Academies, NASAC), Professor Melanija Mitrovi´c (Head of the CAM-FMEN, University of Niš), Professor Mujahid Abbas (Government College University, Lahore), and Professor Madad Khan (Faculty of Sciences, COMSATS University Islamabad, Abbottabad Campus). Paraphasing Barut (A.O. Barut, Foundation of Physics 24(11), Nov. 1994, p. 1571), they are convinced that the first principles of things will never be adequately known. Science is an open ended endeavor, and it can never be closed. We do science without knowing the first principles. It does in fact not start from first principles, nor from the end principles, but from the middle. We not only change theories but also the concepts and entities themselves, and what questions to ask. The foundations of science must be continuously examined and modified; it will always be full of mysteries and surprises. The history of nonassociative algebraic structures can be traced at least to the middle of the nineteenth century. The theory of nonassociative algebraic structures is an enormously broad and greatly advanced area. Interesting new algebraic ideas arise, with challenging opportunities to discover connections to other areas of mathematics, natural sciences, and engineering. Besides, computer-assisted methods proved useful in the development of the theory of nonassociative algebraic structures, e.g., in finding proofs and constructing examples and applications. In the above-mentioned conference, the speakers tried to present a brief overview of the origins of nonassociative algebraic structures, a selection of current research topics and future directions, some generalizations within the framework of classical, fuzzy, and intuitionistic logic, and further applications. The topics developed include the following: • Groupoids (binary systems) and their generalizations (AG-groupoids, quasigroups, loops, neutrosophic groupoids, Smarandanache groupoids) • Nonassociative algebras and their generalizations (left almost algebras) • Ordered nonassociative algebraic structures (logical algebras) • Applications within natural sciences and engineering • Computer-aided development and transformation of the theory of nonassociative algebraic structures • Algebraic geometry and its relations with quiver algebras • Enumerative combinatorics • Representation theory • Fuzzy logic and foundation theory • Fuzzy algebraic structures • Group amalgams • Graph theory

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• Actions of groups on various geometric objects, such as diagram geometric and buildings • Constructive nonassociative algebraic structures Prominent experts in the field coming from respectable universities from all over the world (more than 20 represented countries) took part in this conference. There were 9 keynote, 20 invited, and 18 contributed talks. The speaker abstracts and BIOs were grouped in alphabetical order published on the conference website. We are thankful for their inspiring talks and contributions to the scientific discussion. Cotonou, Benin Niš, Serbia Lahore, Pakistan Abbottabad, Pakistan January 2023

Mahouton Norbert Hounkonnou Melanija Mitrovi´c Mujahid Abbas Madad Khan

Acknowledgments

The editors express their gratitude to Professor Abdenacer Makhlouf for the time he has devoted to reading the book and writing the Foreword. They also thank all the contributors who have agreed to be part of this first volume, and anonymous referees. The editors thank the Faculty of Mechanical Engineering and the Faculty of Civil Engineering and Architecture, University of Niš (Serbia), the International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi (Benin), the Government College University, Lahore, and the Faculty of Sciences, COMSATS University Islamabad, Abbottabad Campus (Pakistan) for their various supports. Special thanks go to the CAM-FMEN technical supporting team for the successful organization of the CaCNAS:FA 2021 conference, and to the ICMPA-UNESCO Chair team for their efforts in preparing this book. Finally, we thank the Springer Nature Project Coordinator, Ms. Saveetha Balasundaram, and the Tex team for their invaluable help in writing this book.

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Contents

1

Quandles, Knots, Quandle Rings and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . Mohamed Elhamdadi and Brooke Jones

2

New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras and Related Algebraic Structures . . . . . Guilai Liu and Chengming Bai

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Some Varieties of Loops (Bol-Moufang and Non-Bol-Moufang Types) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adewale Roland Tunde Sòlárìn, John Olusola Adéníran, Tèmítópé Gbóláhàn Jaiyéo.lá, Abednego Orobosa Isere, and Yakub Tunde Oyebo

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The .Z3 -Graded Extension of the Poincaré Algebra . . . . . . . . . . . . . . . . . . . . 165 Richard Kerner

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Ternary Leibniz Color Algebras and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Ibrahima Bakayoko and Ismail Laraiedh

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(Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary (Hom-)bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Mahouton Norbert Hounkonnou and Gbevewou Damien Houndedji

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Elduque-Myung Type Mutations of Hom-algebras . . . . . . . . . . . . . . . . . . . . . 295 Germán García Butenegro and Sergei Silvestrov

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(Hom-).f -generalized Witt Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Mahouton Norbert Hounkonnou, Bignon Hugues Degbedji, Fridolin Melong, and Melanija Mitrovi´c

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Some Algebraic Properties of Representations of Polynomial Covariance Commutation Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Domingos Djinja, Sergei Silvestrov, and Alex Behakanira Tumwesigye

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Applications of Quasigroups in Cryptography and Coding Theory . . 419 Smile Markovski, Verica Bakeva, Vesna Dimitrova, Aleksandra Mileva, Aleksandra Popovska-Mitrovikj, and Hristina Mihajloska Trpcheska

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Generalized Quadratic Quasigroup Functional Equations . . . . . . . . . . . . 491 Aleksandar Krapež

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A Primer on Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . 533 José F. Cariñena, Héctor Figueroa, and Partha Guha

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Editors and Contributors

About the Authors Dr. Sc in 1992 from Catholic University of Louvain in Belgium, Mahouton Norbert Hounkonnou is a full Professor of Mathematics and Physics at the University of Abomey-Calavi, Benin. His works deal with noncommutative and nonlinear mathematics, and complex systems. He authors and reviews books. Further, he serves as member and associate editor of editorial boards for renowned journals and books in mathematics and mathematical physics, including the Editorial Boards of Mathematics in Mind, Springer (https://www.springer.com/series/15543? detailsPage=aboutTheEditor), Fields Cognitive Science Network (http://www.fields.utoronto.ca/ generalinfo/Fields-Cognitive-Science-Network), the Peer Community In (PCI) Neuroscience, etc. He published over 200 main research papers in outstanding ISI-ranked peer-reviewed journals and international conference proceedings in the fields of mathematics, mathematical physics and complexity. He is a visiting professor at African, Asian, European, and North American Universities. Together with his peers at the international level, he founded the International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair) of the University of Abomey-Calavi where he created multi-university master degree and PhD programs in mathematics xix

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with connections, motivations, or applications to physics, and in physics with important relations to mathematics. The best African students from over 13 French and English-speaking countries are selected to follow these graduate programs, which attract prominent and leading mathematicians and mathematical physicists worldwide who come to give lectures and supervise student’s research, what has substantially increased international collaboration with African, Asian, American, European, and Indian mathematicians. The ICMPAUNESCO Chair presently hosts the international conference and school series on Contemporary Problems in Mathematical Physics, which are held every two years and each year since 1999 and 2005, respectively. These activities have led to a significant network of researchers connected with the ICMPA-UNESCO Chair that benefits from the resources available in mathematics and mathematical physics in Africa. Professor Hounkonnou supervised over 35 PhD and 40 MSc students from various countries and continents, including Belgium, Benin, Burkina-Faso, Burundi, Cameroun, Democratic Republic of Congo, Niger, Nigeria, Senegal, Togo, Zambia, etc. Professor Hounkonnou was awarded a series of recognitions for the excellence of his work such as the Prize of the Third World Academy of Sciences (TWAS) in 1996, the Tokyo University of Science President Award in 2015, and the 2016 World Academy of Sciences C.N.R. Rao Prize for Scientific Research “for his incisive work on noncommutative and nonlinear mathematics and his contributions to world-class mathematics education”. He was a member of UNESCO Scientific Board for International Basic Sciences Programme (IBSP), NANUM 2014 Award Committee Member of the International Congress of Mathematicians (ICM 2014) as reviewer for region Africa, and member of the InterAcademy Partnership working group on Harnessing Science, Engineering and Medicine to Address Africa’s Challenges, etc. He is TWAS research professor in Zambia, the chair of the African Academy of Sciences Commission on Pan-African Sciences Olympiad (2014-present),

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and chair of the African Academy of Sciences Membership Advisory Committee (MAC) in Mathematical Sciences (2013–2021). Professor Hounkonnou is the co-chair of the network of African, European, and Mediterranean Academies for Science Education (AEMASE III). He is the current president of the Network of African Science Academies (NASAC) and former president of the Benin National Academy of Sciences, Arts, and Letters. His membership extends to InterAcademy Partnership Advisory Committee and Science Education Programme (IAP SEP), the International Association of Mathematical Physics, American Mathematical Society, London Mathematical Society, Society for Industrial and Applied Mathematics (SIAM), Academy of Science of South Africa (ASSAf), Hassan-II Academy of Science and Technology, Morocco, African Academy of Sciences (AAS), The World Academy of Sciences (TWAS), Scientific Council of the Centre International de Mathématiques Pures et Appliquées (CIMPA), Scientific Committee of the International Centre for Advanced Training and Research in Physics (CIFRA, Magurele-Bucharest, Romania), as well as other scientific organizations. He is a Knight of the Benin National Order, Doctor Honoris Causa of the Université Toulouse III Paul Sabatier, France, and representative for Africa of the International Mathematical Union Commission for developing countries.

Melanija Mitrovi´c is a full professor at the Department of Mathematics and Informatics of the University of Niš. She received her BSc and MSc degrees at the Faculty of Philosophy, University of Niš, and her PhD degree at the Faculty of Mathematics and Sciences, University of Niš, in 2000. She lectures at all three levels of higher education in Serbia, as well as abroad. She is a member of the Faculty Council since 2015; Quality Board, Committee for the Student Evaluation of Educational Quality (2009–2015); and Deputy Vice head of the Department of Mathematics and

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Informatics (2005–2010). She is the head of the Center of Applied Mathematics of the Faculty of Mechanical Engineering Niš, CAM-FMEN (http:// camfmen.masfak.ni.ac.rs), around which, among other activities, she develops an interdisciplinary research group investigating applications of algebraic structures to problems in engineering science. She holds the status of permanent full professor at the International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Benin (http://www.cipma.net/spip.php?page=rubrique& id_rubrique=27). Her membership includes the Editorial Board of the Book series Mathematics in Mind, Springer (https://www.springer.com/series/ 15543?detailsPage=aboutTheEditor), the Fields Cognitive Science Network (http://www.fields. utoronto.ca/generalinfo/Fields-Cognitive-ScienceNetwork), Peer Community In (PCI) Neuroscience. Major directions of her research, professional work, and expertise focus on foundations of constructive mathematics, classical and constructive algebraic structures, especially within areas of semigroup and semiring theory, Witt and Virasoro algebras, integrable systems, groups, and representations. Her innovating work within the foundations of constructive mathematics (theory of sets with (non-tight) apartness, theory of ordered sets with (non-tight) apartness), and its applications to constructive semigroups with apartness positions her among the pioneers of the constructive mathematics in Serbia. Outside of Serbia, she is recognized as the mother of the novel theory of constructive semigroups with apartness, considered as a new algebraic theory. She publishes her works in renowned outstanding international journals. She is reviewer for renowned journals in mathematics. She is invited as plenary speakers at highly ranking international conferences in a regular basis to communicate about the results of her research. She often acts as the chair of organizing committees and as program committee member of prestigious international conferences. She was the Chair of the Organizing Committee and member of the Program Committee of the Workshop “Theoretical Computer

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Science—from Foundation to Applications”, TCSFP 2009, as well as “Constructive Mathematics: Foundations and Practice”, CMFP 2013—both organized at the Faculty of Mechanical Engineering, University of Niš. In connection with the CMFP 2013 organization, she got IMU/CDC Support (Conference Grant Support) as the first Serbian mathematical conference ever co-sponsored by IMU (International Mathematical Union, Berlin). In addition, CM:FP 2013 was the first conference on constructive mathematics (mathematics with intuitionistic logic) organized at the Western Balkan region. In the period from October 2020 till November 2021, she was the Chair of the Organizing Committee and member of the Program Committee of the following online and/or hybrid international scientific conferences: “Mathematics for human flourishing in the time of COVID19 and post Covid-19” (http://camfmen.masfak. ni.ac.rs/Webinar_Covid.html), MS.2 A.2 M 2021 (http://mathsocart.masfak.ni.ac.rs/), CaCNAS:FA 2021 (http://cacnas.masfak.ni.ac.rs/), ACaCS 2021 (http://acacs2020.masfak.ni.ac.rs/). She is a guest speaker in leading universities worldwide. She serves as an international jury member for doctoral theses. She is author of a well-known scientific monograph, at national level, on semigroup theory. Her works in the field of algebra so far has made a significant contribution to its development, which is reflected in the expansion of existing knowledge (classical semigroup theory and semiring theory) and in the establishment of completely new theory of constructive semigroups with apartness. She received the Award Povelja of the Faculty of Mechanical Engineering Niš in 2019 for her permanent work connected to the promotion of the faculty and University of Niš algebraic group (http://camfmen.masfak.ni.ac.rs/CV/CV_Melanija_ Mitrovic.pdf).

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Prof. Mujahid Abbas is professor and chairperson of Department of Mathematics, Government College University, Lahore, Pakistan. He is also the director-general of Abdus Salam School of Mathematical Sciences, Lahore, Pakistan. He is working as an extraordinary professor in the Department of Mathematics and Applied Mathematics, UP, South Africa. He has served this department as an extraordinary professor and associate professor since August 2015. Prior to these responsibilities, he has served different universities such as Lahore University of Management Sciences, Pakistan, University of Management and Technology, Pakistan, Indiana University Bloomington, USA, University of Birmingham, UK, King Abdulaziz University, Saudi Arabia, King Saud University, Saudi Arabia, King Fahd University of Petroleum and Minerals, Saudi Arabia, Ton Duc Thang University, Vietnam, China Medical University, Taiwan and Abdus Salam International Center of Theoretical Physics, Italy, in different roles such as associate professor, professor, post-doctoral fellow, honorary senior research fellow, visiting professor, consultant, research collaborator, and research scholar. He completed his first PhD in the field of Functional Analysis back in 2005 from Pakistan and second PhD in the field of Soft Set Theory from Universität Politecnica De Valencia, Spain. He has produced three hundred and sixty research papers in internationally acclaimed journals. Many of his publications may be used as a benchmark is evident from the fact that his research work has received 11991 citations so far. An h-index of his research publications is 47 and i-10-index is 251. His cumulative impact factor is 236.144. He initiated several new concepts which were later employed by many researchers to obtain some interesting results in related areas of research. Moreover, he was Highly Cited Researcher in the years 2015–2019 and is NRF rated mathematician. Besides three book chapters which appeared in the books published by Birkhauser, John Wiley & Sons Inc., USA, and CRC Press, Taylor & Francis group, he has authored three books (published) including his recent book titled “Background and Recent Developments of Metric Fixed Point Theory” pub-

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lished by CRC Press, Taylor’s and Francis Group. He has won three consecutive research productivity awards by Council of Science and Technology, Government of Pakistan. Pakistan Academy of Sciences has also awarded him a Gold Medal in recognition of his research contribution. He has supervised 16 Master and 9 PhD students. He has presented his research work through different seminars and conferences in several countries including UK, Italy, Sweden, Turkey, Nigeria, Jordan, Saudi Arabia, Qatar, South Africa, Pakistan, Thailand, Korea, and USA. He is serving as a member of the Editorial and Advisory Boards of several journals and is a member of various scientific committees of national and international conferences. More details can be accessed through the following links: https://scholar.google.com.pk/citations?user= 8H9zFMsAAAAJ&hl=en; www.gcu.edu.pk. Dr. Madad Khan is currently working as an associate professor at the Department of Mathematics COMSATS University Islamabad, Abbottabad, Abbottabad Campus. He remains head of department and graduate program coordinator for 5 years. He has published more than 100 papers in internationally reputed journals. He did postdoctorates from University of Chicago (8th in world ranking), USA, and University of Birmingham (61th in world ranking), UK, in 2015 and 2013. He visited University of Oxford (5th in word ranking) and University of Cambridge, UK. He visited Jeju National University, Korea, for postdoctoral research work. He did PhD in 2008 form QAU, Islamabad, Pakistan. He did MPhil, MSc, and BSc with distinction. He is working on genetic algebra, computational mathematics, fuzzy mathematics, and semigroups/AGgroupoids. He is invited as a speaker at 3rd and 4th High Mile conferences on non-associative Mathematics organized by University of Denver Colorado, USA. He is an invited speaker in 2017 IEEE International Conference on INnovations in Intelligent SysTems and Applications (INISTA 2017) Gdynia, Poland, July 3–5, 2017, invited Speaker in Logic, Algebras and Applications, Jeju National University, Jeju, Korea, January 2018 and invited

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Speaker in International Conference on Uncertainty Mathematics, School of Mathematics, Northwest University Xi’an, China, in 2018, 2019 and 2020. He produced 3 PhDs and 40 MPhils students. Three PhD and five MPhil are currently working under his supervision. He is a member of BoS of six universities including Allama Iqbal University, COMSATS, Hazara, Abottabad, and Sardar Bahadur Khan Women universities and Shaheed Benazir Bhutto University, Sheringal Director. He is an executive member of Pakistan Mathematical Society (www.pakms.org.pk). He is member of selection board of five universities, COMSATS, Hazara, Abottabad, Muzaffar Abbad University Azad universities, and University of Karachi. He is a member of DTRC committees at the Department of Mathematics, Department of Civil Engineering, COMSATS University Islamabad, and Islamia College University Peshawar, Pakistan. He has international academic collaborations with 36 professors in USA, UK, Canada, Italy, Poland, Romania, Albania, Japan, China, Iran, Korea, and Turkey. He organized several national and international conferences. He is external examiner of six universities in Pakistan for graduate theses evaluation. He published four books from USA and Belgium. Fifty copies of my printed books are available in 15 Romanian libraries. His books are available online. His bio data is published in a Book “S. Florentin, The Encyclopedia of Neutrosophic Researchers, Pons Editions, Brussels, Belgium 2016” on its 100 page. This book is available online. He hosted (invited) four professors from Korea, China, Brazil, and USA to Pakistan for joint research work of mutual interest.

Editors and Contributors

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Contributors John Olusola Adéníran Federal University of Agriculture, Abeokuta, Nigeria Chengming Bai Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, China Ibrahima Bakayoko Département de Mathématiques, Université de N’Zérékoré, N’Zérékoré, Republic of Guinea Verica Bakeva Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia Germán García Butenegro Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden José F. Cariñena Departamento de Física Teórica Universidad de Zaragoza, Zaragoza, Spain Bignon Hugues Degbedji International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin Centre International de Recherches et d’Etudes Avancées en Sciences Mathématiques & Informatiques et Applications (CIREASMIA), Cotonou, Benin Vesna Dimitrova Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia Domingos Djinja Department of Mathematics and Informatics, Faculty of Sciences, Eduardo Mondlane University, Maputo, Mozambique Division of Applied Mathematics and Physics, UKK, Mälardalens University, Väster˙as, Sweden Mohamed Elhamdadi University of South Florida, Tampa, FL, USA Héctor Figueroa Departamento de Matemáticas, Universidad de Costa Rica, San Pedro, Costa Rica Partha Guha Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi, UAE SNBNCBS, Kolkata, India Gbevewou Damien Houndedji International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin Mahouton Norbert Hounkonnou, International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin Centre International de Recherches et d’Etudes Avancées en Sciences Mathématiques & Informatiques et Applications (CIREASMIA), Cotonou, Benin Abednego Orobosa Isere Ambrose Alli University, Ekpoma, Nigeria

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Editors and Contributors

Tèmítópé Gbóláhàn Jaiyéo.lá Obafemi Awolowo University, Ile Ife, Nigeria Brooke Jones University of South Florida, Tampa, FL, USA Richard Kerner Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Sorbonne, Universités - CNRS UMR 7600, Tour 23-13, 5-ème étage, Paris, France Aleksandar Krapež Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia Ismail Laraiedh Department of Mathematics, Faculty of Sciences, Sfax University, Sfax, Tunisia Department of Mathematics, College of Sciences and Humanities - Kowaiyia, Shaqra University, Shaqra, Kingdom of Saudi Arabia Guilai Liu Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, China Smile Markovski Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia Fridolin Melong International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin Centre International de Recherches et d’Etudes Avancées en Sciences Mathématiques & Informatiques et Applications (CIREASMIA), Cotonou, Benin Aleksandra Mileva Faculty of Computer Science, Goce Delcev University, Stip, North Macedonia Melanija Mitrovi´c CAM-FMEN, University of Niš, Niš, Serbia Aleksandra Popovska-Mitrovikj Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia Yakub Tunde Oyebo Lagos State University, Ojo, Nigeria Sergei Silvestrov Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden Division of Applied Mathematics and Physics, UKK, Mälardalens University, Väster˙as, Sweden Hristina Mihajloska Trpcheska Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia Alex Behakanira Tumwesigye Department of Mathematics, College of Natural Sciences, Makerere University, Kampala, Uganda Adewale Roland Tunde Sòlárìn National Mathematical Centre, Federal Capital Territory, Abuja, Nigeria

Nebojša’s Flight Through the Algebraic and Heavenly Space

Prof. PhD Dragan Kosti´c, Eng., Dean Prof. PhD Slaviša Trajkovi´c, Eng. and Prof. PhD Snežana Ðori´c-Veljkovi´c, Phys. Behalf of Faculty of Civil Engineering & Architecture http://www.gaf.ni.ac.rs/ In Memoriam Asst. Prof. Nebojša Stevanovi´c, PhD 1962–2009 Abstract We all dreamed of flying as children, and Nebojša made that dream come true. He flew through space, science, and life with a particular ease. Apparently, in his life’s flight, he saw further and could do more. A rare ability for timely action, innovative ideas, communication, and implementation are attributes of great people, and it seems that Nebojša belongs to the greats, whose memory should be preserved. We all remember him as accomplished, both professionally and personally. The interest and all-round talents that he possessed with an adrenaline-fueled need for extreme sports are the hallmarks of his personality, as a predisposition that came true, it seems, simultaneously in several parallel lives. PhD Nebojša Stevanovi´c, assistant professor, was a member of the Department of Mathematics, Physics, and Informatics of the Faculty of Civil Engineering and Architecture in Niš and a member of the paragliding club “Albatros” from Niš. Nebojsa’s life was marked by the desire to conquer both celestial and algebraic spaces. He sailed in the spaces of Abel-Grassmann groupoids and semigroups as well as in the sky like a true athlete. When you find yourself facing a character like Nebojša Stevanovi´c, you find yourself in a dilemma about which of his characteristics to write about. He possessed many great and astonishing talents. Most of all is known about his successful academic career. What is less known about xxix

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him is that he was extremely talented in other activities related to paragliding, car mechanics, and motorcycling. He passed on his love for these sports to his children, Jelena and Nikola, whom he also used to take to competitions. His fatherly love for his children was touching. He knew how to set his children on the right course, from their earliest childhood. Jelena is about to defend her doctoral dissertation at the Faculty of Civil Engineering and Architecture in Niš, and Nikola is about to defend his master’s thesis at the Faculty of Mechanical Engineering in Niš. Picture 1 PhD Nebojša Stevanovi´c, assistant professor and his family (wife Sonja, daughter Jelena, and son Nikola) with his puppy Edi

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Nebojša was born in 1962 in Babušnica, and his parents came from nearby Kaluderovo. Primary and secondary school he finished in Niš. He was the first ¯ generation of “directed education” in the Mathematical and Technical field—a computer programmer. After graduating from the Faculty of Philosophy—Department of mathematics, he became an assistant at the Faculty of Civil Engineering and Architecture in Niš in 1990, and in the same year, he enrolled in postgraduate studies at the Faculty of Science and Mathematics of the University of Novi Sad. He defended his master’s thesis “Abel-Grassmann’s groupoids and semigroups” in 1994 and obtained the academic title of Master of Mathematical Sciences: Disciplines of Algebra and Mathematical Logic. One of his colleagues, a doctoral student in Novi Sad was Andreja Tepavˇcevi´c, now a professor and PhD, who summed up her impressions of Nebojša Stevanovi´c in a few sentences after learning about his untimely death: “Cognition about Nebojša’s death was a shock to me, I got the feeling that after that nothing will be the same ever again. How could die someone who was flying through life?” They were linked by their mutual mentor, prof. Svetozar Mili´c, who supervised Nebojša’s master’s degree, and Andreja’s doctorate. Similar research topics were the subject of their professional discussions, but she was also fascinated by his stories about paragliding. She was convinced that his practice of paragliding had an impact on his mathematical results and the development of the area of nonassociative algebraic structures. Ten years later, at a conference in Novi Sad, she was pleased to see that he had developed his scientific ideas, which showed great potential. “After his untimely death in 2009, nothing is the same for science or for his friends who remember him”.

Picture 2 Common photo of the participants of one of the conferences in Novi Sad

He defended his doctoral dissertation “Groupoids on which the Abel-Grassmann law applies” in 2006 at the Faculty of Science and Mathematics in Niš. Retired Prof. PhD Petar Proti´c, mentor of Nebojša’s scientific and research work, remembers

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when he got him interested in the research in the field of algebra with proposal for developing the topic of his master’s thesis in the semigroup theory: “He was an assistant at the Faculty of Civil Engineering and Architecture in Niš, where I was professor, and since we were engaged in the same scientific field, we were cooperating. At the initiative of Prof. PhD Stojan Bogdanovi´c, I noticed that some ideas and techniques from the theory of semigroups can be applied to this nonassociative algebraic structure. We started studying with the idea of making it the topic of his master’s thesis. The idea we were developing is left-almost semigroups, similar to the theory of semigroups. A test to check whether a groupoid is a left almost semigroup starts from ideals, arrangements, congruences, subclasses, bands, etc. The first thing we agreed on is that the term left weak semigroup is not appropriate because it resembles associativity, and these groupoids are still quite far from that because you have to take the parentheses into consideration. Also, there are finite Abel-Grassman groupoids that do not have idempotents, which is impossible with semigroups. We were looking for a new name, in the monograph on Latin squares by Denes and Keedwell, the law .(ab)c = (cb)a, which is valid for all elements .a, b, c of the groupoid S, was noted as the left Abel-Grassman law. Prof. Bogdanovi´c suggested that we should call this class of semigroups Abel-Grassman groupoids or AG-groupoids, which we accepted. The results we reached during the research were presented at conferences and symposiums, and then published in scientific journals. In the meantime, professor Milan Božinovi´c joined us, so in a way, a small Niš school for Abel-Grassman groupoids was created. Relatively quickly, Stevanovi´c defended his master’s thesis, with chapters that are essentially an original scientific contribution to the study of AG groupoids. As a result of an intensive research, Stevanovi´c completed his doctoral dissertation relatively quickly, where not a single chapter was taken from the master’s thesis. I had the honor and pleasure of being his mentor for the preparation of his master’s thesis and doctoral dissertation. Interest in our work appeared first exactly where the idea of studying left almost semigroups originated, at the University of Islamabad in Pakistan, and soon in some other countries, now mostly under the name of AG groupoids. The ideas we started with made full sense, to our satisfaction, our ideas were improved and new ones were born, thus the school for the study of AG groupoids was born at the University of Islamabad, where several doctoral dissertations related to this issue were defended. Unfortunately, Professor Nebojša Stevanovi´c is not among us, but I, as his friend first and foremost, and mentor, take the liberty to thank everyone who continued and raised the study of AG-groupoids to a higher level on his behalf and mine. Thus, our ideas from the beginning of the nineties of the last century gained full meaning, which means that our work was not in vain.” He (and his coauthors) published his work in numerous renowned journals, participated in international and domestic conferences, reviewed the papers from his area of interest, was quoted many times, and so forth. https://www.researchgate.net/scientific-contributions/Nebojša-Stevanovi´c2023626744 He started his teaching career at the Faculty of Civil Engineering and Architecture in Niš in 1990, at the Department of Mathematics, Physics, and Informatics.

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Picture 3 Detail from the doctoral dissertation defense

The work on the educational process was characterized by meticulousness, systematic presentation, and a good relationship with students and colleagues. His colleague Prof. Dr. Snežana Djori´c-Veljkovi´c testified about his qualities in the relationship with students and colleagues: “Through my colleague Katarina Petkovi´c, after so many years, I received a preparation notebook for practice classes, which shows his thoroughness and responsibility. The combination of circumstances is that we got our doctorates and were elected in close time frames, so we exchanged experiences related to news procedures in the functioning of the University system. He also showed solidarity with his colleagues at the moment of the PhD promotion ceremony when I parked in front of the University building, even though there was a possibility that we would both be late for the promotion”. Collegial cooperation with Prof. Stevanovi´c left a deep mark on all colleagues with whom he collaborated. Prof. Ljubica Velimirovi´c, PhD, is one of the colleagues with whom he worked directly, and which over time grew into a friendship with him and his family: “We met in the nineties when we were both teaching assistants in Mathematics at Faculty of Civil Engineering and Architecture. At that time, we were intensively engaged in science on the preparation of a doctorate. It was important for some internal information that you have someone who is willing and wellintentioned to share with you information about the method of publishing papers, about the method of scientific research using different methods that were then used for the first time at the University, whether it was text processing in Tex, or if it was about computer application, it was necessary to have someone who would help you and exchange information in some way. Nebojša was always correct, very well informed, there were no reservations, he always met you at halfway. He used his good qualities, such as accuracy, correctness, great intelligence, diligence, not

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only to achieve scientific results, points in work, but also to share it all with his colleagues. So, he was a very valuable collaborator. I also loved him as a person, as a friend who always helped both students and colleagues. So, I kept him as a special memory. I think that such colleagues who work with their colleagues in such a way are rare. I will always miss him”.

Picture 4 Promotion of Doctors of Science at the University of Niš in 2006th

According to the students to whom he taught exercises in mathematics subjects at the Faculty of Civil Engineering and Architecture, he was rated as a good lecturer, extremely approachable as a professor, open to students. It was interesting to watch him easily pass over areas that were completely clear to few, with jokes and stories about paragliding. He managed to explain as much as they could understand. Along with a successful teaching and scientific career, Nebojša also conquered the heavenly spaces. It is known that paragliding is the easiest way to realize the dream of flying and find out why birds chirp. Practicing this extremely dangerous sport, he recognized the need for paragliding pilots to be organized through an association, i.e., a club. Together with several enthusiasts, he founded the Paragliding Club “Albatross” and he who gave it its name. Over time, the love for glider flying and kite flying from his childhood grew into an obsession with paragliding. He was the first President of the Paragliding Association of Serbia. Through his personal efforts, some of the national and international competitions were organized in Niš (World Cup 2005, European Championship 2008) on the fields above the village of Si´cevo. He was a very successful competitor as a standard member of the team that had won several first and second places in the National Team Championships. Comrades, friends, and members of the club still remember the plans they could not realize because of the illness that took Nebojša away from us. The symbolism associated with our Nebojša refers to his character traits in multiple ways.

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Picture 5 Presentation of Niš in Lausanne—Competing for the organization of the Europian Cup 2008

It was believed that children bearing the name Nebojša will be brave and fearless and will drive away all evil forces. The name originated from the negation of the word “to fear”, which is why it is believed that all those who bear the name Nebojša do not know fear, nor do they falter in the face of obstacles. The idea to name the paragliding club “Albatross” came from Nebojša as one of the club’s founders. There is an obvious fascination and analogy with one of the largest flying birds that can fly thousands of kilometers without landing using favorable lifting air currents, the albatross consumes very little energy. Charles Baudelaire is known to all of us, he is one of the most famous French and world poets, the most significant for the fact that he laid the foundations of modernism with his symbolist poems. “Albatross” is one of the more famous songs from the “Flowers of Evil” collection. The poet personifies the albatross in the poem by calling it a traveler. At the same time, the motif of the traveler is a symbol of freedom and the great breadth of space that he visits in his lifetime. The accidental association of the symbolism associated with Nebojša Stevanovi´c is only proof that coincidences do not exist. He belonged to a small group of our friends who we consider sincere, consistent, witty . . . and everything that characterizes friends who understand each other even if they do not have to say what they think. In imagination, I was convinced that as a younger colleague, he would talk inspiredly about my work at the moment of my retirement, and in real time, with pain in my soul, I am trying to revive the memory of Nebojša. He was blessed with the qualities that are necessary for the job of a university professor. He was persistent in his work, unremitting researcher, consistent in his objectivity, responsible in fulfilling his obligations, but also funny. The friendship with Nebojša was built over time, thanks to the same inclination toward work, research, responsibility, but also the same sense of humor, which was based on connecting illogicality with abundant use of the richness of the Serbian language. He was of a restless spirit. Mathematics was a scientific field in which he successfully

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Picture 6 One of the Faculty celebrations with some dear colleagues

engaged in research, and in his spare time, in addition to motor mechanics, he had a passion for motorcycling and paragliding. The high level of adrenaline in these extreme sports was his daily need. Everything he did was followed by success, which is popularly said, God created him for everything. He never forgot of his origins or was ashamed of them, and he very often stayed in his native Kaludjerovo. Nebojša’s life unfolded the way he wanted, without stereotypical bounds, but with extreme self-sacrifice. In addition to the interest he had in technology, mathematics, and hanging out with friends, he also showed a preference for philosophy by collecting beautiful texts and presentations. He was tried to express his state of mind in verses. One of his philosophical dilemmas is contained in the following verses. There are always two kinds of people Who always blame for everything others, and who are always to blame for themselves. Who shape the world in their own image And those who hesitate to walk, lest they would step on flower. Some would like to be right when they are wrong, And those who’d die if they wronged someone. Those who are fine to be good, and Those who are fine to be bad, Because they are safe that way. There are always two kinds of people,

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Tell me, which kind are we? Sources http://cacnas.masfak.ni.ac.rs/NS.html https://www.researchgate.net/scientific-contributions/Nebojsa-Stevanovic2023626744 https://ezproxy.nb.rs:2058/nauka_u_srbiji.132.html?autor=Stevanovic%20Nebojsa %20R&samoar=#.Y6qjKxXMJPZ https://drive.google.com/file/d/1VUd0nBRPKEVeRcMg8WXo-mzV35qQ4hxS/ view Digital archives of PhD Nebojša Stevanovi´c and his children

Chapter 1

Quandles, Knots, Quandle Rings and Graphs Mohamed Elhamdadi and Brooke Jones

1.1 Introduction Quandles are in general non-associative algebraic structures whose axioms are modeled on the three Reidemeister moves in knot theory. They were introduced independently by Joyce [49] and Matveev [54] with the goal being to obtain invariants of knots and links. They proved that two knots are equivalent (up to reverse mirror image) if and only if their fundamental quandles are isomorphic as quandles. Thus changing the knot theory problem of distinguishing two knots into the algebra problem of studying isomorphism of quandles. Since then there have been many investigations in the field of quandle theory. For more details on quandles see also [29, 35]. Recently, quandles have been investigated from many different point of views. In [19] a homology theory of set theoretic Yang-Baxter equation was developed and invariants of classical knots and virtual knots were defined using Yang-Baxter cocycles. In [60] second Yang-Baxter homology for the HOMFLYPT polynomial was considered and a homology of the YB operator for the Jones polynomial was investigated in [34, 61]. Quandles were also investigated in relations to Lie algebras [15], Frobenius algebras [17], Hopf algebras [16], representation theory [30], quasigroups and Moufang loops [25]. For a quandle X, the quandle ring .k[X] exhibits properties of the quandle X and the ring of coefficients .k. The study of quandle rings .k[X] is a meeting place of quandle theory and ring theory. The algebraic study of quandle rings was initiated in [14] in an analogous way to the theory of group rings for groups. The augmentation ideal of a quandle ring was introduced and studied, relationships

M. Elhamdadi (O) · B. Jones University of South Florida, Tampa, FL, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_1

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between subquandles of the given quandle and ideals of the associated quandle ring were investigated. In [26] the authors investigated various properties of quandle rings and proved that quandle rings of non-trivial quandles are not powerassociative. They also showed that if .k is Noetherian and X finite, then .k[X] is left and right Noetherian. A decomposition into simple left or right ideals of .k[X] was given when the quandle X is a dihedral quandle. The authors of [26] also studied isomorphisms of quandle rings and gave examples of non-isomorphic quandles with isomorphic quandle rings. Zero-divisors and idempotents in quandle rings have been investigated in [13]. Orderability of quandles was defined and many examples of orderable quandles were given. Furthermore, idempotents in quandle rings of some quandles were computed and applied to determine automorphism groups of some quandle rings. In [32] idempotents in quandle rings were investigated with relation to quandle coverings. Quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space showing the use of quandle rings in knot theory in [33]. This article serves two purposes. The first being to give an introduction the readers who are not so familiar with the topic of quandle theory. The second purpose is to report on new results. We compute second and third homology of disjoint union of quandles using the machinery of spectral sequences. A part of this article focuses on introducing the idea of using graph theory to study quandle theory. The purpose is to establish a connection between graph theory and quandle theory via quandle rings with the hope that it will turn out to be mutually beneficial for these two branches of mathematics. The main idea of using graph theory is that the edges of the graph should encode some binary relation of algebraic meaning in the quandle ring. Here we introduce, for the first time, the zero-divisor graph coming from quandle rings. Precisely, a zero-divisor graph is a directed graph whose vertex set is the nonzero zero-divisors of a quandle ring, wherein there is an edge from a vertex u to a vertex v if their product .u · v is zero. We prove several foundational properties for these graphs. In particular, we show that with .Z2 -coefficients, every vector of even components is a left zero-divisor. Furthermore, we prove that for commutative finite quandles, the graph .r(Z2 [X]) is connected and its diameter .diam(r(Z2 [X])) is bounded above by four. The article is organized as follows. Section 1.2 introduces the basics of combinatorial knot theory and prepare the terrain for the next sections. In Sect. 1.3 we discuss Fox colorings of knots which can serve as an introduction to the theory of dihedral quandles in particular and quandle theory in general. Section 1.4 gives the basics of racks and quandles needed for the article in addition to giving the characterization of the automorphism groups and inner automorphism groups of dihedral quandles [28]. In Sect. 1.5 we discuss the relation between left and right distributive quasigroups and some types of quandles, namely Alexander, Latin and medial quandles [25]. In Sect. 1.6 we review the construction of the homology theory of quandles and give the precise explicit formulas for 2- and 3-cocycles. Section 1.7 explains in detail the machinery of spectral sequence which is a powerful tool of computing homology and cohomology. We then state the results of this section which are new and are appearing for the first time in this article. In Sect. 1.8 we study singular knots and

1 Quandles, Knots, Quandle Rings and Graphs

3

their invariants which was motivated mainly by the theory of Vassiliev invariants [71]. We also review, following [7], the notion of an algebraic structure coming from oriented singular knots and links called singquandles. A motivation for this notion is given by introducing generalized Reidemeister moves, showing how the axioms of singquandles are obtained and explicit examples are given. Section 1.9 studies singquandles coming from group theory. A rich family of singquandles is obtained from groups. We also give, in this section, a variety of examples including a generalization of affine oriented singquandles, as well as an infinite family of non-isomophic singquandles over groups. Section 1.10 deals with topological quandles following [30]. The precise definition is stated and many examples are also given. The notions of units, ideals, kernels, and inner automorphisms in this context were investigated. Furthermore the notions of modules and rack group bundles over topological quandles were investigated and also central extensions of topological quandles were considered. In Sect. 1.12 quandles are “linearized” so to be able to use tools from other areas of mathematics, such as linear algebra. This leads to the notion of quandle rings investigated in this section. It is shown that with the exception of the trivial quandle, quandle rings are never powerassociative. In Sect. 1.13 we give multiple results of quandle ring automorphism groups. Section 1.14 gives the necessary material from graph theory needed for rest of the article. In Sect. 1.15 the most natural graph constructed from a quandle is introduced with some explicit examples. Some properties of the quandle such as being latin or connected but not latin can be read directly from the Quandle graph. Section 1.16 introduces and investigates one of the main objects of study which is the zero-divisor graph from quandle rings. Precisely we define zero-divisor graphs coming from quandle rings and study their properties. We prove that with .Z2 coefficients, every vector of even components is a left zero-divisor. Furthermore, we prove that for commutative finite quandles, the graph .r(Z2 [X]) is connected and its diameter .diam(r(Z2 [X])) is bounded above by four. We also give examples of zero-divisor quotient graph .Z[X] and zero-divisor graph over .Z2 [X] for the 3 quandles of order three. The material of this section is new and is appearing for the first time in this article. In Sect. 1.17 we consider zero-divisor graph automorphisms of quandle rings and investigate their relations with automorphisms of quandle rings and automorphisms of quandles. Explicit examples are also given.

1.2 Basics of Combinatorial Knot Theory In this section we review some basic ingredients of combinatorial knot theory needed for the motivation of quandle theory. First we start with the definition of a knot. Definition 1.1 A knot is an embedding .K : S 1 c→ R3 of a circle into three dimensional space .R3 .

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Fig. 1.1 A diagram of the figure eight knot

Fig. 1.2 A diagram of the trefoil knot

Fig. 1.3 Two diagrams of the Unknot

By an embedding it is meant that the image .K(S 1 ) in .R3 is homeomorphic to .S 1 . In other words, a knot can be thought of as a closed curve in the 3-dimensional space that does not intersect itself. The following two diagrams, Figs. 1.1 and 1.2, show respectively the figure eight knot and trefoil knot. Two knots .K1 and .K2 are considered equivalent if one can be deformed continuously to the other one. Precisely, if there is a continuous map .H : R3 × [0, 1] → R3 such that .H (x, t) is injective for any .t ∈ [0, 1] and .H (K1 , 0) = K1 and .H (K1 , 1) = K2 . Such a map H is called an isotopy. Two knots that are equivalent are said to be of the same knot type. The two diagrams of Fig. 1.3 represent both the unknot. One of the main question of knot theory is that given two knots .K1 and .K2 , are they equivalent? In order to answer this question we need the following theorem of Reidemeister. This theorem allows the passage from knots in the 3-space to knot

1 Quandles, Knots, Quandle Rings and Graphs

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Fig. 1.4 Unoriented Reidemesister moves RI, RII and RIII Fig. 1.5 Positive and negative crossings

diagrams in the plan and thus brings combinatorial tools into the study of knots. This theorem is powerful as it allows constructing invariants of knots for the purpose of distinguishing between knots. Theorem 1.1 (Reidemeister’s Theorem) Two knots .K1 and .K2 are equivalent if and only if there is a finite sequence of planar isotopies and Reidemeister moves that take any projection of the knot .K1 to any projection of the knot .K2 . The following figure, Fig. 1.4, shows respectively the three Reidemester moves RI, RII and RIII. Theses moves are local moves meaning that the changes are made in small disks while outside the disks no change in the projection is happening. Given a knot diagram, an orientation corresponds to a choice of a direction to travel around the knot. Then at each crossing of the diagram, we have either a positive crossing or a negative crossing as defined in Fig. 1.5. Definition 1.2 An n-component link is an embedding of a set of n circles into .R3 . One of the easiest invariant of links is the so-called linking number. We give its definition now.

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Fig. 1.6 Unlink, torus link and Whitehead link

Definition 1.3 Given an oriented 2-component link .L = K1 u K2 . The linking number of L is lk(L) =

E

.

l(τ ),

τ

where the sum is taken over all crossings .τ between .K1 and .K2 . Example 1.1 The linking number does not distinguish between the 2-component unlink and the Whitehead link since both have zero as linking numbers, but it does distinguish between the unlink and the torus link .T (2, 4) since .lk(T (2, 4)) = 2 as can be seen in Fig. 1.6.

1.3 Fox Colorings of Knots This section discusses Fox colorings [37] of knots which can serve as an introduction to the theory of dihedral quandles in particular and quandle theory in general. The 3-coloring invariant is the simplest invariant that distinguishes the trefoil knot from the trivial knot, since each of the three colorings of the unknot uses exactly one color while the trefoil has a non-trivial coloring (a coloring that uses more than one color). The idea of 3-coloring and more generally n-coloring was developed by Ralph Fox. He introduced a diagrammatic definition of colorability of a knot K by .Zn (the integers modulo n) in 1961 article titled “A quick trip through knot theory”[37]. This notion of colorability can be seen as one of the simplest invariant of knots. For a natural number n greater than or equal to 2, a diagram D of a knot K is said to be n-colorable if at every crossing, the sum of the colors of the under-arcs is twice the color of the over arc (modulo n) as seen in Fig. 1.7. It is well known [37] that for a prime p, a knot K is p-colorable if and only if p divides the determinant of K. The problem of finding the minimum number of

1 Quandles, Knots, Quandle Rings and Graphs

7

Fig. 1.7 Fox coloring: + a = 2b

.c

colors for p-colorable knots with p prime less than or equal to 11 was studied in [43, 58, 64]. For example Satoh proved in [64] that any 5-colorable knot admits a nontrivially 5-colored diagram where the coloring assignment uses only 4 of the 5 available colors. For a prime p, let K be a p-colorable knot and let .Cp (K) denotes the minimum number of colors among all diagrams of the knot K. In [57], it was proved that .Cp (K) ≥ log2 (p)+2. This implies that in case of .p = 13, the minimum number of colors of 13-colorable knots is greater than or equal to 5. Precisely, in [27], the following theorem was proved giving that .C13 (K) = 5. Theorem 1.2 ([27]) Any 13-colorable knot has a 13-colored diagram with exactly five colors. Thus, .C13 (K) = 5 for any 13-colorable knot K. This theorem was proved using a sequence of lemmas. In each of the lemmas the coloring scheme of the diagram was decreased by one color. This was accomplished by some specific transformations of knot diagrams. For more details, the reader can consult [27]. In [1] the authors investigated Fox colorings of knots that are 17colorable. Precisely, they proved that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram. The case of the prime number 19 was investigated recently in [42] where it was shown that any 19-colorable knot, at least six colors are enough to color the knot, that is, the minimum number of 19-colorable knot is six, that is .C19 (K) = 6 for any 19colorable knot K. Notice that Fox colorings corresponds to colorings of knots by Dihedral quandles .Rn = Zn with quandle operation .x ∗ y = −x + 2y as can be seen in the following section.

1.4 Some Basics of Racks and Quandles Now that we’ve seen that Fox coloring is a coloring of a knot by the set .Zn of integers modulo n with the binary operation .x ∗ y = −x + 2y, it seems very natural to replace .Zn by a general set X and to replace the binary operation .x ∗y = −x +2y by a more general operation .∗ : X × X → X. This leads to the general notion of a

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M. Elhamdadi and B. Jones

quandle. Depending on what Reidemeister moves we use and axiomatize we obtain a slightly different algebraic structure. Using Reidemeister moves II and III gives the notion of a rack, while using Reidemeister moves I, II and III gives the notion of a quandle. Now we review these two notions. Definition 1.4 A rack is a set .(X, ∗) with a binary operation in which right multiplications .Rx : X → X given by .y |→ y ∗ x are automorphisms. In other words the maps .Rx are bijections and the operation satisfies the rightdistributive property for all .x, y, z ∈ X, (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z).

.

If in addition, all elements are idempotents then .(X, ∗) is called a quandle. A quandle homomorphism between two quandles .(X, >) and .(Y, ∗) is a map .φ : X → Y such that .φ(u > v) = φ(u) ∗ φ(v) for all .u, v ∈ X. A quandle isomorphism is a bijective quandle homomorphism, and two quandles are isomorphic if there exits a quandle isomorphism between them. Some typical examples of quandles are: • Any set X is a quandle with the operation .x ∗ y = x for all .x, y ∈ X. It is called a trivial quandle. • Any group H with conjugation .x ∗ y = yxy −1 , is a quandle called conjugation quandle and denoted by .Conj (H ). • Let n be a positive integer. For elements .x, y ∈ Zn (integers modulo n), define .x ∗ y = −x + 2y. This operation defines a quandle structure called the dihedral quandle and denoted .Rn . Replacing .Zn by any abelian group G, one gets the quandle called Takasaki quandle [69]. • A group .X = G with operation .x ∗ y = yx −1 y is called the core quandle of G, denoted .Core(G). • Any .Z[t, t −1 ]-module M with .x ∗ y = tx + (1 − t)y is a quandle and called an Alexander quandle. • Let G be a group and .φ be an automorphism of G, then define a quandle structure on G by .x∗y = φ(xy −1 )y. Further, let H be a subgroup of G such that .φ(h) = h, for all .h ∈ H . Then .G/H is a a quandle with operation .H x ∗Hy = H φ(xy −1 )y. It is called the homogeneous quandle .(G, H, φ). Now we explain where the axioms of a quandle come from. First we need to explain how coloring of a knot diagram by a given quandle is defined. Let .(X, ∗) be a quandle and let D be a diagram of a knot K. A coloring of D by X consists of assignments of elements of the quandle X to the arcs of the diagram. The colorings at positive and negative crossings are made according to Fig. 1.8. The axiom of idempotency in the definition of a quandle can be seen from Fig. 1.9, while the property of right invertibility can be seen in Figs. 1.10 and 1.11. Precisely, Fig. 1.10 shows the identity .(x ∗ y)∗y = x, while Fig. 1.11 shows the identity .(x∗y) ∗ y = x. The right distributivity property can be seen in Fig. 1.12.

1 Quandles, Knots, Quandle Rings and Graphs Fig. 1.8 Coloring of positive and negative crossings by quandle elements

Fig. 1.9 Reidemeister move I and idempotency

Fig. 1.10 Reidemeister move II and right invertibility

Fig. 1.11 Reidemeister move II and right invertibility

Fig. 1.12 Reidemeister move III and right distributivity

9

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M. Elhamdadi and B. Jones

There are some groups naturally obtained from a quandle. For example, one can consider the whole automorphism group of a quandle X denoted by .Aut (X). This group contains a normal subgroup generated by right multiplications called Inner automorphism group .Inn(X) :=< Rx , x ∈ X > (see [28] for more details on automorphism groups of quandles). For each .x ∈ X, the left multiplication by x is the map .Lx : X → X with .Lx (y) := x ∗ y. The following is a list of some properties and some definitions of quandles. 1. A quandle is connected if .Inn(X) acts transitively on X. 2. A quandle is faithful if the mapping .x |→ Rx is an injective from X to .Inn(X). 3. A quandle X is involutory, or a kei, if the right translations are involutions: .Rx2 = id, for all .x ∈ X. 4. A Latin quandle is a quandle such that for each .a ∈ X, the left translation .La is a bijection. That is, the multiplication table of the quandle is a Latin square. 5. A quandle X is medial if .(a ∗ b) ∗ (c ∗ d) = (a ∗ c) ∗ (b ∗ d) for all .a, b, c, d ∈ X. It is easily seen that every Alexander quandle is medial. It is important to know the automorphism groups and Inner autmorphism groups of certain quandles as they can be used, for example, in computations of invariants of knots such as Fox coloring as we saw in the previous section. For this reason, we give the result for the automorphism group and inner automorphism group of dihedral quandles [28]. In order to state the result we need to recall that the affine group of .Zn is the group of all invertible affine transformations of .Zn , Aff(Zn ) := {fa,b : Zn → Zn , fa,b (x) = ax + b, a ∈ Z× n , b ∈ Z},

.

The element .fa,b is identified with the pair .(a, b) and the group multiplication is given by .(a, b)(c, d) = (ac, ad + b). The identity is .(1, 0) and the inverse is given −1 −1 by .(a, b)−1(= (a ) , −a b). Usually the element .(a, b) is represented in a matrix a b notation as . 0 1 so group multiplication corresponds to multiplication of matrices. Theorem 1.3 ([28]) Let .Rn = Zn be the dihedral quandle with the operation .x ∗ y = −x + 2y (mod n). Then the automorphism group .Aut (Rn ) is isomorphic to the affine group .Aff (Zn ). Since the affine group .Aff (Zn ) is semi-direct product group .Zn x Z× n , we have Corollary 1.1 ([28]) The cardinal of .Aut (Rn ) is .n φ(n), where .φ denotes the Euler function. For the dihedral quandle .Rn = Zn and for each .i ∈ Zn the symmetry .Si given by Si (j ) = 2i − j (mod n), can be thought of as a reflection of a regular n-gon. If n is odd, the axis of symmetry of .Si connects the vertex i to the mid-point of the side opposite to i. If .n = 2m is even, the axis of symmetry of .Si passes through the opposite vertices i and .i + m (mod 2m). From these observations, we have the easy characterization of the inner automorphism group of dihedral quandles given by the following theorem (see [28]).

.

1 Quandles, Knots, Quandle Rings and Graphs

11

Theorem 1.4 ([28]) The inner automorphism group .I nn(Rn ) of the dihedral quandle .Rn is isomorphic to the dihedral group .D m2 of order m where m is the least common multiple of n and 2.

1.5 Quandles and Quasigroups The material in this section is based on [25]. This section discusses the relation between the following types of quandles: Alexander, Latin and medial quandles on one hand and left and right distributive quasigroups on the other hand. Connections between quasigroups and quandles were established in [65]. Algebraic structures that are both right-distributive and also left distributive were investigated by Burstin and Mayer [9] in 1929 where they assumed both left and right multiplications invertible maps. They stated that there are none of orders 2 and 6, observed that the group of automorphisms is transitive, and showed that such a quasigroup is idempotent, since .x ∗ (x ∗ x) = (x ∗ x) ∗ (x ∗ x) and thus .x = x ∗ x by invertibility of the right multiplication by .x ∗ x. Definition 1.5 ([12]) (1) A quasigroup is a set Q with a binary operation .∗ such for all .u ∈ Q the right translation .Ru and left translation .Lu by u are both permutations. (2) If the operation .∗ has an identity element e in Q then the quasigroup is called a loop and denoted .(Q, ∗, e). Quasigroups differ from groups in the sense that they satisfy identities which usually conflict with associativity. Distributive quasigroups have transitive groups of automorphisms but the only group with this property is the trivial group. In [67] it was shown that there are no right-distributive quasigroups whose order is twice an odd number. Right-distributive quasigroups are intimately connected with the binary operation of a conjugation in a group since in a right-distributive quasigroup it holds that .Ry∗z = Rz Ry Rz−1 and the mapping .x |→ Rx is injective. We will see below that distributive quasigroups relate to Moufang loops. Definition 1.6 ([12]) Let .(M, ∗) be a set with a binary operation. It is called a Moufang loop if it is a loop such that the binary operation satisfies one of the following equivalent identities: x ∗ (y ∗ (x ∗ z)) = ((x ∗ y) ∗ x) ∗ z, .

(1.1)

z ∗ (x ∗ (y ∗ x)) = ((z ∗ x) ∗ y) ∗ x, .

(1.2)

(x ∗ y) ∗ (z ∗ x) = (x ∗ (y ∗ z)) ∗ x.

(1.3)

.

While studying projective geometry, Ruth Moufang introduced the so called Moufang planes and non-associative algebraic systems called Moufang loops in

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the first half of this century [55]. Her doctoral thesis was titled “Zur Struktur der projektiven Geometrie der Ebene” (On the structure of the projective geometry of the plane) and supervised by Max Wilhelm Dehn. Moufang loops differ from groups in that they need not be associative. A Moufang loop that is associative is a group. The Moufang identities may be viewed as weaker forms of associativity. The typical examples include groups and the set of nonzero octonions which gives a nonassociative Moufang loop. Theorem 1.5 (Moufang’s Theorem) Let .a, b, c be three elements in a commutative Moufang loop (abbreviated CML) M for which the relation .(a∗b)∗c = a∗(b∗c) holds. Then the subloop generated by them is associative and hence is an Abelian group. A consequence of this theorem is that every two elements in CML generate an Abelian subgroup. Let .(X, ∗) be a right-distributive quasigroup. Then .(x ∗x)∗x = (x ∗x)∗(x ∗x) which implies that each element is idempotent and .(X, ∗) is then a Latin quandle. Fix −1 (x) ∗ .a ∈ X and define the following operation, denoted .+, on X by .x + y := Ra −1 La (y). Then .a + y = y and .y + a = y. Thus .(X, +, a) is a loop. Therefore any right-distributive quasigroup satisfying one of the Moufang identities (1.1), (1.2) and (1.3) is a Moufang loop. Note that .Ra (x) + La (y) = x ∗ y. The Moufang loop is commutative if and only if (u ∗ v) ∗ (w ∗ z) = (u ∗ w) ∗ (v ∗ z)

.

(1.4)

A magma .(X, ∗) that satisfies Eq. (1.4) is said to be medial (Belousov [8]) or abelian (Joyce [49]). The Bruck-Toyoda theorem gives the following characterization of medial quasigroup. Given an Abelian group M, two commuting automorphisms f and g of M and a fixed element a of M, define an operation .∗ on M by .x ∗ y = f (x) + g(y) + a. This quasigroup is called affine quasigroup. It’s clear that .(M, ∗) is a medial qasigroup. The Bruck-Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way. Belousov gave the connection between distributive quasigroups and Moufang loops in the following Theorem 1.6 ([8]) If .(X, ∗) be a distributive quasigroup then for all .a ∈ X, (X, +, a) is a commutative Moufang loop.

.

Now let .(X, ∗) be a Latin quandle (that is right-distributive quasigroup), then the automorphism .φ = Ra satisfies .2φ(a) = a. If the order of a is odd then one can write .φ(a) = 12 a. The map .x |→ 2x being a homomorphism is equivalent to .(x + y) + (x + y) = (x + x) + (y + y), (mediality property). Recall that a magma is a set with a binary operation. We have the following question: do the following three properties imply associativity for a finite magma .(X, +)?

1 Quandles, Knots, Quandle Rings and Graphs

13

1. .(X, +) is a commutative loop with identity element 0. 2. For all .x, y in X we have the identity .(x + y) + (z + z) = (x + z) + (y + z). 3. There is an automorphism f of .(X, +) satisfying .f (x) + f (x) = x for all x. (in other words, the map .x |→ 2x is onto and .(x + x) + (y + y) = (x + y) + (x + y). In fact, if .(X, +) is a loop satisfying condition 2, then .(X, +) is a commutative Moufang loop, necessarily satisfying the other conditions. There exist nonassociative commutative Moufang loops. The smallest order at which such loops occur is 81, and there are, in fact, two such loops of that order. The easier to describe of the two commutative Moufang loops of order 81 is the one of exponent 3. Now consider the following example of a quasigroup. Let .F = Z3 and define the following operation on .F 4 given by, (x0 , x1 , x2 , x3 ) + (y0 , y1 , y2 , y3 ) =

.

(x0 + y0 + (x1 − y1 )(x2 y3 − x3 y2 ), x1 + y1 , x2 + y2 , x3 + y3 ), This is very first known example, published by Bol, who attributed it to Zassenhaus [10]. The construction from loops to quandles requires the maps .x |→ 2x to be bijections as well as a homomorphisms. Is this guaranteed for commutative Moufang loops? Every abelian group is a commutative Moufang loop, so squaring is not always a bijection, of course. For the two examples we mentioned above (loops of order 81), the answer is yes. Any commutative Moufang loop modulo its center will have exponent 3. If you have a commutative Moufang loop which is indecomposable in the sense that it is not a direct product of smaller loops, then it will have order a power of 3. Nonassociativity starts showing up at order 81. Classification of commutative Moufang loops of higher order has not been worked out in detail because of the computational difficulties. Much literature has been about free commutative Moufang loops of exponent 3, because they turn out to be finite and of order .3n . Quandles which are also quasigroups correspond to a class of loops known as Bruck loops. Commutative Moufang loops have been investigated in detail by Bruck and Salby Theorem 1.7 ([12]) If .(X, +) is a commutative Moufang loop then .X = A × B is a direct product of an abelian group A with order prime to 3 and a commutative Moufang loop of order .3k . Latin quandles are right distributive quasigroups and left-distributive Latin quandles are distributive quasigroups. Belousov’s theorem tells us that if .(X, ∗) is left-distributive Latin quandle then .(X, +) is a commutative Moufang loop and then Bruck-Slaby theorem tells us that .(X, ∗) is affine over a commutative Moufang loop, and then medial. The smallest Latin quandle that is not left distributive is of order 15 and was found by David Stanovsky (see [66, p. 29]) using an automatic model builder SEM for all quasigroups satisfying left distributivity, but not mediality. This motivated Jan Vlachy [72] to look for a more theoretical argument that would

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explain the nonexistence of any smaller quasigroups of this kind and proved that there are exactly two non-isomorphic types of these smallest non-right-distributive left-distributive quasigroups with 15 elements. He constructed them explicitly using the Galkin’s representation [39]. In the survey paper [38], page 950, Galkin states that nonmedial quasigroups of order less than 27 appear only in orders 15 and 21 and are given by the following construction: Define a binary operation on .Z3 × Zp by (x, a) ∗ (y, b) = (2y − x, −a + μ(x − y)b + τ (x − y))

.

x, y ∈ Z3 , a, b ∈ Zp ,

where .μ(0) = 2, .μ(1) = μ(2) = −1, and .τ : Z3 → Zp is such that .τ (0) = 0. This construction was generalized by replacing .Zp by any abelian group A in [18]. Let A be an abelian group, also regarded naturally as a .Z-module. Let .μ : Z3 → Z, .τ : Z3 → A be functions. These functions .μ and .τ need not be homomorphisms. Define a binary operation on .Z3 × A by (x, a) ∗ (y, b) = (2y − x, −a + μ(x − y)b + τ (x − y))

.

x, y ∈ Z3 , a, b ∈ A.

Proposition 1.1 ([18]) For any abelian group A, the above operation .∗ defines a quandle structure on .Z3 × A if .μ(0) = 2, .μ(1) = μ(2) = −1, and .τ (0) = 0. This quandle .(Z3 × A, ∗) is called the Galkin quandle and denoted by .G(A, τ ). Lemma 1.1 ([18]) For any abelian group A and .c1 , c2 ∈ A, .G(A, c1 , c2 ) and G(A, 0, c2 − c1 ) are isomorphic.

.

Various properties of Galkin quandles were studied in [18] and their classification in terms of pointed abelian groups was given. We mention a few properties. Each .G(A, c) is connected but not Latin unless A has odd order, .G(A, c) is non-medial unless .3A = 0 We conclude with the folowing properties relating distributivity and mediality to quandles [18]: Alexander quandles are left-distributive and medial. It is easy to check that for a finite Alexander quandle .(M, T ) with .T ∈ Aut(M), the following are equivalent: (1) .(M, T ) is connected, (2) .(1 − T ) is an automorphism of M, and (3) .(M, T ) is Latin. It was also proved by Toyoda [70] that a Latin quandle is Alexander if and only if it is medial. As noted by Galkin, .G(Z5 , 0) and .G(Z5 , 1) are the smallest non-medial Latin quandles and hence the smallest non-Alexander Latin quandles. We note that medial quandles are left-distributive (by idempotency). It is proved in [18] that any left-distributive connected quandle is Latin. This implies, by Toyoda’s theorem, that every medial connected quandle is Alexander and Latin. The smallest Latin quandles that are not left-distributive are the Galkin quandles of order 15. It is known that the smallest left-distributive Latin quandle that is not Alexander is of order 81. In [18] knots with crossing numbers up to twelve were distinguished by Galkin quandles. Notice that a coloring by Galkin quandle induces tricolorability (Fox coloring by the dihedral quandle .Z3 with the operation .x ∗ y = −x + 2y

1 Quandles, Knots, Quandle Rings and Graphs

15

modulo 3) since any Galkin quandle sujects to the dihedral quandle .Z3 . Precisely the following was proved in [18] Proposition 1.2 Let K be a knot with a prime determinant .p > 3. Then K is nontrivially colored by a finite Galkin quandle .G(A, τ ) if and only if p divides .|A|. For more on Galkin quandles and their use in knot theory the reader can consult [18] .

1.6 Cohomology of Quandles This section gives the construction of the homology theory of quandles defined in [23] and we give the precise and explicit formulas for 2-cocycles and 3-cocycles. Let .(X, ∗) be a finite quandle and let A be an abelian group. Let .CnR (X) be the free abelian group generated by n-tuples .(x1 , . . . , xn ) of elements R (X) by of a quandle X. Define a homomorphism .∂n : CnR (X) → Cn−1 ∂n (x1 , x2 , . . . , xn ) =

n E

.

(−1)i [(x1 , x2 , . . . , xi−1 , xi+1 , . . . , xn )

i=2

− (x1 ∗ xi , x2 ∗ xi , . . . , xi−1 ∗ xi , xi+1 , . . . , xn )]

(1.5)

for .n ≥ 2 and .∂n = 0 for .n ≤ 1. Then .C∗R (X) = {CnR (X), ∂n } is a chain complex. Let .CnD (X) be the subset of .CnR (X) generated by n-tuples .(x1 , . . . , xn ) with .xi = xi+1 for some .i ∈ {1, . . . , n − 1} if .n ≥ 2; otherwise let .CnD (X) = 0. If X is a D (X) and .C D (X) = {C D (X), ∂ } is a sub-complex quandle, then .∂n (CnD (X)) ⊂ Cn−1 n ∗ n

of .C∗R (X). Put .Cn (X) = CnR (X)/CnD (X) and .C∗ (X) = {Cn (X), ∂n' }, where .∂n' is the induced homomorphism. Henceforth, all boundary maps will be denoted by .∂n . For an abelian group A, define the chain complex by Q

Q

C∗Q (X; A) = C∗Q (X) ⊗ A,

.

Q

∂ = ∂ ⊗ id.

(1.6)

The nth quandle homology group and the nth quandle cohomology group [20] of a quandle X with coefficient group A are HnQ (X; A) = Hn (C∗Q (X; A)),

.

HQn (X; A) = H n (CQ∗ (X; A)).

(1.7)

For the purpose of computing invariants of knots, it is convenient to dualize the homology to obtain a cohomology theory. The cochaim complex is given by CQ∗ (X; A) = H om(C∗Q (X), A),

.

δ = H om(∂, id).

(1.8)

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In this cohomology theory, low dimensional cocycles are used to construct some invariants of knots and links. Here we give the explicit formulas for 2-cocycles and 3-cocycles: A 2-cocycle is a function .φ : X × X → A such that .φ(x, y) + φ(x ∗ y, z) = φ(x, z) + φ(x ∗ z, y ∗ z), and for all x, .φ(x, x) = 0. A 3-cocycle is a function .ψ : X × X × X → A such that ψ(x, y, z) + ψ(x, z, w) + ψ(x ∗ z, y ∗ z, w)

.

= ψ(x ∗ y, z, w) + ψ(x ∗ w, y ∗ w, z ∗ w) + ψ(x, y, w), and for all .x, y, .ψ(x, x, y) = ψ(x, y, y) = 0. Low dimensional cocycles can be used to construct invariants of knotted spaces. Precisely, 2-cocycles can be used to define an enhanced version of the coloring invariant of knots in the 3-space while 3-cocycles can be used to define an enhanced version of the coloring invariant of surfaces in the 4-space. For more details on this the reader can consult [23, 29].

1.7

A Spectral Sequence in Quandle Homology

The results of this section are new and are appearing for the first time in this article. Often when computing homology one encounter what is called double complex. It is a family of doubly graded abelian groups of the form .(Mi,j , d h , d v ) where h v .dp,q : Mp,q → Mp−1,q and .dp,q : Mp,q → Mp,q−1 and the following condition is satisfied, h v v h dp,q ◦ dp,q + dp−1,q ◦ dp,q = 0.

.

This means that the following diagrams anti-commute

Definition 1.7 Let .M = M∗,∗ be a double complex, then its total complex Tot(M) is the complex with the n-th term

1 Quandles, Knots, Quandle Rings and Graphs

T ot (M)n :=

17

o

.

Mp,q

p+q=n

and differential given by .Dn : T ot (M)n → T ot (M)n−1 and Dn :=

E

.

h v dp,q + dp,q .

p+q=n

It is a routine computation (based on the anticommutativity of the previous diagram) to check that .Dn−1 ◦Dn = 0 and thus the total complex .T ot (m) is in fact a chain complex. We always assume that .Mp,q = 0 whenever p or q is negative thus obtaining what is usually called a first quandrant double complex. Each individual column of the double complex is itself a chain complex with homology 1 Ep,q = Hp (Mp,∗ , d v ).

.

Now the horizontal differential induces a map 1 1 d 1 := (d h )∗ : Ep,q → Ep−1,q ,

.

so one can take the homology of these horizontal complexes to get 2 1 Ep,q := Hp (E∗,q , d) = Hph (Hqv (M∗,∗ )).

.

There is a map 2 2 d 2 : Ep,q → Ep−2,q+1

.

2 be the class of the cycle .x ∈ E 1 (i.e. .d 1 (x) = 0). obtained by letting .[x] ∈ Ep,q p,q This element x is itself the class of some vertical cycle .x˜ ∈ Mp,q (i.e. .d v (x) ˜ = 0). The cycle condition .d 1 (x) = (d h )∗ = 0 is equivalent to .d h (x) ˜ = d v (y) for some .y ∈ Mp−1,q+1 . From all these relations we obtain

d v (d h (y)) = −d h (d v (y)) = −d h (d h (x)) ˜ = 0.

.

Thus .d h (y) is a vertical cycle in .Mp−2,q+1 and determines a class .z = [d h (y)] ∈ 1 . Let compute .d 1 (z). We have Ep−2,q+1 d 1 (z) = (d h )∗ (z) = d∗h (d h (y)) = 0.

.

2 Therefore z is a cycle for .d 1 and thus determines .d 2 [x] := [z] ∈ Ep−1,q+1. Now we give the following definition of spectral sequences.

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Definition 1.8 A spectral sequence .E = {E r , d r } is a sequence of .Z-bigraded r r modules, each with diffrential .d r : Ep,q −→ Ep−r,q+r−1 , of bidegree .(−r, r − 1) r , d r ). and with isomorphisms .E r+1 ∼ H (E = In [23], calculations of cohomological dimensions of some quandles were achieved with the help of a computer. Our work here was motivated by a search for general methods (classifying space, spectral sequences, equivariant homology etc) to shed light on quandle homology. In general when given a sequence of subcomplexes .{Fp C}p∈Z with .Fp−1 C ⊂ Fp C, it is reasonable to try to obtain information about .H∗ (C) in terms of the groups .H∗ (Fp C/Fp−1 C). We use this idea to calculate homology of disjoint union of two quandles. Now we give the definition of a disjoint union of quandles. Definition 1.9 Let .(X, ∗1 ) and .(Y, ∗2 ) be two racks. Consider the .W = X u Y the binary operation

x∗y =

.

⎧ ⎪ ⎪ ⎨x ∗1 y, x ∗2 y, ⎪ ⎪ ⎩x,

if x and y are both in X, if x and y are both in Y, otherwise.

We have the following straightforward lemma stating that the disjoint union of two quandles is a quandle. Lemma 1.2 If X and Y are quandles then W is also a quandle. Now we set up a spectral sequence for disjoint unions of racks and quandles. We use the following terminology for spectral sequences following [11]. Let .Fp C = {Fp Cp+q } be a filtration of a chain complex C, i.e. .· · · ⊂ Fp C ⊂ r r = Fp Cp+q ∩ ∂ −1 (Fp−r Cp+q−1 ) and .Bp,q = Fp Cp+q ∩ Fp+1 C ⊂ · · · . Let .Zp,q r−1 ∂Fp+r−1 Cp+q+1 = ∂Zp+r−1,q−r . Then the r-th term of the spectral sequence is described by r−1 r r r r r r Ep,q = Zp,q /(Bp,q + Zp−1,q+1 ) = Zp,q /(Bp,q + (Fp−1 Cp+q ∩ Zp,q )).

.

Setting .Fp H (C) = Im(H (Fp C) → H (C)), we have ∞ Ep,q = Fp Hp+q (C)/Fp−1 Hp+q (C).

.

Let .X, Y be racks and .W = X u Y be the disjoint union. For simplicity the chain complex is denoted by .C = {Cn } = {Cn (W ; Z)}. Let .Fp Cn be the abelian subgroup of .Cn generated by n-chains of the form .w = (w1 , · · · , wn ) such that at most p entries among .wi ’s are elements of Y . The definition of the boundary homomorphism of quandle homology implies the following Lemma 1.3 The sequence .Fp C = {Fp Cn }n∈Z≥0 is a filtration of C.

1 Quandles, Knots, Quandle Rings and Graphs

19

This filtration gives a spectral sequence that collapses. Lemma 1.4 The spectral sequence for this filtration collapses, i.e., Ep2 = Ep3 = · · · = E ∞ .

.

r has less than p entries Proof If a .(p + q)-chain .w = (w1 , · · · , wp+q ) ∈ Zp,q r−1 r r /(B r + from Y , then .w ∈ Zp−1,q+1 . Therefore, from the description .Ep,q = Zp,q p,q r−1 r ), the group .Ep,q is generated by .(p + q)-chains .w = (w1 , · · · , wp+q ) Zp−1,q+1 with exactly p entries from Y . However, the boundary homomorphism .∂ decreases r the number of entries from Y at most one, which induces the zero map .d r : Ep,q → r Ep−r,q+r−1 for .r ≥ 2. u

We start the analysis of the spectral sequence from the .E 1 -terms as follows. 1 of the spectral sequence is equal to .C (Y ), Lemma 1.5 The term .Ep,0 p i.e., 1 Ep,0 = Cp (Y ).

.

1 Proof In general the .E 1 -terms are described by .Ep,q = Hp+q (Fp C/Fp−1 C), where the boundary homomorphism .∂ is the induced one. Thus we show that .∂ = 0 on the chain complex .Fp C/Fp−1 C at .q = 0. The group .Fp Cp+q /Fp−1 Cp+q is generated by the .(p + q)-chains whose entries have exactly p elements from Y , and q elements from X. Each term (say .w ' ) of .∂w has one less entries than .w = (w1 , · · · , wp+q ), with .wi deleted. If .wi ∈ Y , then this term .w ' is in .Fp−1 Cp+q−1 , hence is zero under .∂ : Fp Cp+q /Fp−1 Cp+q → Fp Cp+q−1 /Fp−1 Cp+q−1 . Any element of .Fp Cp+1 /Fp−1 Cp+1 is represented by .w ∈ Fp Cp+1 such that there is exactly one entry from X. If .wi ∈ X, then this term .w ' has no element from X as its entries, and as X acts on Y trivially, two such terms .w1' and .w2' cancel out with opposite signs. This implies that .∂ = 0 : Fp Cp+1 /Fp−1 Cp+1 → Fp Cp /Fp−1 Cp , 1 = F C /F and .Ep,0 u p p p−1 Cp = Cp (Y ). 1 Now we look at the term .Ep−1,1 which comes from the homology of

1 Lemma 1.6 For .p > 1, .Ep−1,1 = ⊕p H1 (X) ⊗ Cp−1 (Y ).

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M. Elhamdadi and B. Jones

Proof We look for the image of .∂ : Fp−1 Cp+1 /Fp−2 Cp+1 → Fp−1 Cp /Fp−2 Cp . For any .w = (w1 , · · · , wp+1 ) with .wi , wj ∈ X and .wk ∈ Y for .k /= i, j , .∂w has all terms in .Fp−2 Cp except .±{(· · · , wi , · · · ) − (· · · , wi ∗ wj , · · · )}. Hence the subgroup .Gi generated by .w = (w1 , · · · , wp+1 ) with .wi ∈ X and all other terms are from Y contains the subgroup generated by .±{(· · · , wi , · · · ) − (· · · , wi ∗ wj , · · · )} as the image under .∂. This implies that .Gi contributes one factor of .H1 (X) ⊗ Cp−1 (Y ), and there are p such factors. u The chain complex .{∂ : F0 Cq /F−1 Cq → F0 Cq−1 /F−1 Cq−1 } is isomorphic to {∂ : Cq (X) → Cq−1 (X)} since .F−1 C = {0}. This gives the following

.

1 of the spectral sequence is equal to .H (X), i.e., Lemma 1.7 The term .E0,q q 1 E0,q = Hq (X).

.

Next we investigate the .E 2 -terms. Using Lemma 1.5, the chain complex 1 1 1 .{d : E p,0 → Ep−1,0 } is isomorphic to .{∂ : Cp (Y ) → Cp−1 (Y )}. This gives 2 = H (Y ). Lemma 1.8 .Ep,0 p 1 → E1 By Lemma 1.7, the chain complex .{d 1 : Ep,0 p−1,0 } is isomorphic to .{∂ = 0 : Hp (X) → Hp−1 (X)}. This implies the following 2 of the spectral sequence is equal to .H (X), i.e., .E 1 = Lemma 1.9 The term .E0,q q 2,q Hq (X).

We have the following applications: The second homology group of .W = X u Y is given by the following Proposition 1.3 .H2 (X u Y ) = H2 (X) ⊕ H2 (Y ) ⊕ (⊕2 H1 (X) ⊗ H1 (Y )). Proof We have {0} = F−1 H2 (W ) ⊂ F0 H2 (W ) ⊂ F1 H2 (W ) ⊂ F2 H2 (W ) = H2 (W )

.

2 = H (Y ) by Lemma 1.8 and where .H2 (W )/F1 H2 (W ) = E2,0 2 2 = H (X) by Lemma 1.9. .F0 H2 (W ) = E 2 0,2 2 is the homology group at the middle term of The term .E1,1 1 1 1 E0,1 = H1 (X) ← E1,1 = ⊕2 H1 (X) ⊗ C1 (Y ) ← E2,1 = ⊕3 H1 (X) ⊗ C2 (Y )

.

by Lemmas 1.5, 1.6, and 1.7. The last homomorphism is induced by the following terms, where .xi ∈ X and .yj ∈ Y for any .i, j = 1, 2, 3. ∂(x1 , y2 , y3 ) = −(x1 , y2 ) + (x1 , y2 ∗ y3 ),

.

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21

∂(y1 , x2 , y3 ) = −(y1 , x2 ) + (y1 ∗ y3 , x2 ), ∂(y1 , y2 , x3 ) = (y1 , x3 ) − (y1 ∗ y2 , x3 ). 2 = ⊕ H (X) ⊗ H (Y ). This implies that .E1,1 2 1 1 Since .H1 is a free abelian group, 2 F0 H2 (W ) = H2 (X) → F1 H2 (W ) → E1,1 = ⊕2 H1 (X) ⊗ H1 (Y )

.

splits, and we obtain .F1 H2 (W ) = H2 (X) ⊕ (⊕2 H1 (X) ⊗ H1 (Y )). The inclusion-induced homomorphism splits 2 .H2 (W ) → E 2,1 = H2 (Y ) → {0}, so we obtain the result.

u

1 1 Other .E 1 -terms are given by . Ep,0 = Cp (Y ). For .p > 1, . Ep−1,1 = 1 ⊕p H1 (X) ⊗ Cp−1 (Y ) and .E0,q = Hq (X). 2 2 = H (X). As an = Hp (Y ), and .E0,q As for the .E 2 -terms, the PI has . Ep,0 q application, we obtain the second quandle cohomology of the disjoint union

H2 (X u Y ) = H2 (X) ⊕ H2 (Y ) ⊕ (⊕2 H1 (X) ⊗ H1 (Y )).

.

Other .E 2 -term are given by 2 E2,1 = ⊕3 (H1 (X) ⊗ H2 (Y ))/J2 ,

.

2 E1,2 = ⊕3 (H2 (X) ⊗ H1 (Y ))/J1 ,

where .J1 and .J2 are the subgroups generated respectively by the cycles {∂(y1 , y2 , x3 , x4 ) = (y1 , x3 , x4 ) − (y1 ∗ y2 , x3 , x4 ) + (y1 , y2 , x3 )

.

−(y1 ∗ y4 , y2 ∗ y4 , x3 )} and {∂(x1 , x2 , y3 , x4 ) = (x1 , y3 , x4 ) − (x1 ∗ x2 , y3 , x4 ) + (x1 , x2 , y3 )

.

−(x1 ∗ x4 , x2 ∗ x4 , y3 )} The third cohomology group is given by H3 (X u Y ) = H3 (X) ⊕ (⊕3 H2 (X) ⊗ H1 (Y ))/J1 ) ⊕ (⊕3 H1 (X) ⊗ H2 (Y ))/J2 )

.

⊕H3 (Y ). Eventhough this approach is limited to low dimensional ones, at this stage of investigation, we expects more useful filtrations so to lead to more results and homology of quandles. We are investigating the classifying space approach since it’s

22

M. Elhamdadi and B. Jones

related to spectral sequences. This classifying space should have the nice property (as in the group case) that an extension of quandles should induce a fibration of classifying spaces. Problem Given an extension E of a quandle X by a group A: . A → E → X, Q 2 = HpQ (X, Hq (A)) which converge to .Hp+q (E) can a spectral sequence .Ep,q be constructed? Spectral sequences will allow calculation of higher dimensional homology and cohomology.

1.8 Quandles and Singular Knot Theory The study of singular knots and their invariants was motivated mainly by the theory of Vassiliev invariants [71]. Most of the important knot invariants have been extended to singular knot invariants. Fiedler extended the Kauffman state models of the Jones and Alexander polynomials to the context of singular knots [36]. Juyumaya and Lambropoulou constructed a Jones-type invariant for singular links using a Markov trace on a variation of the Hecke algebra [50]. In [52] Kauffman and Vogel defined a polynomial invariant of embedded 4-valent graphs in .R3 extending an invariant for links in .R3 called the Kauffman polynomial [51]. In [44], Henrich and the fourth author investigated singular knots in the context of virtual knot theory, flat virtual knot theory and flat singular virtual knot theory. They introduced algebraic structures called semiquandles, singular semiquandles and virtual singular semiquandles. They also gave an application to distinguishing Vassiliev-type invariants of virtual knots. Other extensions of classical invariants of knots to singular knots can be found in the works of [59, 68]. Recall that a singular link in .S 3 is the image of a smooth immersion of n circles in .S 3 that has finitely many double points, called singular points. An orientation of each circle induces orientation on each component of the link. This gives an oriented singular link. Here we assume that any singular link is oriented, unless specified otherwise. Furthermore, we deal with singular link projections, or diagrams, which are projections of the singular link to the plane such that the information at each crossing is preserved by leaving a little break in the lower strand. Two oriented singular link diagrams are considered equivalent if and only if one can obtain one from the other by a finite sequence of singular Reidemeister moves (see Fig. 1.14). In the case of classical knot theory, the axiomatisation of the Reidemeister moves gives rise to the definition of a quandle. Here we generalize the structure of quandles by considering singular oriented knots and links and color them by some algebraic structures called singquandles. Singquandles can be defined for un-oriented singular knots as was done in [21, 22]. Here for simplicity we focus only on the oriented case obtaing what we call oriented singquandles. The axioms of oriented singquandles are obtained using a generating set of Reidemeister moves on oriented singular links (see Fig. 1.14).

1 Quandles, Knots, Quandle Rings and Graphs

23

Fig. 1.13 Regular and singular crossings

As mentioned earlier, the axioms of quandle structures are deduced from the axiomatization of Reidemeister moves. Inspired by this idea we define new binary operations on a quandle X and derive the axioms of oriented singquandle by considering a generating set of oriented singular Reidemeister moves. The results of this section are based on [7]. A semiarc in a singular link diagram L is an edge between vertices in the link L considered as a 4-valent graph. The oriented singquandle axioms are obtained by assigning elements of the oriented singquandle to semiarcs in an oriented singular link diagram and letting these elements act on each other at crossings as shown in Fig. 1.13. Using the generalized Reidemeister moves for singular knot theory, we need to derive the axioms that the binary operations .∗, .R1 and .R2 in Fig. 1.13 should satisfy. For this purpose, we begin with the generating set of Reidemeister moves given in Fig. 1.14. The proof that this is a generating set can be found in [7]. Using this set of Reidemeister moves, the singquandle axioms can be derived easily. This can be seen in Figs. 1.15, 1.16, and 1.17. It is necessary to have a generating set of oriented singular Reidemeister moves. Such a set is given in Fig. 1.14. The proof that this set is a generating set can be found in [7] where it is shown that only three oriented moves are required to generate all possible Reidemeister move on oriented singular knots. Figures 1.15, 1.16, and 1.17 immediately give us the following definition. Definition 1.10 ([7]) Let .(X, ∗) be a quandle. Let .R1 and .R2 be two maps from .X× X to X. The triple .(X, ∗, R1 , R2 ) is called an oriented singquandle if the following axioms are satisfied: R1 (x ∗¯ y, z) ∗ y = R1 (x, z ∗ y) coming from o4a.

.

R2 (x ∗¯ y, z) = R2 (x, z ∗ y)¯∗y coming from o4a. (y ∗¯ R1 (x, z)) ∗ x = (y ∗ R2 (x, z))¯∗z coming from o4e .

(1.9) (1.10) (1.11)

R2 (x, y) = R1 (y, x ∗ y) coming from o5a.

(1.12)

R1 (x, y) ∗ R2 (x, y) = R2 (y, x ∗ y) coming from o5a

(1.13)

The following examples were given in [7].

24

Fig. 1.14 A generating set of singular Reidemeister moves

Fig. 1.15 The Reidemeister move .o4a and colorings

M. Elhamdadi and B. Jones

1 Quandles, Knots, Quandle Rings and Graphs

25

Fig. 1.16 The Reidemeister move .o4e and colorings

Fig. 1.17 The Reidemeister move .o5a and colorings

Example 1.2 Let .X = Zn with the quandle operation .x ∗ y = ax + (1 − a)y, where a is invertible so that .x ∗¯ y = a −1 x + (1 − a −1 )y. Now let .R1 (x, y) = bx + cy, then by axiom (1.12) we have .R2 (x, y) = acx + (c(1 − a) + b)y. By substituting these expressions into axiom (1.9) we can find the relation .c = 1 − b. Substituting, we find that the following is an oriented singquandle for any invertible a and any b in .Zn : x ∗ y = ax + (1 − a)y.

(1.14)

R1 (x, y) = bx + (1 − b)y.

(1.15)

R2 (x, y) = a(1 − b)x + (1 − a(1 − b))y

(1.16)

.

This example can generalized to give the following Example 1.3 Let .A = Z[t ±1 , v] and let X be a .A-module. Then the operations x ∗ y = tx + (1 − t)y,

.

R1 (x, y) = α(a, b, c)x + (1 − α(a, b, c))y

and

R2 (x, y) = t[1 − α(a, b, c)]x + [1 − t (1 − α(a, b, c))]y

.

where .α(a, b, c) = at +bv +ctv, make X an oriented singquandle which we call an Alexander oriented singquandle. The fact that X is an oriented singquandle follows from Example 1.2 by straightforward substitution.

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Definition 1.11 A coloring of an oriented singular link L is a function .C : R −→ X, where X is a fixed oriented singquandle and R is the set of semiarcs in a fixed diagram of L, satisfying the conditions given in Fig. 1.13. Now the following lemma is immediate from Definition 1.10. Lemma 1.10 ([7]) The set of colorings of a singular link by an oriented singquandle is an invariant of oriented singular links. The set of colorings of a singular link L by an oriented singquandle X will be denoted by .ColX (L). As in the usual context of quandles, the notions of oriented singquandle homomorphisms and isomorphisms are immediate. Definition 1.12 Let .(X, ∗, R1 , R2 ) and .(Y, >, S1 , S2 ) be two oriented singquandles. A map .f : X −→ Y is homomorphism if the following axioms are satisfied : 1. .f (x ∗ y) = f (x) > f (y), 2. .f (R1 (x, y)) = S1 (f (x), f (y)), 3. .f (R2 (x, y)) = S2 (f (x), f (y)). If in addition f is a bijection, then we say that it is an isomorphism, and we say that .(X, ∗, R1 , R2 ) and .(Y, >, S1 , S2 ) are isomorphic. Note that isomorphic oriented singquandles will induce the same set of colorings.

1.9 Oriented Singquandles Over Groups When the underlying set of an oriented singquandle X is a group, one obtains a rich family of structures. Here we give examples including a generalization of affine oriented singquandles, as well as an infinite family of non-isomophic singquandles over groups. The following examples were given in [7]. Example 1.4 Let .X = G be an abelian additive group, with f being a group automorphism and g a group endomorphism. Consider the operations .x ∗ y = f (x) + y − f (y) and .R1 (x, y) = g(y) + x − g(x). First, note that the inverse operation of .∗ is given by .x∗y = f −1 (x) + y − f −1 (y). Now We can deduce from axiom (1.12) that .R2 (x, y) = g(f (x)) + y − g(f (y)). Substituting into the axioms, we find that the axioms are satisfied when .(f ◦ g)(x) = (g ◦ f )(x). Example 1.2 is a particular case of this structure with .f (x) = ax and .g(x) = (1 − b)x. Example 1.5 Let .X = G be a non-abelian multiplicative group with the quandle operation .x ∗ y = y −1 xy. Then a direct computation gives the fact that .(X, ∗, R1 , R2 ) is a singquandle if and only if .R1 and .R2 satisfy the following equations: y −1 R1 (yxy −1 , z)y = R1 (x, y −1 zy).

(1.17)

.

R2 (yxy

−1

, z) = yR2 (x, y

−1

zy)y

−1

.

(1.18)

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27

x −1 R1 (x, z)y[R1 (x, z)]−1 x = z[R2 (x, z)]−1 y[R2 (x, z)]z−1. R2 (x, y) = R1 (y, y −1 xy). [R2 (x, y)]

−1

R1 (x, y)R2 (x, y) = R2 (y, y

−1

xy)

(1.19) (1.20) (1.21)

A straightforward computation gives the following solutions, for all .x, y ∈ G. 1. 2. 3. 4. 5.

R1 (x, y) = x and .R2 (x, y) = y. R1 (x, y) = xyxy −1 x −1 and .R2 (x, y) = xyx −1 . −1 xy and .R (x, y) = y −1 x −1 yxy. .R1 (x, y) = y 2 −1 x −1 yx, and .R (x, y) = x −1 y −1 xy 2 . .R1 (x, y) = xy 2 −1 y)n and .R (x, y) = (y −1 x)n+1 y, where .n ≥ 1. .R1 (x, y) = y(x 2 . .

Next we focus our attention on a subset of some infinite families of oriented singquandle structures in order to show that some in fact are not isomorphic. A direct computation shows that the maps .R1 and .R2 in the following proposition satisfy the five axioms of Definition 1.10 and thus we have Proposition 1.4 ([7]) Let .X = G be a non-abelian group with the binary operation x ∗ y = y −1 xy. Then, for .n ≥ 1, the following maps .R1 and .R2 yield an oriented singquandles structures .(X, ∗, R1 , R2 ) on G:

.

1. .R1 (x, y) = x(xy −1 )n and .R2 (x, y) = y(x −1 y)n , 2. .R1 (x, y) = (xy −1 )n x and .R2 (x, y) = (x −1 y)n y, 3. .R1 (x, y) = x(yx −1 )n+1 and .R2 (x, y) = x(y −1 x)n . Furthermore, in each of the cases (1), (2) and (3), different values of n give also non-isomorphic singquandles. To see that the three solutions are pairwise non-isomorphic singquandles, we compute the set of colorings of a singular knot by each of the three solutions. Consider the singular link given in Fig. 1.18 and color the top arcs by elements x and y of G. Then it is easy to see that the set of colorings is given by Fig. 1.18 An oriented Hopf link

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M. Elhamdadi and B. Jones

ColX (L) = {(x, y) ∈ G × G | x = R1 (y, y −1 xy), y = R2 (y, y −1 xy)}.

.

(1.22)

Using .R1 (x, y) = x(xy −1 )n and .R2 (x, y) = y(x −1 y)n from solution (1), the set of colorings becomes: ColX (L) = {(x, y) ∈ G × G | (x −1 y)n+1 = 1},

.

(1.23)

while the set of colorings of the link L with .R1 (x, y) = (xy −1 )n x and .R2 (x, y) = (x −1 y)n y from solution (2) is: ColX (L) = {(x, y) ∈ G × G | x −1 (x −1 y)n y = 1}.

.

(1.24)

Finally, the set of colorings of the same link with .R1 (x, y) = x(yx −1 )n+1 and −1 x)n from solution (3) is: .R2 (x, y) = x(y ColX (L) = {(x, y) ∈ G × G | (y −1 x)n = 1}.

.

(1.25)

This allows us to conclude that solutions (1), (2), and (3) are pairwise nonisomorphic oriented singquandles. In fact these computations also give that different values of n in any of the three solutions (1), (2) and (3) give non-isomorphic oriented singquandles. We exclude the case of .n = 0 as solutions (1) and (2) become equivalent. Example 1.6 Let .X = Zn with .x ∗ y = αx + (1 − α)y, where .α is invertible in Zn . Assume that .R1 (x, y) = mx + (1 − m)y and .R2 (x, y) = nx + (1 − n)y, then a direct computation gives the fact that .(X, ∗, R1 , R2 ) is an oriented singquandle if and only if .n = 1 − m and thus .R2 (x, y) = (1 − m)x + my giving the condition that .R1 (x, y) = R2 (y, x).

.

We illustrate here how these new structures can be used to distinguish between oriented singular knots and links. Example 1.7 We choose X to be a non-abelian group and consider the onefold conjugation quandle on X along with the binary operation .R1 and .R2 given by .R1 (x, y) = x 2 y −1 and .R2 (x, y) = yx −1 y. By considering solution (5) in Example 1.5 when .n = 1 we know that X is an oriented singquandle. In this example we show how the this oriented singquandle can be used to distinguish between two singular knots that differ only in orientation. Color the two arcs on the top of the knot on the left of Fig. 1.19 by elements x and y. This implies that the coloring space is the set in the diagonal in .G × G. On the other hand, the knot on the righthand side of Fig. 1.19 has the coloring space −1 = yxy −1 }. Since this set is not the same as the diagonal .{(x, y) ∈ G × G, xyx of .G × G, the coloring invariant distinguishes these two oriented singular knots. Example 1.8 Consider the quandle given in the above example with .R1 (x, y) = x(yx −1 )2 and .R2 (x, y) = xy −1 x . This can be obtained from item (3) of

1 Quandles, Knots, Quandle Rings and Graphs

29

Fig. 1.19 Two oriented singular Hopf links with one singular point

Fig. 1.20 Two oriented singular Hopf links .L1 and .L2 with two singular points

Proposition 1.4 by setting .n = 1. If we color the top arcs of the link .L1 on the left of Fig. 1.20 by x and y, then we obtain the following two equations: x = R1 (R1 (x, y), R2 (x, y)), and y = R2 (R1 (x, y), R2 (x, y)).

.

On the other hand, if we color the top arcs of the link .L2 on the right of Fig. 1.20 by x and y and the middle arcs by u and v, then we obtain the following four equations: x = R1 (y, u), v = R2 (y, u), y = R1 (x, v), and u = R2 (x, v),

.

Choose a group G with non-trivial center .Z(G) (such as the group .S 3 ). If .x /= y are both in the center .Z(G) then one obtains the condition that .x 2 = y 2 in the case of 4 4 .L1 , while in the case of .L2 one obtains the condition .x = y . Thus by choosing a 2 4 4 2 2 pair .(x, y) ∈ G such that .x = y but .x /= y , one obtains that the links .L1 and .L2 are distinct.

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1.10 Topological Quandles The study of topological quandles was investigated in [31, 63]. Here we review some of that material mostly from [31]. A topological rack is a rack X which is a topological space such that the map .X × X e (x, y) |→ x ∗ y ∈ X is a continuous. In a topological rack, the right multiplication .Rx : X e y |→ y ∗ x ∈ X is a homeomorphism, for all .x ∈ X. Observe that an ordinary (finite) rack is automatically a topological rack with respect to the discrete topology. Definition 1.13 A quandle (resp. topological quandle) is a rack (resp. topological rack) such that .x ∗ x = x, ∀x ∈ X. Remark 1.1 Suppose that a rack (resp. a topological rack) X is equipped with a binary operation .∗ : X × X e (x, y) |→ x ∗ y ∈ X that is right and left distributive at the same time. Then .(X, ∗) is a quandle (resp. topological quandle). Indeed, for all .x ∈ X, we have Rx∗x (x) = x ∗ (x ∗ x) = (x ∗ x) ∗ (x ∗ x) = Rx∗x (x ∗ x),

.

which implies that .x ∗ x = x. Example 1.9 (The Conjugation Quandle) Let G be a topological group. The operation x ∗ y = yxy −1

.

makes G into a topological quandle which is denoted by .Conj (G) and is called the conjugation quandle of G. In fact, any conjugacy class of G is a topological quandle with this operation. Example 1.10 (The Core Quandle) Let G be a topological group. The operation x ∗ y = yx −1 y

.

defines a topological quandle structure on G. This quandle will be denoted by Core(G) and we call it the core of G. Observe that this operation satisfies .(x ∗y)∗y = x. Any quandle in which this equation is satisfied is called an involutive quandle. .

Example 1.11 (Symmetric Manifold) First recall that a symmetric manifold M is a Riemannian manifold such that each point .x ∈ M is an isolated fixed point of an involtutive isometry .ix : M → M. Given such manifold, every .x ∈ M endows M with the structure of topological quandle by setting .x ∗ y = iy (x). Example 1.12 Let .S n be the unit sphere of .Rn+1 . Then, with respect to the operation

1 Quandles, Knots, Quandle Rings and Graphs

x ∗ y = 2(x · y)y − x,

.

31

x, y ∈ S n ,

where .x · y is the usual scalar product in .Rn+1 , and the topology inherited from n+1 , .S n is a topological quandle. .R Example 1.13 Following the previous example, let .λ and .μ be real numbers, and let .x, y ∈ S n . Then λx ∗ μy = λ[2μ2 (x · y)y − x].

.

In particular, the operation .

± x ∗ ±y = ±(x ∗ y)

provides a structure of topological quandle on the projective space .RPn . Example 1.14 Let G be a topological group and .σ be a homeomorphism of G. Let H be a closed subgroup of G such that .σ (h) = h, for all .h ∈ H . Then .G/H is a quandle with operation [x] ∗ [y] := [σ (x)σ (y)−1 y],

.

where for .x ∈ G, .[x] denotes the class of x in .G/H . For example, one can consider the group G to be the group of rotations .G = SO(2n + 1), .H = SO(2n) and 2n+1 . .G/H = S Definition 1.14 ([30]) Let X be a topological rack or quandle. An element .u ∈ X is 1. a stabiliser if .x ∗ u = x, for all .x ∈ X; 2. totally fixed in X if .u ∗ x = u, for all .x ∈ X; 3. a unit if u is a stabiliser and is totally fixed in X. The set of all stabilisers of X (resp. all totally fixed points in X) is denoted by Stab(X) (resp. .F ix(X))

.

Observe that if u is a stabiliser in the rack X, we have .u ∗ u = u. Moreover, if u is a unit, then .(x ∗ u) ∗ y = x ∗ y for all .x, y ∈ X. Lemma 1.11 Assume the topological rack X admits a non-empty set of units. Then for all arbitrary pair of units .u, v we have u ∗ v = u, v ∗ u = v.

.

Proof Indeed, if u and v are units in X, then by (1) and (2) in the definition 1.14, u we have .u ∗ v = u and .u ∗ v = u. Definition 1.15 The set of all units in a topological racks or quandle X is denoted by .UX . We say that X is unital if .UX is non-empty.

32

M. Elhamdadi and B. Jones

Example 1.15 Let G be a topological group. Then it is easy to check that .UConj (G) is exactly the centre .Z(G) of G. Example 1.16 (Topological Linear Rack) Let G be a topological group and V a continuous representation; i.e., there is a continuous map G × V e (g, v) |→ g · u ∈ V

.

with .g · (h · v) = (gh) · v, for all .g, h ∈ G, v ∈ V . We define a topological rack structure on .G × V as follows: (g, u) ∗ (h, v) := (h−1 gh, h−1 · u), g, h ∈ G, u, v ∈ V .

.

We denote this rack as .G x V . Observe that this rack is unital and .(1, 0) is a unit. The following proposition is immediate. Proposition 1.5 ([30]) Let G be a topological group and V a countinuous representation through the map .π : G → GL(V ). Denote by .V G the subspace of V consisting of invariant vectors under the continuous G-action. Then we have Stab(G x V ) ∼ = [Z(G) ∩ ker(π )] × V , F ix(G x V ) ∼ = Z(G) × V G ,

.

and UGxV ∼ = [Z(G) ∩ ker(π )] × V G .

.

Definition 1.16 Let X and Y be topological racks. A rack morphism from X to Y is a continuous map .f : X → Y such that .f (x ∗ y) = f (x) ∗ f (y), for all .x, y ∈ X. Morphisms of topological quandles are defined in the same way. Isomorphisms of racks or quandles are defined accordingly. If Y is unital, then f is said to be unital if .f (UX ) ⊆ UY . Example 1.17 Given a topological rack X, each element .x ∈ X defines a rack automorphism through .Rx : X e y |→ y ∗ x ∈ X. Moreover, if X is unital, .Rx is a unital morphism. Proposition 1.6 Let G be a topological group. Then every unit element in .Core(G) is a 2-torsion of the group G. In particular, if G is torsion free, .UCore(G) is empty. For instance, .Core(R) has no units. Example 1.18 The classical map .f : R → S 1 given by .f (t) = e2iπ t is a quandle homomorphism from .R with the binary operation .t ∗ t ' = 2t ' − t to the quandle .S 1 with operation .z ∗ z' = z' z−1 z' . Lemma 1.12 ([30]) Let X be a non-unital topological rack. Define the unitarization .X+ of X by adding a one point set .{1} to X and declaring that .x ∗ 1 = x and .1∗x = 1 for all .x ∈ X and endowing it with the topology induced from the inclusion

1 Quandles, Knots, Quandle Rings and Graphs

33

map .X e x → | x ∗ 1 ∈ X+ . Then .X+ is a unital topological rack . Moreover, the inclusion .X → X+ is an injective morphism of topological racks. Remark 1.2 Notice that .u ∈ X is a stabiliser if and only if .Ru is the identity morphism of racks .X → X. Further, u is totally fixed if and only if it is a fixed point of .Rx for every .x ∈ X. It follows that in the Definition and Lemma 1.12, we have changed nothing in the “structure” of X since the added unit 1 may be identified with the identity morphism of the racks .I d : X → X and be considered as a fixed point of all of the morphisms .Rx . Now we introduce the notion of Inner group for topological quandles. Let X be a topological quandle. Notice that if .f, g : X → X are (continuous) quandle morphisms then so is .f g := f ◦ g. If moreover f and g are quandle automorphisms (i.e., quandle homeomorphisms), then so is fg. The set .Aut (X) of quandle automorphisms forms a group under composition. Furthermore, when equipped with the compact-open topology, .Aut (X) is a topological group. Recall that the right translation .Rx : X → X is an automorphism of topological quandle. Proposition 1.7 Define the inner representation of X to be the map .

R : X → Aut (X) x |→ Rx

Then R is continuous. Moreover, for all .u, v ∈ X, we have Ru Rv (·) = Ru (·) ∗ Ru (v)

.

We shall note that the compact-open topology has basis open sets W (K, U ) := {f : X → X rack homomorphism | f (K) ⊂ U } ,

.

where .K ⊂ X is compact and .U ⊂ X is open. We then need the following lemma to prove the proposition 1.7. Lemma 1.13 Let X be a topological rack and let K and U be compact and open subsets of X, respectively. Suppose there exists .x ∈ X such that .K ∗ x ⊂ U . Then there is an open neighbourhood V of x such that .K ∗ V ⊂ U . Proof Since the rack operation .X × X e (y, x) |→ y ∗ x ∈ X is a continuous -x,y and .Vx,y of y and x, map and U open, there exists open neighbourhoods .V respectively, such that -x,y ∗ Vx,y ⊂ U. V

.

-x,y }y∈K is an open cover of the compact subset Now, for a fixed .x ∈ X, the family .{V .K ⊂ X. Hence, there is a finite set .{y0 , · · · , yn } ⊂ K such that

34

M. Elhamdadi and B. Jones

K⊂

n ||

.

-x,yk ∗ Vx,yk ⊂ U. -x,yk , and V V

k=0

It is straightforward that the open neighbourhood Vx :=

n n

.

Vx,yk

k=0

of x satisfies .K ∗ Vx ⊂ U .

u

Proof of Proposition 1.7 Let .W (K, U ) be an open subset in .Aut (X). Then thanks to Lemma 1.13, if x is in the inverse image of .W (K, U ) by R, there is an open neighbourhood .Vx such that .Vx ⊂ R −1 (W (K, U )); hence, .R −1 (W (K, U )) is open in X and R is then continuous. For the second statement, we have, for all .y ∈ X, Rx Rx ' (y) = (y ∗ x ' ) ∗ x = (y ∗ x) ∗ (x ' ∗ x) = Rx (y) ∗ Rx (x ' ),

.

u Definition 1.17 We define the inner automorphism group .I nn(X) of X to be the closure of the subgroup generated by the image of X by R in .Aut (X); i.e., I nn(X) := < R(X) > ⊂ Aut (X).

.

Recall that for any quandle endomomorphism f of X, we have .f Rx = Rf (x) f . Then .I nn(X) is a normal subgroup of .Aut (X) as the closure of a normal subgroup. With the quotient topology, .Aut (X)/I nn(X) is a topological group. Also, since R is continuous, if X is compact, then .I nn(X) is a compactly generated group. Example 1.19 Consider again the core of .R. Then .Aut (Core(R)) is the affine group {( Aff (R) =

.

) } ab , 0 /= a, b ∈ R 01

and the inner group .I nn(Core(R)) = R. Example 1.20 Let .M(/= I2 ) be an invertible two-by-two matrix over the integers Z (i.e. .det (M) = ±1), where .I2 is the identity matrix, and assume that .M 2 /= I2 . The plane .R2 becomes a topological quandle with the operation

.

x ∗ y = Mx + (I2 − M)y.

.

1 Quandles, Knots, Quandle Rings and Graphs

35

It is easily seen that this map is compatible with the projection of .R2 → R2 /Z2 . Let m and n be two vectors of .Z2 . We have (x + m) ∗ (y + n) = x ∗ y + m ∗ n.

.

Since .m ∗ n ∈ Z2 , we obtain a quandle operation on the torus .T 2 = S 1 × S 1 . Lets compute the automorphism group .Aut (T 2 ). First, one notices that any function .fA,B on .R2 such that .fA,B (x) = Ax + B with the condition .MA = AM is a quandle homomorphism. Thus if .A ∈ GL2 (R) and .MA = AM, then .fA,B is an automorphism of the quandle .R2 . In fact we claim that the converse is also true. Precisely if f is a quandle automorphism and we consider the function .g(x) = f (x) − f (0). Then .g(0) = 0 and g satisfies the equation g(Mx + (I2 − M)y) = Mg(x) + (I2 − M)g(y).

.

In particular .g(Mx) = Mg(x), and thus g will be of the form .g(x) = λx, where λ ∈ GL2 (R) and .λM = Mλ. Thus .Aut (T 2 ) is the subgroup of the affine group 2 .Aff (R ) of elements of the form .fA,B for which A commute with M and the inner group .I nn(T 2 ) = R2 . Obviously this example can be generalised to an n-torus with .n ≥ 2. .

Now given a topological abelian group .(G, +) and a continuous automorphism .σ of G. The operation .x ∗y = σ (x)+(I d −σ )(y) makes G into a topological quandle called topological Alexander quandle. In general putting different topologies on the same quandle may result in different topological quandles. In [19], the classification problem of topological quandles on some manifolds was investigated. Precisely, topological Alexander quandle structures, up to isomorphism, were fully classified on the real line and the unit circle (see [19] for more details). In [63] Rubinsztein considered the set of homomorphisms .H om(Q(L), X) from the fundamental quandle .Q(L) of a given knot L to a fixed topological quandle X and proved that it is an invariant of the knot. This set inherits a topological structure (equipped with the compact-open topology). Recently a continuous cohomology theory for topological quandles was introduced by the first named author in [34], and compared to the algebraic theories. Extensions of topological quandles were studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the inverse limit was applied to quandles.

1.11 Relation to Khovanov Homology In 1999, Khovanov [53] introduced a bi-graded homology theory .Khi,j (L) for a diagram of a knot or a link L whose Euler characteristic is the Jones polynomial (see also [6]). Khovanov’s invariant can be seen as a “categorification” of the

36

M. Elhamdadi and B. Jones

Jones polynomial [46]. Khovanov’s invariant proved to be a very powerful invariant since it distinguished between knots which were not distinguishable by the Jones polynomial (for example Khovanov’s invariant is different for the knot .942 and its mirror image while Jones invariant is the same for .942 and its mirror image. The same phenomenon is true for the knot .10125 and its mirror image [62]). Recall that a topological quandle is a quandle with a topology such that the quandle operation is continuous (see the precise definition above). For example spheres have quandle operation .x ∗ y = 2 < x, y > y − x where .< x, y > stands for the usual inner product. This operation induces topological quandle structure on the projective spaces. Conjugacy classes in Lie groups are also examples of topological quandles. In [63] Rubinsztein considered the set of homomorphisms .H om(Q(L), X) from the fundamental quandle .Q(L) of a given knot L to a fixed topological quandle X and proved that it is an invariant of the knot. This set inherits a topological structure (equipped with the compact-open topology). It is called the space of colorings .JX (L) of the knot L by the quandle X. As examples Rubinsztein gave in [63]: 1. The invariant space of the figure eight knot .41 is JS 2 (41 ) = S 2 ∪ RP3 ∪ RP3 ,

.

while the “collapsed” (the singly graded homology obtained by collapsing along m = i − j ) Khovanov homology for .41 is given in [45] in page 57,

.

Khm (41 ) = H m (S 2 ){−1} ⊕ H m (RP3 ){−3} ⊕ H m (RP3 ){0}.

.

2. The invariant space of the knot .52 colored by the sphere .S 2 is JS 2 (52 ) = S 2 ∪ RP3 ∪ RP3 ∪ RP3 ,

.

while the collapsed Khovanov homology for .52 is Khm (52 ) = H m (S 2 ){1} ⊕ H m (RP3 ){5} ⊕ H m (RP3 ){7} ⊕ H m (RP3 ){11}.

.

Here .RP3 stands for the real projective space of dimension three. There are more examples of knots with similarities between Rubinstein’s invariant of knots and Khovanov’s invariant (see the paper of Jacobsson and Rubinsztein [45] for more details). This work of Jacobsson and Rubinsztein is convincing evidence that more investigation of topological quandles is needed, such as “extensions” of the first named author results about cocycle invariants in the context of topological quandles.

1 Quandles, Knots, Quandle Rings and Graphs

37

1.12 Quandle Rings and Power Associativity Since quandles are just sets with a binary operations, it is natural to “linearize” them in order to use tools from other areas of mathematics, such as linear algebra. In parallel with group rings, it is natural to investigate quandle rings. For a quandle .(X, >) and a field .k, we consider a nonassociative algebra .k[X]. Precisely, Let .k[X] E be the set of elements that are uniquely expressible in the form . x∈X ax ex , where .x ∈ X and .ax = 0 for almost all x, (that is .ex represent the basis elements of .k[X]). Addition on .k[X] is defined as usual (

E

.

ax ex ) + (

x∈X

E

bx ex ) =

x∈X

E

(ax + bx )ex ,

x∈X

and the multiplication is given by the following operation, where .x, y ∈ X and ax , ay ∈ k,

.

(

E

.

x∈X

ax ex ) · (

E y∈X

by ey ) =

E

ax by ex>y .

x,y∈X

It turns out that quandle rings are associative if and only if the quandle is trivial. Thus a weaker notion of associativity is needed to study quandle rings of non-trivial rings. The notion is called power associativity. In [14], power associativity of dihedral quandles was investigated and the question of determining the conditions under which the quandle ring .R[X] is power associative was raised. In this section we give a complete solution to this question. Precisely, we prove that quandle rings are never power associative when the quandle is non-trivial and the ring has characteristic zero. But first let’s recall the following definition from [4] Definition 1.18 A ring .k in which every element generates an associative subring is called a Power-associative ring. Example 1.21 Any alternative algebra is power associative. Recall that an algebra A is called alternative if .x · (x · y) = (x · x) · y and .x · (y · y) = (x · y) · y, ∀x, y ∈ A, (for more details see [30]). It is well known [4] that a ring .k of characteristic zero is power-associative if and only if (x · x) · x = x · (x · x) and (x · x) · (x · x) = [(x · x) · x] · x, for all x ∈ k.

.

The following theorem shows that quandle rings are never power associative when the quandle is non-trivial and the ring is of characteristic zero. Theorem 1.8 Let .k be a ring of characteristic zero and let .(X, ∗) be a non-trivial quandle. Then the quandle ring .k[X] is not power associative.

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M. Elhamdadi and B. Jones

By analogy with group rings, the augmentation ideal is defined to Ebe the kernel of the surjective ring homomorphism .e : k[X] → k such that .e( x∈X ax ex ) = E a . It will be denoted by . I . It is two-sided ideal of codimension one in x X x∈X .k[X]. By fixing .x0 ∈ X, one sees that the elements .ex − ex0 , (x ∈ X, x /= x0 ) form a basis for .IX . We also have the isomorphism of rings .k[X]/Ix ∼ = k. Lemma 1.14 E E Let X be a finite quandle of order n and let .k be a ring. Then we have ( ni=1 ei ) · nj=2 αj −1 (e1 − ej ) = 0.

.

u

Proof A straightforward computation gives the result.

Notation: In the rest of the paper, it is important to make a distinction between the product in the quandle, denoted by .∗, and the product, denoted by .·, in the quandle ring.

1.13 Ring Automorphism Groups of Quandle Rings In this section we compute the ring automorphism group of .Z2 [Tn ] of the trivial quandles .Tn for some small values of n (Table 1.1, 1.2, 1.3 and 1.4).

1.14 Relevant Graph Theory Concepts In this section we review some basics of graph theory from [41]. A cycle in a graph G is a sequence .v0 , f1 , v1 , f2 , · · · , fm , vm = v0 where .vi are vertices and .fj +1 = {vj , vj +1 } are edges. Table 1.1 We give the following automorphism groups of the trivial quandles up to order 4 X .T2 .T3 .T4 .R4

.Aut (Z2 [X])

.Aut (Z3 [X])

.Z2

.S3

.S4

.((((Z3

× Z2 × Z2 ) x P SL(3, 2) .Z2 × Z2

.(Z3

.(Z2

× Z 3 ) x Q8 ) x Z 3 ) x Z 2 ) × Z3 × Z3 ) x GL(3, 3) .S3 × S3

Table 1.2 We give the following automorphism groups X

.Aut (Z4 [X])

.Aut (Z5 [X])

.T2

.D8

.D10

.T3

.((((((Z2 ×Z2 ×Z2 ×Z2 )xZ2 )xZ2 )xZ3 )xZ4 )xZ2 )

.(Z5

.T4

.(Z4

× Z4 × Z4 ) x (Z2 × ((Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 ) x P SL(3, 2)) .D8 × D8

.(Z5

.R4

× Z5 ) x (SL(2, 5) x Z2 ) × Z5 × Z5 ) x (Z2 × P SL(3, 5)) .D10 × D10

1 Quandles, Knots, Quandle Rings and Graphs

39

Table 1.3 We give the following automorphism groups X

.Aut (Z6 [X])

.Aut (Z7 [X])

.T2

.D12

.D14

.T3

.((((Z3

× Z 3 ) x Q8 ) x Z 3 ) x Z 2 ) × S4

.(Z7

× Z7 ) x (SL(2, 7) x Z2 )

.D14

× D14

.T4 .R4

.Z2

× Z 2 × S3 × S3

Table 1.4 We give the following automorphism groups X

.Aut (Z8 [X])

.Aut (Z9 [X])

.T2

.D16

.D18

.T3

.(((((((Z2

× (((Z2 × Z2 × Q8 ) x Z2 ) x Z2 )) x Z2 ) x Z2 ) x Z 2 ) x Z 2 ) x Z 2 ) x Z 3 ) x Z 2

.((((((Z9

x Z9 ) x Z3 ) x Z3 ) x Z3 ) x Q8 ) x Z 3 ) x Z 2

.T4 .R4

.D16

× D16

.D18

× D18

Definition 1.19 The distance between two vertices u and v in a graph G is the length of the shortest path from u to v. Definition 1.20 The diameter of a graph G is the maximum distance between any two vertices of G. Definition 1.21 A graph G is connected if it is non-empty and any two vertices of G are linked by a path in G. Definition 1.22 A graph G is complete if every pair of vertices are adjacent. We recall the notions of graph homomorphisms and automorphisms as we will need them later in the article. Definition 1.23 Let G and H be two graphs. A map .f : V (G) → V (H ) is called a graph homomorphism if, for all pairs of vertices .u, v ∈ V (G), .uv ∈ E(G) implies that .f (u)f (v) ∈ E(H ). If furthermore f is a bijection then it is called an isomorphism of graphs. The set of all graph isomorphisms of G form a group called the automorphism group of the graph G denoted by .Autgraph (G). It is clear that this group is a subgroup of the symmetry group .Sym(V (G)) of the vertex set .V (G). For example for the complete graph, the group .Autgraph (Kn ) is isomorphic to the symmetric group .Sn . Definition 1.24 Let .G = (V , E) be a graph. An equivalence relation .R on V induces the quotient graph .(V ' , E ' ) where the vertex set .V ' is the quotient set .V /R and the edge set is .E ' := {([x], [y]) |(x, y) ∈ E}.

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M. Elhamdadi and B. Jones

1.15 Graph of the Quandle The most natural graph to consider from a quandle is the graph obtained from the action of the .I nn(X) on the quandle X.Precisely, the graph is given by vertices being elements of a quandle X and an edge from x to y if there is a .z ∈ X such that .x ∗ z = y. We will call this graph the “Quandle graph”. In the following we give three different examples, one being Latin, one being connected non-Latin, and one non-connected with three orbits. Example 1.22 Let .X = Z2 [t]/(t 2 + t + 1) be the connected quandle of order 4. Since it is latin then its quandle graph is complete.

Example 1.23 Let .X = {1, 2, . . . , 6} be the connected quandle of order 6 given by its right multiplications: .

S1 = (3 5)(4 6),

S2 = (3 6)(4 5),

S3 = (1 5)(2 6),

S4 = (1 6)(2 5),

S5 = (1 3)(2 4),

S6 = (1 4)(2 3),

its Quandle graph is

From the graph, we see that X is connected, but not Latin, and .diam(G) = 2. Example 1.24⎤ Consider the following quandle of order 6 with Cayley table ⎡ 111112 ⎢2 2 2 2 2 4⎥ ⎥ ⎢ ⎥ ⎢ ⎢3 3 3 3 3 1⎥ .⎢ ⎥ ⎢4 4 4 4 4 3⎥ ⎥ ⎢ ⎣5 5 5 5 5 5⎦ 666666

1 Quandles, Knots, Quandle Rings and Graphs

41

Remark 1.3 The following observations are straightforward: 1. The graph G of a quandle X is connected if and only if X is a connected quandle, in particular .diam(G) ≥ 2. 2. The graph G of a quandle X is complete if and only if X is Latin, in particular .diam(G) = 1.

1.16 Zero-Divisor Graph from Quandle Rings By analogy with [2, 3, 5] we define a graph .r(k[X]) on the quandle ring .k[X] by declaring that the set of vertices is .k[X] \ {0} and that there is an (undirected) edge between vertices a and b if and only if .a /= b and .ab = 0. Then when .k[X] is an integral domain. the graph .r(k[X]) will be empty. We need then to avoid upquandles since Proposition 3.3 of [13] states that if X is a up-quandle then .k[X] has no zero-divisors. Let recall the definition of up-quandle from [13]. Definition 1.25 A quandle X is called up-quandle (unique product quandle) if given any two non-empty finite subsets M and N of X, there exists an .x ∈ X with a unique representation .x = y ∗ z for some .y ∈ M and .z ∈ N . It is important to note the case when .k is infinite because we get an infinite number of vertices in the zero-divisor graph. To avoid this, we consider the quotient graph of .r(k[X]), defined in Definition 1.24, denoted by .r ' (k[X]). Example 1.25 Consider the commutative quandle .X = Z5 with the operation .x ∗ y = 3(x + y). The zero-divisor graph .r(Z2 [X]) is undirected and given by

It is useful to note that the top ten vertices are elements of the form .ei + ej with i /= j , the center vertex is .e1 + e2 + e3 + e4 + e5 , and the bottom five vertices are elements of the form .ei + ej + ek with .i, j, k pairwise distinct.

.

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M. Elhamdadi and B. Jones

Proposition 1.8 of order n, then every .E u ∈ Z2 [X] E If .(X, >) is a finite Latin quandle n n suchE that .u = 2k e for some . k ∈ N, . k ≤ L J, is a zero divisor and . u· i l l=1 i=1 ei = 2 n 0 = i=1 ei · u. E2k n Proof En Since .u = Enl=1 eil for some .k ∈ N, .k ≤ L 2 J and .(X, ∗) is Latin, we have .u · u i=1 ei = 2k i=1 ei = 0 mod 2. Lemma 1.15 If .(X, ∗) is a finite commutative latin quandle of order n, then for ei + ej , ek + el ∈ Z2 [X], .(ei + ej ) · (ek + el ) /= 0 for all .i /= j , .k /= l and therefore .diam(r(Z2 [X])) > 1. .

Proof Suppose .(ei + ej ) · (ek + el ) = 0 for some .i /= j, k /= l, which implies (ei ∗ ek ) + (ei ∗ el ) + (ej ∗ ek ) + (ej ∗ el ) = 0. Since X is Latin, .ei ∗ ek = ej ∗ el and .ei ∗ el = ej ∗ ek . Now, consider

.

(ei ∗ ej ) ∗ (ek ∗ el ) = [ei ∗ (ek ∗ el )] ∗ [ej ∗ (ek ∗ el )] = [(ei ∗ ek ) ∗ (ei ∗ el )] ∗ [(ej ∗ ek ) ∗ (ej ∗ el )] .

= [(ei ∗ ek ) ∗ (ei ∗ el )] ∗ [(ei ∗ el ) ∗ (ei ∗ ek )]

(1.26)

= [ei ∗ (ek ∗ el )] ∗ [ei ∗ (ek ∗ el )] = ei ∗ (ek ∗ el ), which gives .ei ∗ej = ei , but X is Latin and so .(ei +ej )·(ek +el ) /= 0. By Proposition 4.3, .ei + ej and .ek + el are zero divisors in .Z2 [X]. Since .(ei + ej ) · (ek + el ) /= 0, there is no edge between them and thus .diam(r(Z2 [X])) > 1. u Theorem 1.9 Let .(X, ∗) be a finite commutative Latin quandle. The zero-divisor graph .r(Z2 [X]) is connected and .2 ≤ diam(r(Z2 [X])) ≤ 4. Proof Let .(X, ∗) be a finite commutative Latin quandle and .u, v be vertices in r(Z2 [X]). Since .u, v ∈ Z2 [X], we have three cases. E even number of .eiE ’s. By Proposition 4.3, .u · ( ni=1 ei ) = Case E 1 If .u, v are sum of E 0 = ( ni=1 ei ) · u and .v · ( Eni=1 ei ) = 0 = ( ni=1 ei ) · v. Thus, we have a path from u to v that passes through . ni=1 ei .

.

Case 2 If u is sum ofEeven and v is sum of odd number of .ei ’s. We know there is an edge between u and . ni=1 ei . Now, since v is a zero-divisor, .∃w ∈ Z2 [X] \ {0} such that .v · w = 0 = w · v. If w is a sum of an odd number of .ei ’s, then .v · w is a sum of an odd number of .ei ’s which cannot equal zero in .Z2 [X]. This gives Ethat w must be a sum of an even number of .ei ’s, which E we know has a edge with . ni=1 ei .Thus, there is a path from u to v such that .u, f1 , ni−1 ei , f2 , w = v. Case 3 If u and v both sum of odd number of .ei ’s. There exists .w, z ∈ Z2 [X] \ {0} that are sums of an even number of .ei ’s such that .u·w = w·u = 0 and .v·z = E z·v = 0. n If .w = z, we’re done. Otherwise, since w and z each have an edge with . i=1 ei , En we have a path from u to v such that .u, f1 , w, f2 , i−1 ei , f3 , z = v.

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We have thus also showed that .diam(r(Z2 [X])) ≤ 4, and also by Lemma 4.3, we have .2 ≤ diam(r(Z2 [X])) ≤ 4. u In the following three examples, we give Zero-divisor quotient graphs over .Z[X] and Zero-divisor graphs over .Z2 [X] for the quandles of order three. ⎤ ⎡ 111 Example 1.26 Let X be the quandle with Cayley table .⎣ 2 2 2 ⎦ 333 The following graph is .r ' (Z[X]), with zero divisors .u = αe1 + βe2 + γ e3 and .v = δ(−e1 + e2 ) + λ(−e1 + e3 ).

The following graph is .r(Z2 [X]), where the top vertex is .e1 +e2 +e3 , the middle row of vertices are .e1 + e2 , .e1 + e3 , and .e2 + e3 , and the bottom row of vertices are .e1 , .e2 , .e3 .

⎤ 112 Example 1.27 Let X be the quandle with Cayley table .⎣ 2 2 1 ⎦ 333 The following graph is .r ' (Z[X]), with zero divisors .u = α(−e1 + e2 ), .v = β(−e1 + e2 ) + γ (e1 + e3 ), .w = ξ e1 + ζ e2 + ηe3 , .x = λ(e1 + e2 ) + μe3 , .y = σ (−e1 + e2 ) + τ (−e1 + e3 ). ⎡

The following graph is .r(Z2 [X]), where the top vertex is .e1 +e2 +e3 , the middle row of vertices are .e1 + e2 , .e1 + e3 , and .e2 + e3 , and the bottom row of vertices are .e1 , .e2 , .e3 .

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⎤ 132 Example 1.28 Let X be the quandle with Cayley table .⎣ 3 2 1 ⎦ 213 The following graph is .r ' (Z[X]), with zero divisors .u = α(e1 + e2 + e3 ) and .v = β(−e1 + e2 ) + γ (−e1 + e3 ). ⎡

The following graph is .r(Z2 [X]), where the top vertex is .e1 + e2 + e3 and the bottom vertices are .e1 + e2 , .e1 + e3 , and .e2 + e3 .

1.17 Automorphisms of Zero-Divisor Graphs of Quandle Rings In this section we consider zero-divisor graph automorphisms of quandle rings and their relations with automorphisms of quandle rings and automorphisms of quandles. Proposition 1.9 If X is a finite latin quandle, then .Autring (Z[X]) Autquandle (X).

∼ =

Proof Since X is latin, conjecture 3.10 in [32] states that .Z[X] has only trivial idempotents. If .φ is a ring automorphism of .Z[X], then .φ(ei ) = ej for some j , which proves the assertion. u Example 1.29 Consider the quandle X of six elements given in Example 1.23. In [28] this quandle is denoted by .Q72 and its quandle automorphism group is computed as .Aut (X) ∼ = S4 . Note that, if .φ is a ring automorphism of .Z[Q72 ], then each .φ(ei ) is an idempotent. Hence, we have .φ(ei ) = αe1 + (1 − α)e2 or .φ(ei ) = βe3 + (1 − β)e4 or .φ(ei ) = γ e5 + (1 − γ )e6 . The following three examples give the automorphism groups of the quandles of order three, their ring automorphism groups and also their graph automorphism groups. Example 1.30 Consider the trivial quandle .T3 on three elements. Since every permutation of the set .T3 is a quandle homomorphism then .Aut (T3 ) ∼ = S3 . The ring automorphism is given by Table 1.1 as .Aut (Z2 [T3 ]) ∼ = S4 and the graph automorphism group is .Z2 . This is due to the zero-divisor quotient graph having

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only two vertices with one directed edge, and so we have the identity automorphism, and an automorphism sending u to v and v to u. Example 1.31 Now consider the three elements quandle .X = {1, 2, 3} with orbit decomposition .{1, 2} u {3}. Since .{1, 2} is trivial subquandle we get that .Aut (Z[X]) ∼ = Z2 ∼ = Aut (Z[X]). The graph automorphism group is the trivial group. Example 1.32 Consider the three element dihedral quandle .R3 . It’s straightforward to see that .Aut (R3 ) ∼ = S3 . Since it this quandle is latin we have .Aut (T3 ) ∼ = Aut (Z[R3 ]). For a similar reasoning as in Example 1.31, the graph automorphism group is .Z2 . Acknowledgments The authors would like to thank the organizers of CaCNAS:FA 2021 conference where stimulating ideas of this article started. The authors also would like to thank Professor Dmytro Savchuk for fruitful discussions and help with GAP. Mohamed Elhamdadi was partially supported by Simons Foundation collaboration grant 712462.

References 1. Abchir, H., Elhamdadi, M. and Lamsifer, S., On the minimum number of Fox colorings of knots, Grad. J. Math., vol 5, no 2, 2020. 2. Abdollahi, A. and Taheri, Z. Zero divisors and units with small supports in group algebras of torsion-free groups, Comm. Algebra, 46, no. 2, 2018, 887–925. volume=46, 3. Akbari, S.; Maimani, H. R., and Yassemi, S. When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra, 270, no 1, 2003, 169–180. 4. Albert, A. A., Power-associative rings, Trans. Amer. Math. Soc., vol 64, 1948, 552–593. 5. Anderson, David F., and Livingston, Philip S., The zero-divisor graph of a commutative ring, J. Algebra, 217, no 2, 1999, 434–447. 6. Bar-Natan D., On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337–370. 7. Bataineh K., Elhamdadi, M., and Generating sets of Reidemeister moves of oriented singular links and quandles, J. Knot Theory Ramifications, vol 27, no 14, 2018, 1850064, 15. 8. Belousov, V. D., The structure of distributive quasigroups, (in Russian) Mat. Sb. (N.S.) 50 (92) 1960, pp 267–298. 9. Burstin, C. and Mayer, W., Distributive Gruppen, J. Reine Angew. Math. 160 (1929) 111–130. 10. Bol G., Gewebe und gruppen, Math. Ann. 114 (1937), no. 1, 414–431. 11. . Brown, K. S., Cohomology of groups, Graduate Texts in Mathematics, vol 87, SpringerVerlag, 1994. 12. Bruck, H., A survey of binary systems, Ergeb. Math. Grenzgeb. Neue Folge, Heft 20. Reihe: Gruppentheorie, Springer-Verlag, Berlin, 1958. 13. Bardakov, V.G., Passi, I.B.S. and Singh, M. Zero-divisors and idempotents in quandle rings, Osaka Journal of Math. arXiv:2001.06843, 2020. 14. Bardakov, V.G., Passi, I.B.S. and Singh, M. Quandle rings, J. Algebra Appl., vol 18, no 8, 2019, 1950157, 23. 15. Carter, J. S., Crans, A., Elhamdadi, M. and Saito, M. Cohomology of the adjoint of Hopf algebras, J. Gen. Lie Theory Appl., vol 2, no 1, 2008, 19–34. 16. Carter, J. S., Crans, A., Elhamdadi, M. and Saito, M. Cohomology of categorical selfdistributivity, J. Homotopy Relat. Struct., vol 3, no 1, 2008, 13–63.

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Chapter 2

New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras and Related Algebraic Structures Guilai Liu and Chengming Bai

2.1 Introduction This paper aims to introduce and interpret new splittings of operations of Poisson algebras and transposed Poisson algebras in terms of their representations, giving various related algebraic structures.

2.1.1 Classical Splitting of Operations of Lie Algebras and Associative Algebras There are many algebraic structures having a property of “splitting operations”, that is, expressing each product of an algebraic structure as the sum or the (anti)commutator of the sum of a string of operations. The typical examples are pre-Lie algebras and dendriform algebras which illustrate the spitting of operations of Lie algebras and associative algebras respectively “in a coherent way”. Definition 2.1 A pre-Lie algebra is a pair .(A, ◦), such that A is a vector space, and .◦ : A ⊗ A → A is a bilinear operation satisfying (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z), ∀x, y, z ∈ A.

.

(2.1)

G. Liu · C. Bai (O) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, China e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_2

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Pre-Lie algebras, also called left-symmetric algebras, originated from diverse areas of study, including convex homogeneous cones [38], affine manifolds and affine structures on Lie groups [24], and deformation of associative algebras [21]. They also appear in many fields in mathematics and mathematical physics, such as symplectic and Kähler structures on Lie groups [12, 29], vertex algebras [6], quantum field theory [13] and operads [11], see [3, 9] and the references therein. Pre-Lie algebras are Lie-admissible algebras, that is, the commutator of a preLie algebra is a Lie algebra. Hence the operation of a pre-Lie algebra expresses a kind of splitting the Lie bracket of a Lie algebra. Moreover, the left multiplication operators of a pre-Lie algebra give a representation of the commutator Lie algebra, characterizing the so-called “coherent way”. Definition 2.2 A dendriform algebra is a triple .(A, >, ≺), such that A is a vector space, and .>, ≺: A ⊗ A → A are bilinear operations satisfying x > (y > z) = (x·y) > z, (x ≺ y) ≺ z = x ≺ (y·z), (x > y) ≺ z = x > (y ≺ z), (2.2)

.

where .x · y = x > y + x ≺ y, for all .x, y, z ∈ A. In particular, for a dendriform algebra .(A, >, ≺), if x > y = y ≺ x, ∀x, y ∈ A,

.

(2.3)

then .(A, * :=>) is called a Zinbiel algebra. The notion of dendriform algebras was introduced by Loday in the study of algebraic K-theory [31], and they appear in a lot of fields in mathematics and physics, such as arithmetic [32], combinatorics [34], Hopf algebras [10, 22, 23, 35, 37], homology [18, 19], operads [33], Lie and Leibniz algebras [19] and quantum field theory [17]. The sum of two bilinear operations in a dendriform algebra .(A, >, ≺) gives an associative algebra .(A, ·). Hence dendriform algebras have a property of splitting the associativity, that is, expressing the product of an associative algebra as the sum of two bilinear operations. Such a decomposition or splitting of the product of an associative algebra is coherent in the sense that the left and right multiplication operators of a dendriform algebra give a representation of the sum associative algebra. Note that in this sense, pre-Lie algebras and dendriform algebras play similar roles in the splitting of operations of Lie algebras and associative algebras respectively. Furthermore, there is a general theory on the splitting of operations in the above sense (the so-called coherent way) in terms of operads in [4]. The notions of successors and trisuccessors were introduced to interpret the splitting of operations into the sum of two or three pieces respectively. In this sense, the operad of pre-Lie algebras is the successor of the operad of Lie algebras and the operad of dendriform algebras is the successor of the operad of associative algebras. To avoid the possible confusion, we refer to this kind of splitting as the classical splitting, that is, the operations of pre-Lie algebras and dendriform algebras give the classical splitting of operations of Lie algebras and associative algebras respectively.

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2.1.2 Second Splitting of Operations of Lie Algebras and Associative Algebras There is another approach of splitting operations introduced as the “anti-structures” of the successors’ algebras. The first example is anti-pre-Lie algebras introduced in [30], giving another splitting of operations of Lie algebras. Definition 2.3 An anti-pre-Lie algebra is a pair .(A, ◦), such that A is a vector space, and .◦ : A ⊗ A → A is a bilinear operation satisfying x ◦ (y ◦ z) − y ◦ (x ◦ z) = [y, x] ◦ z,

(2.4)

[x, y] ◦ z + [y, z] ◦ x + [z, x] ◦ y = 0,

(2.5)

.

.

for all .x, y, z ∈ A, where the operation .[−, −] : A ⊗ A → A is defined by [x, y] = x ◦ y − y ◦ x, ∀x, y ∈ A.

.

(2.6)

Anti-pre-Lie algebras are characterized as the Lie-admissible algebras whose negative left multiplication operators give representations of their commutator Lie algebras, justifying the notion due to the comparison with pre-Lie algebras. Hence in this sense, the operations of anti-pre-Lie algebras give a splitting of operations of Lie algebras as a kind of “anti-structures” of pre-Lie algebras. Similarly, the notion of anti-dendriform algebras was introduced in [20] as the anti-structures of dendriform algebras, whose operations give a splitting of operations of associative algebras which is different from the classical splitting. Definition 2.4 An anti-dendriform algebra is a triple .(A, >, ≺), such that A is a vector space, and .>, ≺: A ⊗ A → A are bilinear operations satisfying x > (y > z) = −(x · y) > z = −x ≺ (y · z) = (x ≺ y) ≺ z,

.

(x > y) ≺ z = x > (y ≺ z),

.

(2.7) (2.8)

where .x ·y = x > y +x ≺ y, for all .x, y, z ∈ A. In particular, for an anti-dendriform algebra .(A, >, ≺), if x > y = y ≺ x, ∀x, y ∈ A,

.

(2.9)

then .(A, * :=>) is called an anti-Zinbiel algebra. Anti-dendriform algebras keep the property of splitting associativity, that is, the sum of the two bilinear operations in an anti-dendriform algebra .(A, >, ≺) gives an associative algebra .(A, ·). However it is the negative left and right multiplication operators of an anti-dendriform algebra that compose a representation of the sum

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associative algebra, instead of the left and right multiplication operators doing so for a dendriform algebra. To avoid the possible confusion, we refer to this kind of splitting as the second splitting, that is, the operations of anti-pre-Lie algebras and anti-dendriform algebras give the second splitting of operations of Lie algebras and associative algebras respectively.

2.1.3 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras Poisson algebras arose in the study of Poisson geometry [8, 27, 39], and are closely related to a lot of topics in mathematics and physics. Definition 2.5 A Poisson algebra is a triple .(A, ·, [−, −]), where .(A, ·) is a commutative associative algebra, .(A, [−, −]) is a Lie algebra, and they are compatible in the sense of the Leibniz rule: [z, x · y] = [z, x] · y + x · [z, y], ∀x, y, z ∈ A.

.

(2.10)

The notion of transposed Poisson algebras was introduced in [5] as the dual notion of Poisson algebras, which exchanges the roles of the two bilinear operations in the Leibniz rule defining Poisson algebras. They closely relate to a lot of other algebraic structures such as Novikov-Poisson algebras [40] and 3-Lie algebras [16] and further studies are given in [7, 15, 26, 41]. Definition 2.6 A transposed Poisson algebra is a triple .(A, ·, [−, −]), where (A, ·) is a commutative associative algebra, .(A, [−, −]) is a Lie algebra, and they are compatible in the sense of the transposed Leibniz rule:

.

2z · [x, y] = [z · x, y] + [x, z · y], ∀x, y, z ∈ A.

.

(2.11)

The notion of pre-Poisson algebras was introduced in [1] to give the classical splitting of operations of Poisson algebras, that is, they are the algebraic structures that combine pre-Lie algebras and Zinbiel algebras satisfying certain compatible conditions. In this paper, we extend this classical splitting of operations of Poisson algebras to a wide extent, by introducing new splittings of operations of both Poisson algebras and transposed Poisson algebras. Note that both Poisson algebras and transposed Poisson algebras have two bilinear operations, and hence variations of splitting operations become possible. In fact, due to the existence of two bilinear operations for Poisson algebras and transposed Poisson algebras, we consider the new splittings as “mixed splittings” in the sense that the commutative associative products and Lie brackets are splitted

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in different manners respectively. More explicitly, the commutative associative products and Lie brackets in Poisson algebras and transposed Poisson algebras are splitted interlacedly in three manners: the classical splitting, the second splitting and the un-splitting, giving variations of splitting operations. Since the algebraic structures corresponding to the classical splitting and the second splitting are characterized in terms of representations, we also characterize the algebraic structures corresponding to the new splittings of operations of Poisson algebras and transposed Poisson algebras in terms of representations. Note that a representation of a Poisson algebra has a natural dual representation. Hence the characterization of algebraic structures corresponding to the new splittings of operations of Poisson algebras in terms of representations of Poisson algebras on the spaces themselves is the same as that in terms of representations of Poisson algebras on the dual spaces. However, the situation is different for transposed Poisson algebras, that is, one should consider the characterization of the algebraic structures corresponding to the new splittings of operations of transposed Poisson algebras in terms of representations of transposed Poisson algebras on the spaces themselves and representations of transposed Poisson algebras on the dual spaces respectively. Such a phenomenon is partly due to the fact that there might not exist automatically dual representations for representations of transposed Poisson algebras (see Proposition 2.35), exhibiting an obvious difference between Poisson algebras and transposed Poisson algebras. Therefore there are 8 algebraic structures interpreted in terms of representations of Poisson algebras illustrating the mixed splittings of operations of Poisson algebras respectively, including the known pre-Poisson algebras. For illustrating the mixed splittings of operations of transposed Poisson algebras, there are 8 algebraic structures interpreted in terms of representations of transposed Poisson algebras on the spaces themselves and another 8 algebraic structures interpreted in terms of representations of transposed Poisson algebras on the dual spaces. Note that some of them also correspond to the Poisson algebras and transposed Poisson algebras with nondegenerate bilinear forms satisfying certain conditions respectively.

2.1.4 Organization of the Paper This paper is organized as follows. In Sect. 2.2, we recall some facts on pre-Lie algebras and Zinbiel algebras as well as anti-pre-Lie algebras and anti-Zinbiel algebras, as the algebraic structures corresponding to the splittings of operations of Lie algebras and commutative associative algebras, which are interpreted in terms of representations of Lie algebras and commutative associative algebras respectively. In Sect. 2.3, we introduce 8 algebraic structures respectively corresponding to the mixed splittings of operations of Poisson algebras interlacedly in three manners, in terms of representations of Poisson algebras. Some of them are closely related to the Poisson algebras with nondegenerate bilinear forms satisfying certain conditions.

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In Sect. 2.4, we introduce 8 algebraic structures respectively corresponding to the mixed splittings of operations of transposed Poisson algebras interlacedly in three manners, in terms of the representations of transposed Poisson algebras on the spaces themselves. In Sect. 2.5, we introduce 8 algebraic structures respectively corresponding to the mixed splittings of operations of transposed Poisson algebras interlacedly in three manners, in terms of the representations of transposed Poisson algebras on the dual spaces. Some of them are closely related to the transposed Poisson algebras with nondegenerate bilinear forms satisfying certain conditions. Throughout this paper, unless otherwise specified, all the vector spaces and algebras are finite-dimensional over a field of characteristic zero, although many results and notions remain valid in the infinite-dimensional case.

2.2 Splittings of Operations of Lie Algebras and Commutative Associative Algebras and Related Algebraic Structures We recall some facts on pre-Lie algebras and anti-pre-Lie algebras exhibiting the classical splitting and the second splitting of operations of Lie algebras respectively, which are interpreted in terms of representations of Lie algebras. Similarly, we do so for commutative associative algebras by recalling some facts on Zinbiel algebras and anti-Zinbiel algebras.

2.2.1 Pre-Lie Algebras and Anti-Pre-Lie Algebras Recall some basic facts on representations of Lie algebras. A representation of a Lie algebra .(g, [−, −]) is a pair .(ρ, V ), such that V is a vector space and .ρ : g → gl(V ) is a Lie algebra homomorphism for the natural Lie algebra structure on .gl(V ) = End(V ). In particular, the linear map .ad : g → gl(g) defined by .ad(x)(y) = [x, y] for all .x, y ∈ g, gives a representation .(ad, g), called the adjoint representation of .(g, [−, −]). For a vector space V and a linear map .ρ : g → gl(V ), the pair .(ρ, V ) is a representation of a Lie algebra .(g, [−, −]) if and only if .g ⊕ V is a (semi-direct product) Lie algebra by defining the multiplication on .g ⊕ V by [(x, u), (y, v)] = ([x, y], ρ(x)v − ρ(y)u), ∀x, y ∈ g, u, v ∈ V .

.

(2.12)

We denote it by .g Xρ V . Let A and V be vector spaces. For a linear map .ρ : A → End(V ), we set ∗ ∗ .ρ : A → End(V ) by

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

= −, ∀x ∈ A, u ∈ V , v ∗ ∈ V ∗ .

.

55

(2.13)

Here .< , > is the usual pairing between V and .V ∗ . If .(ρ, V ) is a representation of a Lie algebra .(g, [−, −]), then .(ρ ∗ , V ∗ ) is also a representation of .(g, [−, −]). In particular, .(ad∗ , g∗ ) is a representation of .(g, [−, −]). Recall that a bilinear form .B on a Lie algebra .(g, [−, −]) is called invariant if B([x, y], z) = B(x, [y, z]), ∀x, y, z ∈ g.

.

(2.14)

Suppose that .(g, [−, −]) is a Lie algebra. Then the natural nondegenerate symmetric bilinear form .Bd on .g ⊕ g∗ defined by Bd ((x, a ∗ ), (y, b∗ )) = + , ∀x, y ∈ g, a ∗ , b∗ ∈ g∗

.

(2.15)

is invariant on the Lie algebra .g Xad∗ g∗ . For a vector space A with a bilinear operation .◦ : A ⊗ A → A, .(A, ◦) is called a Lie-admissible algebra if the bilinear operation .[−, −] : A ⊗ A → A defined by [x, y] = x ◦ y − y ◦ x, ∀x, y ∈ A

.

(2.16)

equips A with a Lie algebra structure. In this case, .(A, [−, −]) is called the subadjacent Lie algebra of .(A, ◦). For a vector space A together with a bilinear operation .◦ : A ⊗ A → A, denote a linear map .L◦ : A → End(A) by L◦ (x)y := x ◦ y, ∀x, y ∈ A.

.

(2.17)

There is the following characterization of pre-Lie algebras. Proposition 2.1 ([3, 9]) Let A be a vector space together with a bilinear operation ◦ : A ⊗ A → A. Then the following conditions are equivalent:

.

1. .(A, ◦) is a pre-Lie algebra. 2. .(A, ◦) is a Lie-admissible algebra such that .(L◦ , A) is a representation of the sub-adjacent Lie algebra .(A, [−, −]). 3. There is a Lie algebra structure on .A ⊕ A defined by [(x, a), (y, b)] = (x ◦ y − y ◦ x, x ◦ b − y ◦ a), ∀x, y, a, b ∈ A.

.

(2.18)

If a Lie algebra .(g, [−, −]) is the sub-adjacent Lie algebra of a pre-Lie algebra (g, ◦), then .(g, ◦) is called a compatible pre-Lie algebra of .(g, [−, −]). Let .B be a nondegenerate skew-symmetric bilinear form on a Lie algebra .(g, [−, −]). If .B satisfies .

B([x, y], z) + B([y, z], x) + B([z, x], y) = 0, ∀x, y, z ∈ g,

.

(2.19)

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then we say .B is a symplectic form [12, 28] on .(g, [−, −]), and we call the triple (g, [−, −], B) a symplectic Lie algebra.

.

Proposition 2.2 ([12, 25]) Let .(g, [−, −], B) be a symplectic Lie algebra. Then there exists a compatible pre-Lie algebra .(g, ◦) of .(g, [−, −]) defined by B(x ◦ y, z) = −B(y, [x, z]), ∀x, y, z ∈ g.

.

(2.20)

Conversely, let .(A, ◦) be a pre-Lie algebra and .(A, [−, −]) be the sub-adjacent Lie algebra. Then the natural nondegenerate skew-symmetric bilinear form .Bp defined by Bp ((x, a ∗ ), (y, b∗ )) = − , ∀x, y ∈ A, a ∗ , b∗ ∈ A∗

.

(2.21)

is a symplectic form on the Lie algebra .A XL∗◦ A∗ . Similarly, anti-pre-Lie algebras are also characterized in terms of representations of the sub-adjacent Lie algebras. Proposition 2.3 ([30]) Let A be a vector space together with a bilinear operation ◦ : A ⊗ A → A. Then the following conditions are equivalent:

.

1. .(A, ◦) is an anti-pre-Lie algebra. 2. .(A, ◦) is a Lie-admissible algebra such that .(−L◦ , A) is a representation of the sub-adjacent Lie algebra .(A, [−, −]). 3. There is a Lie algebra structure on .A ⊕ A defined by [(x, a), (y, b)] = (x ◦ y − y ◦ x, y ◦ a − x ◦ b), ∀x, y, a, b ∈ A.

.

(2.22)

Similarly, if a Lie algebra .(g, [−, −]) is the sub-adjacent Lie algebra of an antipre-Lie algebra .(g, ◦), then .(g, ◦) is called a compatible anti-pre-Lie algebra of .(g, [−, −]). Recall that a symmetric bilinear form .B on a Lie algebra .(g, [−, −]) is called a commutative 2-cocycle [14] if Eq. (2.19) holds, which in the nondegenerate case is the “symmetric” version of a symplectic form on the Lie algebra .(g, [−, −]). Proposition 2.4 ([30]) Let .B be a nondegenerate commutative 2-cocycle on a Lie algebra .(g, [−, −]). Then there exists a compatible anti-pre-Lie algebra .(g, ◦) of .(g, [−, −]) defined by B(x ◦ y, z) = B(y, [x, z]), ∀x, y, z ∈ g.

.

(2.23)

Conversely, let .(A, ◦) be an anti-pre-Lie algebra and .(A, [−, −]) be the subadjacent Lie algebra. Then the natural nondegenerate symmetric bilinear form .Bd defined by Eq. (2.15) is a commutative 2-cocycle on the Lie algebra .A X−L∗◦ A∗ .

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

57

2.2.2 Zinbiel Algebras and Anti-Zinbiel Algebras Recall some basic facts on representations of commutative associative algebras. A representation of a commutative associative algebra .(A, ·) is a pair .(μ, V ), where V is a vector space and .μ : A → End(V ) is a linear map satisfying μ(x · y) = μ(x)μ(y), ∀x, y ∈ A.

.

(2.24)

For a commutative associative algebra .(A, ·), .(L· , A) is a representation of .(A, ·), called the adjoint representation of .(A, ·). In fact, .(μ, V ) is a representation of a commutative associative algebra .(A, ·) if and only if the direct sum .A ⊕ V of vector spaces is a (semi-direct product) commutative associative algebra by defining the multiplication on .A ⊕ V by (x, u) · (y, v) = (x · y, μ(x)v + μ(y)u), ∀x, y ∈ A, u, v ∈ V .

.

(2.25)

We denote it by .A Xμ V . If .(μ, V ) is a representation of a commutative associative algebra .(A, ·), then .(−μ∗ , V ∗ ) is also a representation of .(A, ·). In particular, .(−L∗· , A∗ ) is a representation of .(A, ·). Recall that a bilinear form .B on a (commutative) associative algebra .(A, ·) is called invariant if B(x · y, z) = B(x, y · z), ∀x, y, z ∈ A.

.

(2.26)

Let .(A, ·) be a commutative associative algebra. Then the natural nondegenerate symmetric bilinear form .Bd defined by Eq. (2.15) is invariant on the commutative associative algebra .A X−L∗· A∗ . For a vector space A together with a bilinear operation .* : A ⊗ A → A, if the bilinear operation .· : A ⊗ A → A defined by x · y = x * y + y * x, ∀x, y ∈ A

.

(2.27)

equips A with a commutative associative algebra structure, then we say .(A, ·) is the sub-adjacent commutative associative algebra of .(A, *). Zinbiel algebras and anti-Zinbiel algebras play a similar role for commutative associative algebras as pre-Lie algebras and anti-pre-Lie algebras do for Lie algebras respectively. The notion of Zinbiel algebras is rewritten in a more straightforward manner as follows. Definition 2.7 ([31]) Let A be a vector space together with a bilinear operation .* : A ⊗ A → A. .(A, *) is called a Zinbiel algebra if x * (y * z) = (x * y) * z + (y * x) * z, ∀x, y, z ∈ A.

.

(2.28)

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Proposition 2.5 ([2]) Let A be a vector space together with a bilinear operation .* : A ⊗ A → A. Then the following conditions are equivalent: 1. .(A, *) is a Zinbiel algebra. 2. .(A, ·) with the bilinear operation .· defined by Eq. (2.27) is a commutative associative algebra and .(L* , A) is a representation of .(A, ·). 3. There is a commutative associative algebra structure on .A ⊕ A defined by (x, a) · (y, b) = (x * y + y * x, x * b + y * a), ∀x, y, a, b ∈ A.

.

(2.29)

If a commutative associative algebra .(A, ·) is the sub-adjacent commutative associative algebra of a Zinbiel algebra .(A, *), then .(A, *) is called a compatible Zinbiel algebra of .(A, ·). Recall that a skew-symmetric bilinear form .B on a (commutative) associative algebra is called a Connes cocycle [2] if B(x · y, z) + B(y · z, x) + B(z · x, y) = 0, ∀x, y, z ∈ A.

.

(2.30)

Proposition 2.6 ([2]) Let .B be a nondegenerate Connes cocycle on a commutative associative algebra .(A, ·). Then there is a compatible Zinbiel algebra .(A, *) of .(A, ·) defined by B(x * y, z) = B(y, x · z), ∀x, y, z ∈ A.

.

(2.31)

Conversely, let .(A, *) be a Zinbiel algebra and .(A, ·) be the sub-adjacent commutative associative algebra. Then the natural nondegenerate skew-symmetric bilinear form .Bp defined by Eq. (2.21) is a Connes cocycle on the commutative associative algebra .A X−L∗* A∗ . Similarly, the notion of anti-Zinbiel algebras is rewritten in a more straightforward manner as follows. Definition 2.8 Let A be a vector space together with a bilinear operation * : A ⊗ A → A. .(A, *) is called an anti-Zinbiel algebra if

.

x * (y * z) = −(x * y + y * x) * z = x * (z * y), ∀x, y, z ∈ A.

.

(2.32)

Proposition 2.7 ([20]) Let A be a vector space together with a bilinear operation * : A ⊗ A → A. Then the following conditions are equivalent:

.

1. .(A, *) is an anti-Zinbiel algebra. 2. .(A, ·) with the bilinear operation .· defined by Eq. (2.27) is a commutative associative algebra and .(−L* , A) is a representation of .(A, ·). 3. There is a commutative associative algebra structure on .A ⊕ A defined by (x, a) · (y, b) = (x * y + y * x, −x * b − y * a), ∀x, y, a, b ∈ A.

.

(2.33)

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

59

Similarly, if a commutative associative algebra .(A, ·) is the sub-adjacent commutative associative algebra of an anti-Zinbiel algebra .(A, *), then .(A, *) is called a compatible anti-Zinbiel algebra of .(A, ·). A symmetric bilinear form .B on a (commutative) associative algebra .(A, ·) is called a commutative Connes cocycle [20] if Eq. (2.30) holds. Proposition 2.8 ([20]) Let .B be a nondegenerate commutative Connes cocycle on a commutative associative algebra .(A, ·). Then there is a compatible anti-Zinbiel algebra .(A, *) of .(A, ·) defined by B(x * y, z) = −B(y, x · z), ∀x, y, z ∈ A.

.

(2.34)

Conversely, let .(A, *) be an anti-Zinbiel algebra and .(A, ·) be the sub-adjacent commutative associative algebra. Then the natural nondegenerate symmetric bilinear form .Bd defined by Eq. (2.15) is a commutative Connes cocycle on the commutative associative algebra .A XL∗* A∗ .

2.3 Mixed Splittings of Operations of Poisson Algebras and Related Algebraic Structures At first we recall some facts on representations of Poisson algebras. Then we introduce 8 algebraic structures respectively corresponding to the mixed splitting of the commutative associative products and Lie brackets of Poisson algebras interlacedly in three manners: the classical splitting, the second splitting and the unsplitting, in terms of representations of Poisson algebras. Finally the relationships between Poisson algebras with nondegenerate bilinear forms satisfying certain conditions and some algebraic structures are given. Definition 2.9 A representation of a Poisson algebra .(A, ·, [−, −]) is a triple (μ, ρ, V ), such that .(μ, V ) is a representation of the commutative associative algebra .(A, ·), .(ρ, V ) is a representation of the Lie algebra .(A, [−, −]), and the following compatible conditions hold:

.

ρ(x · y) = μ(x)ρ(y) + μ(y)ρ(x),

(2.35)

μ([x, y]) = ρ(x)μ(y) − μ(y)ρ(x),

(2.36)

.

.

for all .x, y ∈ A. Let .(A, ·, [−, −]) be a Poisson algebra. Then .(L· , ad, A) is a representation of .(A, ·, [−, −]), called the adjoint representation of .(A, ·, [−, −]). Moreover, .(μ, ρ, V ) is a representation of a Poisson algebra .(A, ·, [−, −]) if and only if the direct sum .A ⊕ V of vector spaces is a (semi-direct product) Poisson algebra by

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defining the multiplications on .A ⊕ V by Eqs. (2.25) and (2.12) respectively. We denote it by .A Xμ,ρ V . Proposition 2.9 ([36]) Let .(A, ·, [−, −]) be a Poisson algebra. If .(μ, ρ, V ) is a representation of .(A, ·, [−, −]), then .(−μ∗ , ρ ∗ , V ∗ ) is also a representation of .(A, ·, [−, −]). Hence we get the following conclusion. Corollary 2.1 Let .(A, ·, [−, −]) be a Poisson algebra. Then .(−L∗· , ad∗ , A∗ ) is a representation of .(A, ·, [−, −]), and the natural nondegenerate symmetric bilinear form .Bd defined by Eq. (2.15) on the resulting Poisson algebra .A X−L∗· ,ad∗ A∗ is invariant on both the commutative associative algebra .A X−L∗· A∗ and the Lie algebra .A Xad∗ A∗ . Next we introduce 8 algebraic structures corresponding to the mixed splitting of the commutative associative products and Lie brackets of Poisson algebras interlacedly in three manners: the classical splitting, the second splitting and the un-splitting, in terms of representations of Poisson algebras. Note that due to Proposition 2.9, the characterization of these algebraic structures in terms of representations of Poisson algebras on the dual spaces is the same as that on the spaces themselves. Before we introduce these various algebraic structures, we give the following “principle” to name them. 1. Every algebraic structure here is named by 3 capital letters. 2. The first letter is unified to be “P” since these algebras are related to Poisson algebras. 3. The second letter denotes the operation corresponding to the splitting of the commutative associative products. Explicitly, the capital letters “C”, “Z” and “A” respectively denote the operations of commutative associative algebras, Zinbiel algebras and anti-Zinbiel algebras, corresponding to the un-splitting, the classical splitting and the second splitting. 4. The third letter denotes the operation corresponding to the splitting of the Lie brackets. Explicitly, the capital letters “L”, “P” and “A” respectively denote the operations of Lie algebras, pre-Lie algebras and anti-pre-Lie algebras, corresponding to the un-splitting, the classical splitting and the second splitting. Note that the PZP algebras combining Zinbiel algebras and pre-Lie algebras are exactly the pre-Poisson algebras introduced in [1].

2.3.1 PCP Algebras Definition 2.10 A PCP algebra is a triple (A, ·, ◦), such that (A, ·) is a commutative associative algebra, (A, ◦) is a pre-Lie algebra, and the following equations hold:

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61

(x · y) ◦ z = x · (y ◦ z) + y · (x ◦ z),

(2.37)

(x ◦ y − y ◦ x) · z = x ◦ (y · z) − y · (x ◦ z),

(2.38)

z ◦ (x · y) − z · (x ◦ y) − z · (y ◦ x) = 0,

(2.39)

.

.

.

for all x, y, z ∈ A. Proposition 2.10 Let (A, ·, [−, −]) be a Poisson algebra and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (L· , L◦ , A) is a representation of (A, ·, [−, −]), then (A, ·, ◦) is a PCP algebra. Conversely, let (A, ·, ◦) be a PCP algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a Poisson algebra with a representation (L· , L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible PCP algebra of (A, ·, [−, −]). Proof Since (L· , L◦ , A) is a representation of (A, ·, [−, −]), we get Eqs. (2.37)– (2.38). Moreover, by Eq. (2.38), we have x ◦ (y · z) − y · (x ◦ z) = −y ◦ (x · z) + x · (y ◦ z), ∀x, y, z ∈ A.

.

(2.40)

Thus for all x, y, z ∈ A, we have 0 = [z, x · y] + [x, z] · y + [y, z] · x

.

(2.38)

= z ◦ (x · y) − (x · y) ◦ z + x ◦ (z · y) − z · (x ◦ y) + y ◦ (z · x) − z · (y ◦ x)

(2.40)

= z ◦ (x · y)−(x · y) ◦ z+x · (y ◦ z)+y · (x ◦ z)−z · (x ◦ y)−z · (y ◦ x)

(2.37)

= z ◦ (x · y) − z · (x ◦ y) − z · (y ◦ x).

Hence Eq. (2.39) holds. Thus (A, ·, ◦) is a PCP algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.2 Let A be a vector space with two bilinear operations ·, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, ·, ◦) is a PCP algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (L· , L◦ , A), where [−, −] is defined by Eq. (2.16). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by (x, a) · (y, b) = (x · y, x · b + a · y), ∀x, y, a, b ∈ A,

.

and the Lie bracket [−, −] is defined by Eq. (2.18).

(2.41)

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2.3.2 PCA Algebras Definition 2.11 A PCA algebra is a triple (A, ·, ◦), such that (A, ·) is a commutative associative algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.37) and the following equations hold: (x ◦ y − y ◦ x) · z = y · (x ◦ z) − x ◦ (y · z),

.

z ◦ (x · y) + z · (x ◦ y) + z · (y ◦ x) − 2(x · y) ◦ z = 0,

.

(2.42) (2.43)

for all x, y, z ∈ A. Proposition 2.11 Let (A, ·, [−, −]) be a Poisson algebra and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (L· , −L◦ , A) is a representation of (A, ·, [−, −]), then (A, ·, ◦) is a PCA algebra. Conversely, let (A, ·, ◦) be a PCA algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a Poisson algebra with a representation (L· , −L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible PCA algebra of (A, ·, [−, −]). Proof Since (L· , −L◦ , A) is a representation of (A, ·, [−, −]), we get Eqs. (2.37) and (2.42). By Eq. (2.42), Eq. (2.40) holds. Thus for all x, y, z ∈ A, we have 0 = [z, x · y] + [x, z] · y + [y, z] · x

.

(2.42)

= z ◦ (x · y) − (x · y) ◦ z − x ◦ (z · y) + z · (x ◦ y) − y ◦ (z · x) + z · (y ◦ x)

(2.40)

= z ◦ (x · y) − (x · y) ◦ z − x · (y ◦ z) − y · (x ◦ z) + z · (x ◦ y) + z · (y ◦ x)

(2.37)

= z ◦ (x · y) + z · (x ◦ y) + z · (y ◦ x) − 2(x · y) ◦ z.

Hence Eq. (2.43) holds. So (A, ·, ◦) is a PCA algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.3 Let A be a vector space with two bilinear operations ·, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, ·, ◦) is a PCA algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (L· , −L◦ , A), where [−, −] is defined by Eq. (2.16). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.41) and the Lie bracket [−, −] is defined by Eq. (2.22). Proposition 2.12 Let (A, ·, [−, −]) be a Poisson algebra. Suppose that B is a nondegenerate symmetric bilinear form on A such that it is invariant on (A, ·) and

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63

a commutative 2-cocycle on (A, [−, −]). Then there is a compatible PCA algebra (A, ·, ◦) in which ◦ is defined by Eq. (2.23). Conversely, let (A, ·, ◦) be a PCA algebra and the sub-adjacent Poisson algebra be (A, ·, [−, −]). Then there is a Poisson algebra A X−L∗· ,−L∗◦ A∗ , and the natural nondegenerate symmetric bilinear form Bd defined by Eq. (2.15) is invariant on the commutative associative algebra A X−L∗· A∗ and a commutative 2-cocycle on the Lie algebra A X−L∗◦ A∗ . Proof Since (A, ·, [−, −]) is a Poisson algebra, the following equation holds: [x, y · z] + [y, z · x] + [z, x · y] = 0, ∀x, y, z ∈ A.

(2.44)

.

Let B be a nondegenerate symmetric bilinear form on A such that it is invariant on (A, ·) and a commutative 2-cocycle on (A, [−, −]). Then .

B((x · y) ◦ z − x · (y ◦ z) − y · (x ◦ z), w) (2.23),(2.26)

=

(2.44)

B(z, [x · y, w] − [y, x · w] − [x, y · w]) = 0,

B((x ◦ y − y ◦ x) · z − y · (x ◦ z) − x ◦ (y · z), w) (2.23),(2.26)

=

(2.10)

B(z, [x, y] · w − [x, y · w] − y · [x, w]) = 0.

Hence Eqs. (2.37) and (2.42) hold by the nondegeneracy of B. Thus Eq. (2.43) holds such that (A, ·, ◦) is a PCA algebra. Conversely, let (A, ·, ◦) be a PCA algebra and the sub-adjacent Poisson algebra be (A, ·, [−, −]). Then (L· , −L◦ , A) is a representation of (A, ·, [−, −]). By Proposition 2.9, (−L∗· , −L∗◦ , A∗ ) is also a representation of (A, ·, [−, −]). Thus there is a Poisson algebra structure AX−L∗· ,−L∗◦ A∗ . It is straightforward to show that Bd is invariant on the commutative associative algebra A X−L∗· A∗ and a commutative 2-cocycle on the Lie algebra A X−L∗◦ A∗ . u n

2.3.3 PZL Algebras Definition 2.12 A PZL algebra is a triple (A, *, [−, −]), such that (A, *) is a Zinbiel algebra, (A, [−, −]) is a Lie algebra, and the following equations hold: [x * y + y * x, z] = x * [y, z] + y * [x, z],

(2.45)

[x, y] * z = [x, y * z] − y * [x, z],

(2.46)

.

.

for all x, y, z ∈ A. Proposition 2.13 Let (A, ·, [−, −]) be a Poisson algebra and (A, *) be a compatible Zinbiel algebra of (A, ·). If (L* , ad, A) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a PZL algebra. Conversely, let (A, *, [−, −]) be a PZL algebra

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and (A, ·) be the sub-adjacent commutative associative algebra of (A, *). Then (A, ·, [−, −]) is a Poisson algebra with a representation (L* , ad, A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, *, [−, −]), and (A, *, [−, −]) is a compatible PZL algebra of (A, ·, [−, −]). Proof We only prove the latter. For all x, y, z ∈ A, we have .

[z, x · y] + [x, z] · y + [y, z] · x (2.46)

= [z, x * y] + [z, y * x] + [x, z] * y + y * [x, z] + [y, z] * x + x * [y, z] = 0. Thus (A, ·, [−, −]) is a Poisson algebra. Moreover, by Eqs. (2.45) and (2.46), (L* , ad, A) is a representation of (A, ·, [−, −]). u n Hence we get the following conclusion. Corollary 2.4 Let A be a vector space with two bilinear operations *, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, [−, −]) is a PZL algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (L* , ad, A), where · is defined by Eq. (2.27). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by [(x, a), (y, b)] = ([x, y], [x, b] + [a, y]), ∀x, y, a, b ∈ A.

.

(2.47)

2.3.4 PZP Algebras or Pre-Poisson Algebras Definition 2.13 ([1]) A pre-Poisson algebra or a PZP algebra is a triple (A, *, ◦), such that (A, *) is a Zinbiel algebra, (A, ◦) is a pre-Lie algebra, and the following equations hold: .

(x * y + y * x) ◦ z = x * (y ◦ z) + y * (x ◦ z),

(2.48)

(x ◦ y − y ◦ x) * z = x ◦ (y * z) − y * (x ◦ z),

(2.49)

.

for all x, y, z ∈ A. Proposition 2.14 ([1]) Let (A, ·, [−, −]) be a Poisson algebra, (A, *) be a compatible Zinbiel algebra of (A, ·) and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (L* , L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a pre-Poisson algebra. Conversely, let (A, *, ◦) be a pre-Poisson algebra, (A, ·) be

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

65

the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a Poisson algebra with a representation (L* , L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible pre-Poisson algebra of (A, ·, [−, −]). Hence we get the following conclusion. Corollary 2.5 Let A be a vector space with two bilinear operations *, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a pre-Poisson algebra. 2. The triple (A, ·, [−, −])is a Poisson algebra with a representation (L* , L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by Eq. (2.18). Proposition 2.15 Let (A, ·, [−, −]) be a Poisson algebra. Suppose that B is a nondegenerate skew-symmetric bilinear form on A such that it is a Connes cocycle on (A, ·) and a symplectic form on (A, [−, −]). Then there is a compatible prePoisson algebra (A, *, ◦) in which * and ◦ are respectively defined by Eqs. (2.31) and (2.20). Conversely, let (A, *, ◦) be a pre-Poisson algebra and the sub-adjacent Poisson algebra be (A, ·, [−, −]). Then there is a Poisson algebra A X−L∗* ,L∗◦ A∗ , and the natural nondegenerate skew-symmetric bilinear form Bp defined by Eq. (2.21) is a Connes cocycle on the commutative associative algebra A X−L∗* A∗ and a symplectic form on the Lie algebra A XL∗◦ A∗ . Proof It is similar to the proof of Proposition 2.12.

u n

2.3.5 PZA Algebras Definition 2.14 A PZA algebra is a triple (A, *, ◦), such that (A, *) is a Zinbiel algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.48) and the following equations hold: (x ◦ y − y ◦ x) * z = −x ◦ (y * z) + y * (x ◦ z),

.

(2.50)

z◦(x *y +y *x)+z*(x ◦y +y ◦x)−y *(z◦x)−x *(z◦y)−x ◦(z*y)−y ◦(z*x) = 0, (2.51) for all x, y, z ∈ A.

.

Proposition 2.16 Let (A, ·, [−, −]) be a Poisson algebra, (A, *) be a compatible Zinbiel algebra of (A, ·) and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (L* , −L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a PZA algebra. Conversely, let (A, *, ◦) be a PZA algebra, (A, ·) be the sub-adjacent

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commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a Poisson algebra with a representation (L* , −L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible PZA algebra of (A, ·, [−, −]). Proof Since (L* , −L◦ , A) is a representation of (A, ·, [−, −]), we get Eqs. (2.48) and (2.50). Thus for all x, y, z ∈ A, we have 0 = [z, x · y] + [x, z] · y + [y, z] · x

.

= z ◦ (x · y) − (x · y) ◦ z + [x, z] * y + y * [x, z] + [y, z] * x + x * [y, z] (2.50)

= z ◦ (x · y) − (x · y) ◦ z + z * (x ◦ y) − x ◦ (z * y) + y * (x ◦ z) − y * (z ◦ x)

+z * (y ◦ x) − y ◦ (z * x) + x * (y ◦ z) − x * (z ◦ y) (2.48)

= z ◦ (x * y) + z ◦ (y * x) + z * (x ◦ y) − x ◦ (z * y)

−y * (z ◦ x) + z * (y ◦ x) − y ◦ (z * x) − x * (z ◦ y). Hence Eq. (2.51) holds and thus (A, *, ◦) is a PZA algebra. The converse part is proved similarly. n u Hence we get the following conclusion. Corollary 2.6 Let A be a vector space with two bilinear operations *, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a PZA algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (L* , −L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by Eq. (2.22).

2.3.6 PAL Algebras Definition 2.15 A PAL algebra is a triple (A, *, [−, −]), such that (A, *) is an anti-Zinbiel algebra, (A, [−, −]) is a Lie algebra, and Eq. (2.46) and the following equation hold: [z, x * y + y * x] = x * [y, z] + y * [x, z], ∀x, y, z ∈ A.

.

(2.52)

Proposition 2.17 Let (A, ·, [−, −]) be a Poisson algebra and (A, *) be a compatible anti-Zinbiel algebra of (A, ·). If (−L* , ad, A) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a PAL algebra. Conversely, let (A, *, [−, −])

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

67

be a PAL algebra and (A, ·) be the sub-adjacent commutative associative algebra of (A, *). Then (A, ·, [−, −]) is a Poisson algebra with a representation (−L* , ad, A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, *, [−, −]), and (A, *, [−, −]) is a compatible PAL algebra of (A, ·, [−, −]). Proof It is similar to the proof of Proposition 2.13.

u n

Hence we get the following conclusion. Corollary 2.7 Let A be a vector space with two bilinear operations *, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, [−, −]) is a PAL algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (−L* , ad, A), where · is defined by Eq. (2.27). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.33) and the Lie bracket [−, −] is defined by Eq. (2.47). Proposition 2.18 Let (A, ·, [−, −]) be a Poisson algebra. Suppose that B is a nondegenerate symmetric bilinear form on A such that it is a commutative Connes cocycle on (A, ·) and invariant on (A, [−, −]). Then there is a compatible PAL algebra (A, *, [−, −]) in which * is defined by Eq. (2.34). Conversely, let (A, *, [−, −]) be a PAL algebra and the sub-adjacent Poisson algebra be (A, ·, [−, −]). Then there is a Poisson algebra A XL∗* ,ad∗ A∗ , and the natural nondegenerate symmetric bilinear form Bd defined by Eq. (2.15) is a commutative Connes cocycle on the commutative associative algebra A XL∗* A∗ and invariant on the Lie algebra A Xad∗ A∗ . Proof It is similar to the proof of Proposition 2.12.

u n

2.3.7 PAP Algebras Definition 2.16 A PAP algebra is a triple (A, *, ◦), such that (A, *) is an antiZinbiel algebra, (A, ◦) is a pre-Lie algebra, and Eq. (2.49) and the following equations hold: .

x * (y ◦ z) + y * (x ◦ z) = 0,

(2.53)

(x * y) ◦ z + (y * x) ◦ z = 0,

(2.54)

.

for all x, y, z ∈ A. Proposition 2.19 Let (A, ·, [−, −]) be a Poisson algebra, (A, *) be a compatible anti-Zinbiel algebra of (A, ·) and (A, ◦) be a compatible pre-Lie algebra of

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(A, [−, −]). If (−L* , L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a PAP algebra. Conversely, let (A, *, ◦) be a PAP algebra, (A, ·) be the subadjacent commutative associative algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a Poisson algebra with a representation (−L* , L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible PAP algebra of (A, ·, [−, −]). Proof Since (−L* , L◦ , A) is a representation of (A, ·, [−, −]), we get Eqs. (2.49) and the following equation: (x * y + y * x) ◦ z = −x * (y ◦ z) − y * (x ◦ z), ∀x, y, z ∈ A.

.

(2.55)

Thus for all x, y, z ∈ A, we have 0 = [z, x · y] + [x, z] · y + [y, z] · x

.

= z ◦ (x * y) + z ◦ (y * x) − (x * y) ◦ z − (y * x) ◦ z + (x ◦ z − z ◦ x) * y +y * (x ◦ z − z ◦ x) + (y ◦ z − z ◦ y) * x + x * (y ◦ z − z ◦ y) (2.49),(2.55)

=

−2(x * y) ◦ z − 2(y * x) ◦ z.

Hence Eq. (2.54) holds, and by Eq. (2.55), Eq. (2.53) holds. Thus (A, *, ◦) is a PAP algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.8 Let A be a vector space with two bilinear operations *, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a PAP algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (−L* , L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.33) and the Lie bracket [−, −] is defined by Eq. (2.18).

2.3.8 PAA Algebras Definition 2.17 A PAA algebra is a triple (A, *, ◦), such that (A, *) is an antiZinbiel algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.55) and the following equations hold: (x ◦ y − y ◦ x) * z = y * (x ◦ z) − x ◦ (y * z),

.

(2.56)

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

z ◦ (x * y + y * x) − (x * y + y * x) ◦ z − x * (z ◦ y) − y * (z ◦ x) = 0,

.

69

(2.57)

for all x, y, z ∈ A. Proposition 2.20 Let (A, ·, [−, −]) be a Poisson algebra, (A, *) be a compatible anti-Zinbiel algebra of (A, ·) and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (−L* , −L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a PAA algebra. Conversely, let (A, *, ◦) be a PAA algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a Poisson algebra with a representation (−L* , −L◦ , A). In this case, we say (A, ·, [−, −]) is the subadjacent Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible PAA algebra of (A, ·, [−, −]). Proof Since (−L* , −L◦ , A) is a representation of (A, ·, [−, −]), we get Eqs. (2.55) and (2.56). Thus for all x, y, z ∈ A, we have 0 = [z, x · y] + [x, z] · y + [y, z] · x

.

= z ◦ (x * y) + z ◦ (y * x) − (x * y) ◦ z − (y * x) ◦ z + (x ◦ z − z ◦ x) * y +y * (x ◦ z − z ◦ x) + (y ◦ z − z ◦ y) * x + x * (y ◦ z − z ◦ y) (2.55),(2.56)

=

2(z ◦ (x * y + y * x) − (x * y + y * x) ◦ z − x * (z ◦ y) − y * (z ◦ x)).

Hence Eq. (2.57) holds, and thus (A, *, ◦) is a PAA algebra. The converse part is proved similarly. n u Hence we get the following conclusion. Corollary 2.9 Let A be a vector space with two bilinear operations *, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a PAA algebra. 2. The triple (A, ·, [−, −]) is a Poisson algebra with a representation (−L* , −L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.33) and the Lie bracket [−, −] is defined by Eq. (2.22). Proposition 2.21 Let (A, ·, [−, −]) be a Poisson algebra. Suppose that B is a nondegenerate symmetric bilinear form on A such that it is a commutative Connes cocycle on (A, ·) and a commutative 2-cocycle on (A, [−, −]). Then there is a compatible PAA algebra (A, *, ◦) in which * and ◦ are respectively defined by Eqs. (2.34) and (2.23). Conversely, let (A, *, ◦) be a PAA algebra and the sub-adjacent Poisson algebra be (A, ·, [−, −]). Then there is a Poisson algebra A XL∗* ,−L∗◦ A∗ , and the natural nondegenerate symmetric bilinear form Bd defined

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Table 2.1 Splittings of Poisson algebras

Algebras PCP PCA

Notations .(A, ·, ◦) .(A, ·, ◦)

Representations of Poisson algebras on the spaces themselves .(L· , L◦ , A) .(L· , −L◦ , A)

Representations of Poisson algebras on the dual spaces ∗ ∗ ∗ .(−L· , L◦ , A ) ∗ ∗ ∗ .(−L· , −L◦ , A ) ∗

∗

, A∗ )

PZL PrePoisson PZA PAL

.(A, *, [−, −])

.(L* , ad, A)

.(A, *, ◦)

.(L* , L◦ , A)

∗ ∗ ∗ .(−L* , L◦ , A )

.(A, *, ◦)

.(L* , −L◦ , A)

.(A, *, [−, −])

.(−L* , −L◦ , A

.(−L* , ad, A)

∗ ∗ ∗ .(L* , ad , A )

PAP PAA

.(A, *, ◦)

.(−L* , L◦ , A)

.(A, *, ◦)

.(−L* , −L◦ , A)

.(L* , L◦ , A

.(−L* , ad

∗

∗

∗

∗

∗)

∗)

∗ ∗ ∗ .(L* , −L◦ , A )

Corresponding nondegenerate bilinear forms on Poisson algebras – invariant, commutative 2-cocycle – Connes cocycle, symplectic form – commutative Connes cocycle, invariant – commutative Connes cocycle, commutative 2-cocycle

by Eq. (2.15) is a commutative Connes cocycle on the commutative associative algebra A XL∗* A∗ and a commutative 2-cocycle on the Lie algebra A X−L∗◦ A∗ . Proof It is similar to the proof of Proposition 2.12.

u n

2.3.9 Summary We summarize some facts on the 8 algebraic structures in the previous subsections respectively corresponding to the mixed splittings of operations of Poisson algebras in Table 2.1.

2.4 Mixed Splittings of Operations of Transposed Poisson Algebras in Terms of Representations on the Spaces Themselves and Related Algebraic Structures We introduce 8 algebraic structures respectively corresponding to the mixed splitting of the commutative associative products and Lie brackets of transposed Poisson algebras interlacedly in three manners: the classical splitting, the second splitting and the un-splitting, in terms of representations of transposed Poisson algebras on the spaces themselves.

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

71

Definition 2.18 A representation of a transposed Poisson algebra .(A, ·, [−, −]) is a triple .(μ, ρ, V ), such that .(μ, V ) is a representation of the commutative associative algebra .(A, ·), .(ρ, V ) is a representation of the Lie algebra .(A, [−, −]), and the following compatible conditions hold: 2μ(x)ρ(y) = ρ(x · y) + ρ(y)μ(x),

(2.58)

2μ([x, y]) = ρ(x)μ(y) − ρ(y)μ(x),

(2.59)

.

.

for all .x, y ∈ A. Let .(A, ·, [−, −]) be a transposed Poisson algebra. Then .(L· , ad, A) is a representation of .(A, ·, [−, −]), called the adjoint representation of .(A, ·, [−, −]). Moreover, .(μ, ρ, V ) is a representation of a transposed Poisson algebra .(A, ·, [−, −]) if and only if the direct sum .A ⊕ V of vector spaces is a (semidirect product) transposed Poisson algebra by defining the multiplications on .A ⊕ V by Eqs. (2.25) and (2.12) respectively. We denote it by .A Xμ,ρ V . Unlike the case of Poisson algebras in Proposition 2.9, for a representation ∗ ∗ ∗ .(μ, ρ, V ) of a transposed Poisson algebra .(A, ·, [−, −]), .(−μ , ρ , V ) is not necessarily a representation of .(A, ·, [−, −]) (see Proposition 2.35). Thus for transposed Poisson algebras, we shall divide into two cases according to the representations of transposed Poisson algebras on the spaces themselves and the dual spaces respectively. Next we introduce 8 algebraic structures in the rest of this section corresponding to the mixed splitting of the commutative associative products and Lie brackets of transposed Poisson algebras interlacedly in three manners: the classical splitting, the second splitting and the un-splitting, in terms of representations of transposed Poisson algebras on the spaces themselves, whereas another 8 algebraic structures are introduced in the next section in terms of representations of transposed Poisson algebras on the dual spaces. Before we introduce these various algebraic structures, we give the following “principle” to name them. 1. Every such algebraic structure in this section and the next section is named by 4 capital letters. 2. The first letter is unified to be “T” since these algebras are related to transposed Poisson algebras. 3. The second letter denotes the operation corresponding to the splitting of the commutative associative products. Explicitly, the capital letters “C”, “Z” and “A” respectively denote the operations of commutative associative algebras, Zinbiel algebras and anti-Zinbiel algebras, corresponding to the un-splitting, the classical splitting and the second splitting. 4. The third letter denotes the operation corresponding to the splitting of the Lie brackets. Explicitly, the capital letters “L”, “P” and “A” respectively denote the operations of Lie algebras, pre-Lie algebras and anti-pre-Lie algebras, corresponding to the un-splitting, the classical splitting and the second splitting.

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5. The last letter is “O” in the case of the representations of transposed Poisson algebras on the spaces themselves and “D” in the case of the representations of transposed Poisson algebras on the dual spaces.

2.4.1 TCPO Algebras Definition 2.19 A TCPO algebra is a triple (A, ·, ◦), where (A, ·) is a commutative associative algebra, (A, ◦) is a pre-Lie algebra, and 2x · (y ◦ z) = (x · y) ◦ z + y ◦ (x · z),

(2.60)

2z · (x ◦ y − y ◦ x) = x ◦ (z · y) − y ◦ (z · x),

(2.61)

.

.

for all x, y, z ∈ A. Proposition 2.22 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (L· , L◦ , A) is a representation of (A, ·, [−, −]), then (A, ·, ◦) is a TCPO algebra. Conversely, let (A, ·, ◦) be a TCPO algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L· , L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible TCPO algebra of (A, ·, [−, −]). Proof We only prove the latter. Let x, y, z ∈ A. We have (2.60)

(2.61)

(x·y)◦z−(x·z)◦y = 2x·(y◦z)−y◦(x·z)−2x·(z◦y)+z◦(x·y) = 0.

.

(2.62)

Then 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.61)

= x ◦ (y · z) − y ◦ (x · z) + y ◦ (z · x)

−(z · x) ◦ y − x ◦ (z · y) + (z · y) ◦ x = (z · y) ◦ x − (z · x) ◦ y (2.62)

= 0.

Thus (A, ·, [−, −]) is a transposed Poisson algebra, and by Eqs. (2.60) and (2.61), u n (L· , L◦ , A) is a representation of (A, ·, [−, −]). Hence we get the following conclusion. Corollary 2.10 Let A be a vector space with two bilinear operations ·, ◦ : A⊗A → A. Then the following conditions are equivalent:

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

73

1. (A, ·, ◦) is a TCPO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L· , L◦ , A), where [−, −] is defined by Eq. (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.41) and the Lie bracket [−, −] is defined by Eq. (2.18).

2.4.2 TCAO Algebras Definition 2.20 A TCAO algebra is a triple (A, ·, ◦), where (A, ·) is a commutative associative algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.60) and the following equation hold: 2z · (x ◦ y − y ◦ x) = −x ◦ (z · y) + y ◦ (z · x), ∀x, y, z ∈ A.

.

(2.63)

Proposition 2.23 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (L· , −L◦ , A) is a representation of (A, ·, [−, −]), then (A, ·, ◦) is a TCAO algebra. Conversely, let (A, ·, ◦) be a TCAO algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L· , −L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible TCAO algebra of (A, ·, [−, −]). Proof We only prove the latter. Let x, y, z ∈ A. We have (2.60)

(x · y) ◦ z − (x · z) ◦ y = 2x · (y ◦ z) − y ◦ (x · z) − 2x · (z ◦ y) + z ◦ (x · y)

.

(2.63)

= 2z ◦ (x · y) − 2y ◦ (x · z).

(2.64)

Then 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.63)

= y ◦ (x · z) − x ◦ (y · z) + y ◦ (z · x)

−(z · x) ◦ y − x ◦ (z · y) + (z · y) ◦ x = 2y ◦ (z · x) − 2x ◦ (y · z) − (z · x) ◦ y +(z · y) ◦ x (2.64)

= 0.

Thus (A, ·, [−, −]) is a transposed Poisson algebra, and by Eqs. (2.60) and (2.63), (L· , −L◦ , A) is a representation of (A, ·, [−, −]). u n

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Hence we get the following conclusion. Corollary 2.11 Let A be a vector space with two bilinear operations ·, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, ·, ◦) is a TCAO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L· , −L◦ , A), where [−, −] is defined by Eq. (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.41) and the Lie bracket [−, −] is defined by Eq. (2.22).

2.4.3 TZLO Algebras Definition 2.21 A TZLO algebra is a triple (A, *, [−, −]), such that (A, *) is a Zinbiel algebra, (A, [−, −]) is a Lie algebra, and the following equation holds: x * [y, z] = [x, y] * z = [x, y * z] = 0, ∀x, y, z ∈ A.

.

(2.65)

Proposition 2.24 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, *) be a compatible Zinbiel algebra of (A, ·). If (L* , ad, A) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a TZLO algebra. Conversely, let (A, *, [−, −]) be a TZLO algebra and (A, ·) be the sub-adjacent commutative associative algebra of (A, *). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L* , ad, A). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, [−, −]), and (A, *, [−, −]) is a compatible TZLO algebra of (A, ·, [−, −]). Proof Since (L* , ad, A) is a representation of (A, ·, [−, −]), the following equations hold: .

2x * [y, z] = [x * y + y * x, z] + [y, x * z], .

(2.66)

2[x, y] * z = [x, y * z] − [y, x * z],

(2.67)

for all x, y, z ∈ A. Thus for all x, y, z ∈ A, we have 0 = 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.66),(2.67)

=

[x, y * z] − [y, x * z] + [z · x, y] + [x, z * y] − [z · x, y] − [x, z · y]

= −[y, x * z]. By Eqs. (2.66) and (2.67) again, we get Eq. (2.65). Hence (A, *, [−, −]) is a TZLO u n algebra. The converse part is proved similarly.

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

75

Remark 2.1 Let (A, *, [−, −]) be a TZLO algebra. Then the sub-adjacent transposed Poisson algebra (A, ·, [−, −]) is trivial in the sense that [x, y · z] = x · [y, z] = 0, ∀x, y, z ∈ A.

.

(2.68)

Note that in this case, it is also a Poisson algebra [5]. Moreover we get the following conclusion. Corollary 2.12 Let A be a vector space with two bilinear operations *, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, [−, −]) is a TZLO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L* , ad, A), where · is defined by Eq. (2.27). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by Eq. (2.47).

2.4.4 TZPO Algebras Definition 2.22 A TZPO algebra is a triple (A, *, ◦), such that (A, *) is a Zinbiel algebra, (A, ◦) is a pre-Lie algebra, and the following equations hold: .

2x * (y ◦ z) = (x * y + y * x) ◦ z + y ◦ (x * z),

(2.69)

2(x ◦ y − y ◦ x) * z = x ◦ (y * z) − y ◦ (x * z),

(2.70)

.

for all x, y, z ∈ A. Remark 2.2 In fact, TZPO algebras might be named as “pre-transposed Poisson algebras” since the operad of TZPO algebras is the successor of the operad of transposed Poisson algebras, illustrating the classical splitting of operations of transposed Poisson algebras. Proposition 2.25 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible Zinbiel algebra of (A, ·) and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (L* , L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a TZPO algebra. Conversely, let (A, *, ◦) be a TZPO algebra, (A, ·) be the subadjacent commutative associative algebra of (A, *) and (A, [−, −]) be the subadjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L* , L◦ , A). In this case, we say (A, ·, [−, −]) is the subadjacent transposed Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible TZPO algebra of (A, ·, [−, −]). Proof We only prove the latter. For all x, y, z ∈ A, we have

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2[x, y] · z − [z · x, y] − [x, z · y] (2.69),(2.70)

=

x ◦ (y * z) − y ◦ (x * z) + (z · x) ◦ y + x ◦ (z * y) − (z · y) ◦ x

−y ◦ (z * x) − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x = 0. Hence (A, ·, [−, −]) is a transposed Poisson algebra, and by Eqs. (2.69)–(2.70), u n (L* , L◦ , A) is a representation of (A, ·, [−, −]). Example 2.1 Let (A, *) be a Zinbiel algebra. Suppose P is a derivation of (A, *), that is, P satisfies P (x * y) = P (x) * y + x * P (y), ∀x, y ∈ A.

.

(2.71)

Then (A, ◦) is a pre-Lie algebra, where x ◦ y = P (x) * y − x * P (y), ∀x, y ∈ A.

.

(2.72)

Moreover, (A, *, ◦) is a TZPO algebra. Note that for the sub-adjacent transposed Poisson algebra (A, ·, [−, −]), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16), the following equation holds: [x, y] = P (x) · y − x · P (y), ∀x, y ∈ A.

.

(2.73)

Moreover we get the following conclusion. Corollary 2.13 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TZPO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L* , L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by Eq. (2.18).

2.4.5 TZAO Algebras Definition 2.23 A TZAO algebra is a triple (A, *, ◦), such that (A, *) is a Zinbiel algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.69) and the following equations hold: (x ◦ y) * z − (y ◦ x) * z = 0,

.

(2.74)

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

x ◦ (y * z) − y ◦ (x * z) = 0,

.

77

(2.75)

for all x, y, z ∈ A. Proposition 2.26 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible Zinbiel algebra of (A, ·) and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (L* , −L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a TZAO algebra. Conversely, let (A, *, ◦) be a TZAO algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L* , −L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, ◦), and (A, *, [−, −]) is a compatible TZAO algebra of (A, ·, [−, −]). Proof Since (L* , −L◦ , A) is a representation of (A, ·, [−, −]), Eq. (2.69) and the following equation hold: 2(x ◦ y − y ◦ x) * z = y ◦ (x * z) − x ◦ (y * z), ∀x, y, z ∈ A.

.

(2.76)

Thus for all x, y, z ∈ A, we have 0 = 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.69),(2.76)

=

y ◦ (x * z) − x ◦ (y * z) + (z · x) ◦ y + x ◦ (z * y) − (z · y) ◦ x

−y ◦ (z * x) − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x = 2y ◦ (x * z) − 2x ◦ (y * z). Hence Eq. (2.75) holds. Substituting Eq. (2.75) into Eq. (2.76), we get Eq. (2.74). u n Thus (A, *, ◦) is a TZAO algebra. The converse part is proved similarly. Hence we get the following conclusion. Corollary 2.14 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TZAO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L* , −L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by Eq. (2.22).

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2.4.6 TALO Algebras Definition 2.24 A TALO algebra is a triple (A, *, [−, −]), such that (A, *) is an anti-Zinbiel algebra, (A, [−, −]) is a Lie algebra, and Eq. (2.65) holds. Proposition 2.27 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, *) be a compatible anti-Zinbiel algebra of (A, ·). If (−L* , ad, A) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a TALO algebra. Conversely, let (A, *, [−, −]) be a TALO algebra and (A, ·) be the sub-adjacent commutative associative algebra of (A, *). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L* , ad, A). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, [−, −]), and (A, *, [−, −]) is a compatible TALO algebra of (A, ·, [−, −]). Proof Since (−L* , ad, A) is a representation of (A, ·, [−, −]), Eq. (2.67) and the following equation hold: 2x * [y, z] = [z, x * y + y * x] + [y, x * z], ∀x, y, z ∈ A.

.

(2.77)

Thus for all x, y, z ∈ A, we have 0 = 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.77),(2.67)

=

[x, y * z] − [y, x * z] − [z · x, y] + [x, z * y] − [z · x, y] − [x, z · y]

= [y, x · z + z * x]. Hence we get [y, x · z + z * x] = 0,

(2.78)

[y, z · x + x * z] = 0.

(2.79)

[y, z · x] = 0.

(2.80)

.

.

Adding them together, we get .

Combining it with Eq. (2.78), we get [x, y * z] = 0.

.

(2.81)

Then by Eqs. (2.67) and (2.77), we get Eq. (2.65). Thus (A, *, [−, −]) is a TALO algebra. The converse part is proved similarly. u n Remark 2.3 For a TALO algebra (A, *, [−, −]), the sub-adjacent transposed Poisson algebra (A, ·, [−, −]) is also trivial in the sense of Eq. (2.68). Hence in this case, it is a Poisson algebra.

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79

Moreover we get the following conclusion. Corollary 2.15 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, [−, −]) is a TALO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L* , ad, A), where · is defined by Eq. (2.27). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.29) and the Lie bracket [−, −] is defined by Eq. (2.47).

2.4.7 TAPO Algebras Definition 2.25 A TAPO algebra is a triple (A, *, ◦), such that (A, *) is an antiZinbiel algebra, (A, ◦) is a pre-Lie algebra, and Eq. (2.70) and the following equations hold: 2x * (y ◦ z) = −(x * y + y * x) ◦ z + y ◦ (x * z),

.

(z * x + x * z) ◦ y − (z * y + y * z) ◦ x = 0,

.

(2.82) (2.83)

for all x, y, z ∈ A. Proposition 2.28 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible anti-Zinbiel algebra of (A, ·) and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (−L* , L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a TAPO algebra. Conversely, let (A, *, ◦) be a TAPO algebra, (A, ·) be the sub-adjacent commutative associative algebra and (A, [−, −]) be the subadjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L* , L◦ , A). In this case, we say (A, ·, [−, −]) is the subadjacent transposed Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible TAPO algebra of (A, ·, [−, −]). Proof Since (−L* , L◦ , A) is a representation of (A, ·, [−, −]), Eqs. (2.70) and (2.82) hold. Thus for all x, y, z ∈ A, we have 0 = 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.82),(2.70)

=

x ◦ (y * z) − y ◦ (x * z) − (z · x) ◦ y + x ◦ (z * y) + (z · y) ◦ x

−y ◦ (z * x) − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x = 2(z · y) ◦ x − 2(z · x) ◦ y.

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Hence Eq. (2.83) holds, and thus (A, *, ◦) is a TAPO algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.16 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TAPO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L* , L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.33) and the Lie bracket [−, −] is defined by Eq. (2.18).

2.4.8 TAAO Algebras Definition 2.26 A TAAO algebra is a triple (A, *, ◦), such that (A, *) is an antiZinbiel algebra, (A, ◦) is an anti-pre-Lie algebra, and Eqs. (2.76),(2.82) and the following equation hold: (z*x+x*z)◦y−(z*y+y*z)◦x+x◦(y*z)−y◦(x*z) = 0, ∀x, y, z ∈ A.

.

(2.84)

Proposition 2.29 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible anti-Zinbiel algebra of (A, ·) and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (−L* , −L◦ , A) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a TAAO algebra. Conversely, let (A, *, ◦) be a TAAO algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L* , −L◦ , A). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible TAAO algebra of (A, ·, [−, −]). Proof Since (−L* , −L◦ , A) is a representation of (A, ·, [−, −]), Eqs. (2.76) and (2.82) hold. Thus for all x, y, z ∈ A, we have 0 = 2[x, y] · z − [z · x, y] − [x, z · y]

.

(2.76),(2.82)

=

y ◦ (x * z) − x ◦ (y * z) − (z · x) ◦ y + x ◦ (z * y) + (z · y) ◦ x

−y ◦ (z * x) − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x = −2(z · x) ◦ y + 2(z · y) ◦ x − 2x ◦ (y * z) + 2y ◦ (x * z).

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Table 2.2 Splittings of transposed Poisson algebras on the spaces themselves Algebras

Notations

TCPO TCAO TZLO TZPO TZAO TALO TAPO TAAO

.(A, ·, ◦) .(A, ·, ◦) .(A, *, [−, −]) .(A, *, ◦) .(A, *, ◦) .(A, *, [−, −]) .(A, *, ◦) .(A, *, ◦)

Representations of transposed Poisson algebras on the spaces themselves .(L· , L◦ , A) .(L· , −L◦ , A) .(L* , ad, A) .(L* , L◦ , A) .(L* , −L◦ , A) .(−L* , ad, A) .(−L* , L◦ , A) .(−L* , −L◦ , A)

Hence Eq. (2.84) holds, and thus (A, *, ◦) is a TAAO algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.17 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TAAO algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L* , −L◦ , A), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A in which the commutative associative product · is defined by Eq. (2.33) and the Lie bracket [−, −] is defined by Eq. (2.22).

2.4.9 Summary We summarize some facts on the 8 algebraic structures in the previous subsections respectively corresponding to the mixed splittings of operations of transposed Poisson algebras in terms of representations of transposed Poisson algebras on the spaces themselves in Table 2.2.

2.5 Mixed Splittings of Operations of Transposed Poisson Algebras in Terms of Representations on the Dual Spaces and Related Algebraic Structures We introduce 8 algebraic structures respectively corresponding to the mixed splitting of the commutative associative products and Lie brackets of transposed

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Poisson algebras interlacedly in three manners: the classical splitting, the second splitting and the un-splitting, in terms of representations of transposed Poisson algebras on the dual spaces. The relationships between transposed Poisson algebras with nondegenerate bilinear forms satisfying certain conditions and some algebraic structures are given. Recall that for a Lie algebra .(g, [−, −]), a pair .(ρ, V ) is a representation if and only if .(ρ ∗ , V ∗ ) is a representation. Hence by Propositions 2.1 and 2.3, we have the following equivalent characterizations of pre-Lie algebras and anti-pre-Lie algebras in terms of the representations of Lie algebras on the dual spaces respectively. Proposition 2.30 Let A be a vector space together with a bilinear operation .◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. .(A, ◦) is a pre-Lie algebra. 2. .(A, ◦) is a Lie-admissible algebra such that .(L∗◦ , A∗ ) is a representation of the sub-adjacent Lie algebra .(A, [−, −]). 3. There is a Lie algebra structure on .A ⊕ A∗ defined by [(x, a ∗ ), (y, b∗ )] = (x◦y−y◦x, L∗◦ (x)b∗ −L∗◦ (y)a ∗ ), ∀x, y ∈ A, a ∗ , b∗ ∈ A∗ . (2.85)

.

Proposition 2.31 Let A be a vector space together with a bilinear operation .◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. .(A, ◦) is an anti-pre-Lie algebra. 2. .(A, ◦) is a Lie-admissible algebra such that .(−L∗◦ , A∗ ) is a representation of the sub-adjacent Lie algebra .(A, [−, −]). 3. There is a Lie algebra structure on .A ⊕ A∗ defined by [(x, a ∗ ), (y, b∗ )] = (x◦y−y◦x, L∗◦ (y)a ∗ −L∗◦ (x)b∗ ), ∀x, y ∈ A, a ∗ , b∗ ∈ A∗ . (2.86)

.

Similarly, for a commutative associative algebra .(A, ·), .(μ, V ) is a representation if and only if .(−μ∗ , V ∗ ) is a representation. Hence by Propositions 2.5 and 2.7, we have the following equivalent characterization of Zinbiel algebras and anti-Zinbiel algebras in terms of the representations of commutative associative algebras on the dual spaces respectively. Proposition 2.32 Let A be a vector space together with a bilinear operation .* : A ⊗ A → A. Then the following conditions are equivalent: 1. .(A, *) is a Zinbiel algebra. 2. .(A, ·) with the bilinear operation defined by Eq. (2.27) is a commutative associative algebra, and .(−L∗* , A∗ ) is a representation of .(A, ·). 3. There is a commutative associative algebra structure on .A ⊕ A∗ defined by (x, a ∗ )·(y, b∗ ) = (x *y +y *x, −L∗* (x)b∗ −L∗* (y)a ∗ ), ∀x, y ∈ A, a ∗ , b∗ ∈ A∗ . (2.87)

.

2 New Splittings of Operations of Poisson Algebras and Transposed Poisson Algebras

83

Proposition 2.33 Let A be a vector space together with a bilinear operation .* : A ⊗ A → A. Then the following conditions are equivalent: 1. .(A, *) is an anti-Zinbiel algebra. 2. .(A, ·) with the bilinear operation defined by Eq. (2.27) is a commutative associative algebra, and .(L∗* , A∗ ) is a representation of .(A, ·). 3. There is a commutative associative algebra structure on .A ⊕ A∗ defined by (x, a ∗ ) · (y, b∗ ) = (x * y + y * x, L∗* (x)b∗ + L∗* (y)a ∗ ), ∀x, y ∈ A, a ∗ , b∗ ∈ A∗ . (2.88)

.

For a representation .(μ, ρ, V ) of a transposed Poisson algebra .(A, ·, [−, −]), (−μ∗ , ρ ∗ , V ∗ ) is not necessarily a representation of .(A, ·, [−, −]). In fact, we have

.

Proposition 2.34 Let .(A, ·, [−, −]) be a transposed Poisson algebra, .(μ, V ) be a representation of .(A, ·) and .(ρ, V ) be a representation of .(A, [−, −]). Then ∗ ∗ ∗ .(−μ , ρ , V ) is a representation of .(A, ·, [−, −]) if and only if 2ρ(y)μ(x) = ρ(x · y) + μ(x)ρ(y),

(2.89)

2μ([x, y]) = μ(x)ρ(y) − μ(y)ρ(x),

(2.90)

.

.

for all .x, y ∈ A. Proof Let .x, y ∈ A, u∗ ∈ V ∗ , v ∈ V . Then we have = , = . Hence the conclusion follows.

u n

Proposition 2.35 Let .(A, ·, [−, −]) be a transposed Poisson algebra and .(μ, ρ, V ) be a representation of .(A, ·, [−, −]). Then .(−μ∗ , ρ ∗ , V ∗ ) is a representation of .(A, ·, [−, −]) if and only if μ([x, y]) = 0, ρ(x · y) = μ(x)ρ(y), ∀x, y ∈ A.

.

(2.91)

In particular, .(−L∗· , ad∗ , A∗ ) is a representation of .(A, ·, [−, −]) if and only if Eq. (2.68) holds. Proof By the assumption that Eqs. (2.58) and (2.59) hold, it is straightforward to show that Eqs. (2.89) and (2.90) hold if and only if Eq. (2.91) holds. Hence the u n conclusion follows from Proposition 2.34.

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Hence we get the following conclusion. Corollary 2.18 Let A be a vector space with two bilinear operations .·, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. .(A, ·) is a commutative associative algebra, .(A, [−, −]) is a Lie algebra, and Eq. (2.68) holds. 2. .(A, ·, [−, −]) is a transposed Poisson algebra with a representation ∗ ∗ ∗ .(−L· , ad , A ). 3. There is a transposed Poisson algebra structure on .A ⊕ A∗ in which the commutative associative product is defined by (x, a ∗ ) · (y, b∗ ) = (x · y, −L∗· (x)b∗ − L∗· (y)a ∗ ),

.

(2.92)

and the Lie bracket is defined by [(x, a ∗ ), (y, b∗ )] = ([x, y], ad∗ (x)b∗ − ad∗ (y)a ∗ ),

.

(2.93)

for all .x, y ∈ A, a ∗ , b∗ ∈ A∗ . Proposition 2.36 Let .(A.·, [−, −]) be a transposed Poisson algebra. Suppose that there is a nondegenerate symmetric bilinear from .B on A such that it is invariant on both .(A, ·) and .(A, [−, −]). Then Eq. (2.68) holds. Conversely, suppose that .(A, ·, [−, −]) is a transposed Poisson algebra and Eq. (2.68) holds. Then there is a transposed Poisson algebra .A X−L∗· ,ad∗ A∗ , and the natural nondegenerate symmetric bilinear form .Bd defined by Eq. (2.15) is invariant on both the commutative associative algebra .A X−L∗· A∗ and the Lie algebra .A Xad∗ A∗ . Proof It follows from a direct checking.

u n

Next we introduce 8 algebraic structures in the rest of this section corresponding to the mixed splitting of the commutative associative products and the Lie brackets of transposed Poisson algebras interlacedly in three manners: the classical splitting, the second splitting and the un-splitting, in terms of representations of transposed Poisson algebras on the dual spaces. We still use the principle given in the previous section to name them.

2.5.1 TCPD Algebras Definition 2.27 A TCPD algebra is a triple (A, ·, ◦), such that (A, ·) is a commutative associative algebra, (A, ◦) is a pre-Lie algebra, and the following equations hold: 2x ◦ (y · z) = (z · x) ◦ y + z · (x ◦ y),

.

(2.94)

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2(x ◦ y) · z − 2(y ◦ x) · z = x · (y ◦ z) − y · (x ◦ z),

(2.95)

3y ◦ (z · x) − 3x ◦ (z · y) − (z · x) ◦ y + (z · y) ◦ x = 0,

(2.96)

.

.

for all x, y, z ∈ A. Proposition 2.37 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (−L∗· , L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), then (A, ·, ◦) is a TCPD algebra. Conversely, let (A, ·, ◦) be a TCPD algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗· , L∗◦ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible TCPD algebra of (A, ·, [−, −]). Proof Since (−L∗· , L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.94)–(2.95). Thus for all x, y, z ∈ A, we have 0 = 2z · [x, y] − [z · x, y] − [x, z · y]

.

(2.95)

= x · (y ◦ z) − y · (x ◦ z) − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x

(2.94)

= 3y ◦ (z · x) − 3x ◦ (z · y) − (z · x) ◦ y + (z · y) ◦ x.

Hence Eq. (2.96) holds, and thus (A, ·, ◦) is a TCPD algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.19 Let A be a vector space with two bilinear operations ·, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, ·, ◦) is a TCPD algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗· , L∗◦ , A∗ ), where [−, −] is defined by Eq. (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.92) and the Lie bracket [−, −] is defined by Eq. (2.85).

2.5.2 Anti-Pre-Lie-Poisson Algebras or TCAD Algebras Definition 2.28 ([30]) An anti-pre-Lie Poisson algebra or a TCAD algebra is a triple (A, ·, ◦), such that (A, ·) is a commutative associative algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.94) and the following equation hold: 2(x ◦ y) · z − 2(y ◦ x) · z = y · (x ◦ z) − x · (y ◦ z), ∀x, y, z ∈ A.

.

(2.97)

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Proposition 2.38 ([30]) Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (−L∗· , −L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), then (A, ·, ◦) is an anti-pre-Lie Poisson algebra. Conversely, let (A, ·, ◦) be an anti-pre-Lie Poisson algebra and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗· , −L∗◦ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible anti-pre-Lie Poisson algebra of (A, ·, [−, −]). Example 2.2 ([30]) Let (A, ·) be a commutative associative algebra with a derivation P . Then there is an anti-pre-Lie algebra (A, ◦) defined by x ◦ y = P (x · y) + P (x) · y, ∀x, y ∈ A.

.

(2.98)

Moreover, (A, ·, ◦) is an anti-pre-Lie Poisson algebra and for the sub-adjacent transposed Poisson algebra (A, ·, [−, −]), the following equation holds: [x, y] = P (x) · y − x · P (y), ∀x, y ∈ A.

.

(2.99)

Moreover we get the following conclusion. Corollary 2.20 Let A be a vector space with two bilinear operations ·, ◦ : A⊗A → A. Then the following conditions are equivalent: 1. (A, ·, ◦) is an anti-pre-Lie Poisson algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗· , −L∗◦ , A∗ ), where [−, −] is defined by Eq. (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ , in which the commutative associative product · is defined by Eq. (2.92) and the Lie bracket [−, −] is defined by Eq. (2.86). Proposition 2.39 ([30]) Let (A.·, [−, −]) be a transposed Poisson algebra. Suppose that there is a nondegenerate symmetric bilinear from B on A such that it is invariant on (A, ·) and a commutative 2-cocycle on (A, [−, −]). Then there is a compatible anti-pre-Lie Poisson algebra (A, ·, ◦) in which ◦ is defined by Eq. (2.23). Conversely, let (A, ·, ◦) be an anti-pre-Lie Poisson algebra and the sub-adjacent transposed Poisson algebra be (A, ·, [−, −]). Then there is a transposed Poisson algebra AX−L∗· ,−L∗◦ A∗ , and the natural nondegenerate symmetric bilinear form Bd defined by Eq. (2.15) is invariant on the commutative associative algebra AX−L∗· A∗ and a commutative 2-cocycle on the Lie algebra A X−L∗◦ A∗ .

2.5.3 TZLD Algebras Definition 2.29 A TZLD algebra is a triple (A, *, [−, −]), such that (A, *) is a Zinbiel algebra, (A, [−, −]) is a Lie algebra, and the following equations hold:

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2[y, x * z] = [x * y + y * x, z] + x * [y, z],

(2.100)

2[x, y] * z = x * [y, z] − y * [x, z],

(2.101)

.

.

2z * [x, y] + [y, x * z + z * x] − [x, y * z + z * y] = 0,

.

(2.102)

for all x, y, z ∈ A. Proposition 2.40 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, *) be a compatible Zinbiel algebra of (A, ·). If (−L∗* , ad∗ , A∗ ) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a TZLD algebra. Conversely, let (A, *, [−, −]) be a TZLD algebra and (A, ·) be the sub-adjacent commutative associative algebra of (A, *). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗* , ad∗ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, [−, −]), and (A, *, [−, −]) is a compatible TZLD algebra of (A, ·, [−, −]). Proof Since (−L∗* , ad∗ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.100) and (2.101). By Eq. (2.100), we have x * [y, z] − y * [x, z] = 2[y, x * z] − 2[x, y * z], ∀x, y, z ∈ A.

.

(2.103)

Thus for all x, y, z ∈ A, we have 0 = 2z · [x, y] − [z · x, y] − [x, z · y]

.

(2.100),(2.101)

=

x * [y, z] − y * [x, z] + 2z * [x, y]

+z * [x, y] − 2[x, z * y] − z * [y, x] + 2[y, z * x] = x * [y, z] − y * [x, z] + 4z * [x, y] − 2[x, z * y] + 2[y, z * x] (2.103)

= 4z * [x, y] + 2[y, x * z + z * x] − 2[x, y * z + z * y].

Hence Eq. (2.103) holds, and thus (A, *, [−, −]) is a TZLD algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.21 Let A be a vector space with two bilinear operations ·, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, [−, −]) is a TZLD algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗* , ad∗ , A∗ ), where · is defined by Eq. (2.27). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.87) and the Lie bracket [−, −] is defined by Eq. (2.93).

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2.5.4 TZPD Algebras Definition 2.30 A TZPD algebra is a triple (A, *, ◦), such that (A, *) is a Zinbiel algebra, (A, ◦) is a pre-Lie algebra, and the following equations hold: .

2y ◦ (x * z) = (x * y + y * x) ◦ z + x * (y ◦ z),

(2.104)

2(x ◦ y − y ◦ x) * z = x * (y ◦ z) − y * (x ◦ z),

(2.105)

y ◦ (x * z + z * x) − x ◦ (y * z + z * y) + z * (x ◦ y − y ◦ x) = 0,

(2.106)

.

.

for all x, y, z ∈ A. Proposition 2.41 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible Zinbiel algebra of (A, ·) and (A, ◦) be a compatible pre-Lie algebra of (A, [−, −]). If (−L∗* , L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a TZPD algebra. Conversely, let (A, *, ◦) be a TZPD algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the subadjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗* , L∗◦ , A∗ ). In this case, we say (A, ·, [−, −]) is the subadjacent transposed Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible TZPD algebra of (A, ·, [−, −]). Proof Since (−L∗* , L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.104)–(2.105). By Eq. (2.104), we have x * (y ◦ z) − y * (x ◦ z) = 2y ◦ (x * z) − 2x ◦ (y * z), ∀x, y, z ∈ A.

.

(2.107)

Thus for all x, y, z ∈ A, we have 0 = 2z · [x, y] − [z · x, y] − [x, z · y]

.

= 2[x, y] * z + 2z * [x, y] − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x (2.105)

= x * (y ◦ z) − y * (x ◦ z) + 2z * (x ◦ y) − 2z * (y ◦ x) − (z * x) ◦ y

−(x * z) ◦ y + y ◦ (z * x) + y ◦ (x * z) − x ◦ (z * y) − x ◦ (y * z) +(y * z) ◦ x + (z * y) ◦ x (2.104),(2.107)

=

3y ◦ (x * z + z * x) − 3x ◦ (y * z + z * y) + 3z * (x ◦ y − y ◦ x).

Hence Eq. (2.106) holds, and thus (A, *, ◦) is a TZPD algebra. The converse part is u n proved similarly. Hence we get the following conclusion.

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Corollary 2.22 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TZPD algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗* , L∗◦ , A∗ ), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.87) and the Lie bracket [−, −] is defined by Eq. (2.85). Proposition 2.42 Let (A, ·, [−, −]) be a transposed Poisson algebra. Suppose that B is a nondegenerate skew-symmetric bilinear form on A such that it is a Connes cocycle on (A, ·) and a symplectic form on (A, [−, −]). Then there is a compatible TZPD algebra (A, *, ◦) in which * and ◦ are respectively defined by Eqs. (2.31) and (2.20). Conversely, let (A, *, ◦) be a TZPD algebra and the sub-adjacent transposed Poisson algebra be (A, ·, [−, −]). Then there is a transposed Poisson algebra A X−L∗* ,L∗◦ A∗ , and the natural nondegenerate skew-symmetric bilinear form Bp defined by Eq. (2.21) is a Connes cocycle on the commutative associative algebra A X−L∗* A∗ and a symplectic form on the Lie algebra A XL∗◦ A∗ . Proof It is similar to the proof of Proposition 2.39 given in [30].

u n

2.5.5 TZAD Algebras Definition 2.31 A TZAD algebra is a triple (A, *, ◦), such that (A, *) is a Zinbiel algebra, (A, ◦) is an anti-pre-Lie algebra, and Eq. (2.104) and the following equations hold: 2(x ◦ y − y ◦ x) * z = y * (x ◦ z) − x * (y ◦ z),

(2.108)

x ◦(y *z)−y ◦(x *z)−3x ◦(z*y)+3y ◦(z*x)+3z*(x ◦y −y ◦x) = 0,

(2.109)

.

.

for all x, y, z ∈ A. Proposition 2.43 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible Zinbiel algebra of (A, ·) and (A, ◦) be a compatible anti-pre-Lie algebra of (A, [−, −]). If (−L∗* , −L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), then (A, *, ◦) is a TZAD algebra. Conversely, let (A, *, ◦) be a TZAD algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗* , −L∗◦ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, ·, ◦), and (A, ·, ◦) is a compatible TZAD algebra of (A, ·, [−, −]).

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Proof Since (−L∗* , −L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.104) and (2.108). Thus for all x, y, z ∈ A, we have 0 = 2z · [x, y] − [z · x, y] − [x, z · y]

.

= 2[x, y] * z + 2z * [x, y] − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x (2.108)

= −x * (y ◦ z) + y * (x ◦ z) + 2z * (x ◦ y) − 2z * (y ◦ x) − (z * x) ◦ y

−(x * z) ◦ y + y ◦ (z * x) + y ◦ (x * z) − x ◦ (z * y) − x ◦ (y * z) +(y * z) ◦ x + (z * y) ◦ x (2.104),(2.107)

=

x ◦ (y * z − 3z * y) − y ◦ (x * z − 3z * x) + 3z * (x ◦ y − y ◦ x).

Hence Eq. (2.109) holds, and thus (A, *, ◦) is a TZAD algebra. The converse part u n is proved similarly. Hence we get the following conclusion. Corollary 2.23 Let A be a vector space with two bilinear operations ·, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TZAD algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (−L∗* , −L∗◦ , A∗ ), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.87) and the Lie bracket [−, −] is defined by Eq. (2.86).

2.5.6 TALD Algebras Definition 2.32 A TALD algebra is a triple (A, *, [−, −]), such that (A, *) is an anti-Zinbiel algebra, (A, [−, −]) is a Lie algebra, and Eq. (2.101) and the following equations hold: 2[y, x * z] = x * [y, z] − [x * y + y * x, z],

(2.110)

x * [y, z] − y * [x, z] + 2[x, z * y] − 2[y, z * x] = 0,

(2.111)

.

.

for all x, y, z ∈ A. Proposition 2.44 Let (A, ·, [−, −]) be a transposed Poisson algebra and (A, *) be a compatible anti-Zinbiel algebra of (A, ·). If (L∗* , ad∗ , A∗ ) is a representation of

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(A, ·, [−, −]), then (A, *, [−, −]) is a TALD algebra. Conversely, let (A, *, [−, −]) be a TALD algebra and (A, ·) be the sub-adjacent commutative associative algebra of (A, *). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L∗* , ad∗ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, [−, −]), and (A, *, [−, −]) is a compatible TALD algebra of (A, ·, [−, −]). Proof Since (L∗* , ad∗ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.101) and (2.110). Thus for all x, y, z ∈ A, we have 2z · [x, y] − [z · x, y] − [x, z · y]

.

(2.110),(2.101)

=

x * [y, z] − y * [x, z] + 2z * [x, y]

−z * [x, y] + 2[x, z * y] + z * [y, x] −2[y, z * x] = x * [y, z] − y * [x, z] + 2[x, z * y] −2[y, z * x]. Hence Eq. (2.111) holds, and thus (A, *, [−, −]) is a TALD algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.24 Let A be a vector space with two bilinear operations *, [−, −] : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, [−, −]) is a TALD algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L∗* , ad∗ , A∗ ), where · is defined by Eq. (2.27). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.88) and the Lie bracket [−, −] is defined by Eq. (2.93). Proposition 2.45 Let (A, ·, [−, −]) be a transposed Poisson algebra. Suppose that B is a nondegenerate symmetric bilinear form on A such that it is a commutative Connes cocycle on (A, ·) and invariant on (A, [−, −]). Then there is a compatible TALD algebra (A, *, [−, −]) in which * is defined by Eq. (2.34). Conversely, let (A, *, [−, −]) be a TALD algebra and the sub-adjacent transposed Poisson algebra be (A, ·, [−, −]). Then there is a transposed Poisson algebra A XL∗* ,ad∗ A∗ , and the natural nondegenerate symmetric bilinear form Bd defined by Eq. (2.15) is a commutative Connes cocycle on the commutative associative algebra A XL∗* A∗ and invariant on the Lie algebra A Xad∗ A∗ . Proof It is similar to the proof of Proposition 2.39 given in [30].

u n

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2.5.7 TAPD Algebras Definition 2.33 A TAPD algebra is a triple (A, *, ◦), such that (A, *) is an antiZinbiel algebra, (A, ◦) is a pre-Lie algebra, and Eq. (2.105) and the following equations hold: 2y ◦ (x * z) = x * (y ◦ z) − (x * y + y * x) ◦ z,

(2.112)

x ◦(z *y) −y ◦(z *x) +z *(x ◦y −y ◦x) +3y ◦(x *z) −3x ◦(y *z) = 0,

(2.113)

.

.

for all x, y, z ∈ A. Proposition 2.46 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible anti-Zinbiel algebra of (A, ·) and (A, ◦) be a compatible preLie algebra of (A, [−, −]). If (L∗* , L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a TAPD algebra. Conversely, let (A, *, ◦) be a TAPD algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L∗* , L∗◦ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible TAPD algebra of (A, ·, [−, −]). Proof Since (L∗* , L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.105) and (2.112). By Eqs. (2.112) and (2.107) holds. Thus for all x, y, z ∈ A, we have 0 = 2z · [x, y] − [z · x, y] − [x, z · y]

.

= 2[x, y] * z + 2z * [x, y] − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x (2.105)

= x * (y ◦ z) − y * (x ◦ z) + 2z * (x ◦ y) − 2z * (y ◦ x) − (z * x) ◦ y

−(x * z) ◦ y + y ◦ (z * x) + y ◦ (x * z) − x ◦ (z * y) − x ◦ (y * z) +(y * z) ◦ x + (z * y) ◦ x (2.112),(2.107)

=

x ◦ (z * y − 3y * z) − y ◦ (z * x − 3x * z) + z * (x ◦ y − y ◦ x).

Hence Eq. (2.113) holds, and thus (A, *, ◦) is a TAPD algebra. The converse part is proved similarly. u n Hence we get the following conclusion. Corollary 2.25 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TAPD algebra. 2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L∗* , L∗◦ , A∗ ), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16).

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3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.88) and the Lie bracket [−, −] is defined by Eq. (2.85).

2.5.8 TAAD Algebras Definition 2.34 A TAAD algebra is a triple (A, *, ◦), such that (A, *) is an antiZinbiel algebra, (A, ◦) is an anti-pre-Lie algebra, and Eqs. (2.108), (2.112) and the following equation hold: x ◦(y *z+z*y)−y ◦(x *z+z*x)+z*(x ◦y −y ◦x) = 0, ∀x, y, z ∈ A.

.

(2.114)

Proposition 2.47 Let (A, ·, [−, −]) be a transposed Poisson algebra, (A, *) be a compatible anti-Zinbiel algebra of (A, ·) and (A, ◦) be a compatible anti-preLie algebra of (A, [−, −]). If (L∗* , −L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), then (A, *, [−, −]) is a TAAD algebra. Conversely, let (A, *, ◦) be a TAAD algebra, (A, ·) be the sub-adjacent commutative associative algebra of (A, *) and (A, [−, −]) be the sub-adjacent Lie algebra of (A, ◦). Then (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L∗* , −L∗◦ , A∗ ). In this case, we say (A, ·, [−, −]) is the sub-adjacent transposed Poisson algebra of (A, *, ◦), and (A, *, ◦) is a compatible TAAD algebra of (A, ·, [−, −]). Proof Since (L∗* , −L∗◦ , A∗ ) is a representation of (A, ·, [−, −]), we get Eqs. (2.108) and (2.112). Thus for all x, y, z ∈ A, we have 0 = 2z · [x, y] − [z · x, y] − [x, z · y]

.

= 2[x, y] * z + 2z * [x, y] − (z · x) ◦ y + y ◦ (z · x) − x ◦ (z · y) + (z · y) ◦ x (2.108)

= y * (x ◦ z) − x * (y ◦ z) + 2z * (x ◦ y) − 2z * (y ◦ x) − (z * x) ◦ y

−(x * z) ◦ y + y ◦ (z * x) + y ◦ (x * z) − x ◦ (z * y) − x ◦ (y * z) +(y * z) ◦ x + (z * y) ◦ x (2.112),(2.107)

=

x ◦ (y * z + z * y) − y ◦ (x * z + z * x) + z * (x ◦ y − y ◦ x).

Hence Eq. (2.114) holds, and thus (A, *, ◦) is a TAAD algebra. The converse part u n is proved similarly. Hence we get the following conclusion. Corollary 2.26 Let A be a vector space with two bilinear operations *, ◦ : A ⊗ A → A. Then the following conditions are equivalent: 1. (A, *, ◦) is a TAAD algebra.

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2. The triple (A, ·, [−, −]) is a transposed Poisson algebra with a representation (L∗* , −L∗◦ , A∗ ), where · and [−, −] are respectively defined by Eqs. (2.27) and (2.16). 3. There is a transposed Poisson algebra structure on A ⊕ A∗ in which the commutative associative product · is defined by Eq. (2.88) and the Lie bracket [−, −] is defined by Eq. (2.86). Proposition 2.48 Let (A, ·, [−, −]) be a transposed Poisson algebra. Suppose that B is a nondegenerate symmetric bilinear form on A such that it is a commutative Connes cocycle on (A, ·) and a commutative 2-cocycle on (A, [−, −]). Then there is a compatible TAAD algebra (A, *, ◦) in which * and ◦ are respectively defined by Eqs. (2.34) and (2.23) . Conversely, let (A, *, ◦) be a TAAD algebra and the subadjacent transposed Poisson algebra be (A, ·, [−, −]). Then there is a transposed Poisson algebra A XL∗* ,−L∗◦ A∗ , and the natural nondegenerate symmetric bilinear form Bd defined by Eq. (2.15) is a commutative Connes cocycle on the commutative associative algebra A XL∗* A∗ and a commutative 2-cocycle on the Lie algebra A X−L∗◦ A∗ . Proof It is similar to the proof of Proposition 2.39 given in [30].

u n

2.5.9 Summary We summarize some facts on the 8 algebraic structures in the previous subsections respectively corresponding to the mixed splittings of operations of transposed Poisson algebras in terms of the representations of transposed Poisson algebras on the dual spaces in Table 2.3.

Table 2.3 Splittings of transposed Poisson algebras on dual spaces Algebras

Notations

TCPD TCAD

.(A, ·, ◦) .(A, ·, ◦)

Representations of transposed Poisson algebras on the dual spaces ∗ ∗ ∗ .(−L· , L◦ , A ) ∗ ∗ ∗ .(−L· , −L◦ , A )

TZLD TZPD

.(A, *, [−, −])

.(−L* , ad

.(A, *, ◦)

.(−L* , L◦ , A

TZAD TALD

.(A, *, ◦)

.(−L* , −L◦ , A

TAPD TAAD

.(A, *, ◦)

.(A, *, [−, −])

.(A, *, ◦)

∗

∗

∗

∗

∗

, A∗ ) ∗) ∗

∗)

∗ ∗ ∗ .(L* , ad , A ) ∗

∗

.(L* , L◦ , A

∗)

∗ ∗ ∗ .(L* , −L◦ , A )

Corresponding nondegenerate bilinear forms on transposed Poisson algebras – invariant, commutative 2-cocycle – Connes cocycle, symplectic form – commutative Connes cocycle, invariant – commutative Connes cocycle, commutative 2-cocycle

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Acknowledgments This work is partially supported by NSFC (11931009, 12271265, 12261131498), the Fundamental Research Funds for the Central Universities and Nankai Zhide Foundation. The authors thank Professor Li Guo for valuable discussion.

References 1. M. Aguiar, Pre-Poisson Algebras, Lett. Math. Phys. 54 (2000) 263–277. 2. C. Bai, Double constructions of Frobenius, Connes cocycles and their duality, J. Noncommut. Geom. 4 (2010) 475–530. 3. C. Bai, An introduction to pre-Lie algebras, in: Algebra and Applications 1: Nonssociative Algebras and Categories, Wiley Online Library (2021) 245–273. 4. C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operations, Manin products and Rota-Baxter operators, Int. Math. Res. Not. 2013 (2013) 485–524. 5. C. Bai, R. Bai, L. Guo and Y. Wu, Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras, J. Algebra 632 (2023) 535–566. 6. B. Bakalov and V. Kac, Field algebras, Int. Math. Res. Not. 2003 (2003) 123–159. 7. P. D. Beites, B. L. M. Ferreira and I. Kaygorodov, Transposed Poisson structures, arXiv: 2207.00281. 8. K. H. Bhaskara and K. Viswanath, Poisson Algebras and Poisson Manifolds, Pitman Res. Notes Math. Ser. 174, Longman Scientific & Technical, Harlow (1988). 9. D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (2006) 323–357. 10. F. Chapoton, Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces, J. Pure Appl. Algebra 168 (2002) 1–18. 11. F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001 (2001), 395–408. 12. B. Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math.Soc. 197 (1974) 145–159. 13. A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math, Phys. 199 (1998), 395–408. 14. A. Dzhumadil’daev and P. Zusmanovich, Commutative 2-cocycles on Lie algebras, J. Algebra 324 (2010) 732–748. 15. B. L. M. Ferreira, I. Kaygorodov and V. Lopatkin, 12 -derivations of Lie algebras and transposed Poisson algebras, RACSAM 115 (2021) 142. 16. V. T. Filippov, Lie algebras satisfying identities of degree 5, Algebra and Logic 34 (1996) 379–394. 17. L. Foissy, Les algèbres de Hopf des arbres enracinés décorés II, Bull. Sci. Math. 126 (2002) 249–288. 18. A. Frabetti, Dialgebra homology of associative algebras, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 135–140. 19. A. Frabetti, Leibniz homology of dialgebras of matrices, J. Pure Appl. Algebra 129 (1998) 123–141. 20. D. Gao, G. Liu and C. Bai, Anti-dendriform algebras, new splitting of operations and Novikov type algebras, arXiv:2209.08962. 21. M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963) 267– 288. 22. R. Holtkamp, Comparison of Hopf algebras on trees, Arch. Math. (Basel) 80 (2003) 368–383. 23. R. Holtkamp, On Hopf algebra structures over free operads, Adv. Math. 207 (2006) 544–565. 24. J.-L. Koszul, Domaines bornés homogènes et orbites de groupes de transformation affines, Bull. Soc. Math. France 89 (1961) 515–533. 25. B. A. Kupershmidt, Non-abelian phase spaces, J. Phys. A: Math. Gen. 27 (1994) 2801–2810.

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26. I. Laraiedh and S. Silvestrov, Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras, arXiv:2106.03277 27. A. Lichnerowicz, Les variétiés de Poisson et leurs algèbras de Lie associées (French), J. Diff. Geom. 12 (1977) 253–300. 28. A. Lichnerowicz, Les variétiés de Jacobi et leurs algèbras de Lie associées, J. Math. Pures Appl. 57 (1978) 453–488. 29. A. Lichnerowicz and A. Medina, On Lie groups with left-invariant symplectic or Kählerian structures, Lett. Math. Phys. 16 (1988), 225–235. 30. G. Liu and C. Bai, Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras, J. Algebra 609 (2022) 337–379. 31. J.-L. Loday, Cup product for Leibniz cohomology and dual Leibniz algebras, in: Math. Scand. Vol.77, Univ. Louis Pasteur, Strasbourg, 1995, pp. 189–196. 32. J.-L. Loday, Arithmetree, J. Algebra 258 (2002) 275–309. 33. J.-L. Loday, Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr. 9, Soc. Math. France, Paris (2004) 155–172. 34. J.-L. Loday and M. Ronco, Order structure on the algebra of permutations and of planar binary trees, J. Algebraic Combin. 15 (2002) 253–270. 35. J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, Comtep. Math. 346 (2004) 369–398. 36. X. Ni and C. Bai, Poisson bialgebras, J. Math. Phys. 54 (2013) 023515. 37. M. Ronco, Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra 254 (2002) 152–172. 38. E. B. Vinberg, Convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963) 340–403. 39. A. Weinstein, Lecture on Symplectic Manifolds, CBMS Regional Conference Series in Mathematics 29, Amer. Math. Soc., Providence, R.I., 1979. 40. X. Xu, Novikov-Poisson algebras, J. Algebra 190 (1997) 253–279. 41. L. Yuan and Q. Hua, 12 -(bi)derivations and transposed Poisson algebra structures on Lie algebras, Linear Multilinear Algebra 70 (2022) 7672–7701.

Chapter 3

Some Varieties of Loops (Bol-Moufang and Non-Bol-Moufang Types) Adewale Roland Tunde Sòlárìn, John Olusola Adéníran, Tèmítópé Gbóláhàn Jaiyéo.lá, Abednego Orobosa Isere, and Yakub Tunde Oyebo

3.1 Bol Loops and Their Constructions 3.1.1 Groupoids, Quasigroups and Loops Quasigroups and loops are studied in four research areas; algebra, geometry, topology and combinatorics. We shall be discussing them in the direction of algebra. Let G be a non-empty set. Define a binary operation “.·” on G. If .x · y ∈ G for all .x, y ∈ G, then the pair .(G, ·) or .G(·) or G is called a groupoid or Magma. If each of the equations: a·x =b

.

and

y·a =b

(3.1)

A. R. T. Sòlárìn (O) National Mathematical Centre, Federal Capital Territory, Abuja, Nigeria J. O. Adéníran Federal University of Agriculture, Abeokuta, Nigeria e-mail: [email protected] T. G. Jaiyéo.lá Obafemi Awolowo University, Ile Ife, Nigeria e-mail: [email protected] A. O. Isere Ambrose Alli University, Ekpoma, Nigeria e-mail: [email protected] Y. T. Oyebo Lagos State University, Ojo, Nigeria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_3

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has unique solutions in G for x and y respectively, then .(G, ·) is called a quasigroup. Hence, there exist the maps: left translation (multiplication) and right translation (multiplication) .La : x |→ a · x and .Ra : y |→ y · a respectively for any .x, y ∈ G. A quasigroup is therefore an algebraic structure having a binary multiplication .x· y usually written xy which satisfies the conditions that for any .a, b in the quasigroup the equations in (3.1) have unique solutions for each x and y lying in the quasigroup. If there exists a unique element .e ∈ G such that for all .x ∈ G, .x · e = x = e · x, called the identity element, then .(G, ·) is called a loop. We write xy instead of .x · y, and stipulate that “.·” has lower priority than juxtaposition among factors to be multiplied. For instance, .x · yz stands for .x(yz). Let .(G, ·) be a quasigroup. Let .x ∈ G, and let .e1 , e2 ∈ G such that .x · e1 = x = e2 · x. So, .xe12 = xe1 = x, from which we get .e12 = e1 . Let .a ∈ G be such that 2 .e1 a = x. Then .e2 e1 a = e2 x = x = e1 a, so that .e2 e1 = e1 = e , or .e2 = e1 . Set 1 2 2 .e = e1 . For any .y ∈ G, we have .ey = e y, so .y = ey. Similarly, .ye = ye implies .y = ye. This shows the existence of an identity element .e ∈ G that is unique. Thus, an associative quasigroup is a loop. Furthermore, for every .∈ G, there are unique elements y and z such that .xy = zx = e. Hence, .y = ey = (zx)y = z(xy) = ze = z. This shows that x has a unique two-sided inverse .x −1 = y = z. Therefore, .(G, ·) is a group.

A group is a quasigroup and a loop. But the converse is not necessarily true. However, if a quasigroup is associative, then it will be a loop and hence a group. Thus, it is at times informally said that a loop is a group without associativity.

It can now be seen that a groupoid .(G, ·) is a quasigroup if its left and right translation mappings are bijections or permutations. Since the left and right translation mappings of a loop are bijective, then the inverse mappings .L−1 x and −1 exist. Let .Rx x\y = yL−1 x = yLx

.

and

x\y = z ⇐⇒ x · z = y

x/y = xRy−1 = xRy and note that and

x/y = z ⇐⇒ z · y = x.

Note: .R does not necessarily mean the set of real numbers in this earlier expression and Sect. 3.3 of this chapter. Hence, .(G, \) and .(G, /) are also quasigroups. Using the operations “.\” and “./”, the definition of a loop can be stated as follows. Definition 3.1 A loop .(G, ·, /, \, e) is a set G together with three binary operations “.·”, “./”, “.\” and one nullary operation e such that (i) .x · (x\y) = y, .(y/x) · x = y for all .x, y ∈ G,

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(ii) .x\(x · y) = y, .(y · x)/x = y for all .x, y ∈ G and (iii) .x\x = y/y or .e · x = x = x · e for all .x, y ∈ G. We also stipulate that “./” and “.\” have higher priority than “.·” among factors to be multiplied. For instance, .x · y/z and .x · y\z stand for .x · (y/z) and .x · (y\z) respectively. In a loop .(G, ·) with identity element e, the left inverse element of .x ∈ G is the element .xJλ = x λ ∈ G and the right inverse element of .x ∈ G is the element ρ .xJρ = x ∈ G such that xλ · x = e

.

and

x · xρ = e

respectively

Jλ and .Jρ are respectively called the left and right inverse maps. All mappings are assumed to be single-valued. If T is a mapping of a set G into itself or some other set and if .x ∈ G, then xT denotes the unique image of x under T .

.

Definition 3.2 A set .|| of permutations on a set G is the representation of a loop (G, ·) if and only if

.

(i) .I ∈ || (identity mapping), (ii) .|| is transitive on G (i.e. for all .x, y ∈ G, there exists a unique .π ∈ || such that .xπ = y), (iii) if .α, β ∈ || and .αβ −1 fixes one element of G, then .α = β. The left and right representation of a loop G is denoted by ||λ (G, ·) = ||λ (G)

.

and

||ρ (G, ·) = ||ρ (G)

respectively.

We shall be discussing the construction of some Bol-Moufang type of loops (with the aid of their representations) in Sects. 3.1 and 3.2 as well as the construction of some non-Bol-Moufang type of loops (without the aid of their representations) in Sect. 3.3. Some basic text books on quasigroup and loop theory, and their applications are Pflugfelder [137], Bruck [23], Chein, Pflugfelder and Smith [31], Dene and Keedwell [37], Goodaire, Jespers and Milies [48], Vasantha Kandasamy [169], Jaiyéo.lá [73] and Shcherbacov [154].

3.1.2 Some Important Subloops of a Loop Let .(G, ·) be a loop. A non-empty subset H of G is a subloop of .(G, ·) if .(H, ·) is a loop. The left nucleus of G is defined and denoted by Nλ (G, ·) = {a ∈ G : ax · y = a · xy ∀ x, y ∈ G}.

.

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The right nucleus of G is defined and denoted by Nρ (G, ·) = {a ∈ G : y · xa = yx · a ∀ x, y ∈ G}.

.

The middle nucleus of G is defined and denoted by Nμ (G, ·) = {a ∈ G : ya · x = y · ax ∀ x, y ∈ G}.

.

The nucleus of G is defined and denoted by N(G, ·) = Nλ (G, ·) ∩ Nρ (G, ·) ∩ Nμ (G, ·).

.

Each of these four subsets of G, i.e. .Nλ (G, ·), .Nρ (G, ·), .Nμ (G, ·), and .N(G, ·), are subloops of .(G, ·) and are often called the nuclei of .(G, ·). The centrum of G is defined and denoted by .C(G, ·) = {a ∈ G : ax = xa ∀ x ∈ G}. The center of G denoted by .Z(G, ·) = N(G, ·) ∩ C(G, ·). Let .a, b and c be three elements of a loop G. The loop commutator of a and b and the loop associator of .a, b and c are the unique elements .(a, b) and .(a, b, c) of G which satisfy ab = (ba) · (a, b)

.

and

(ab)c = a(bc) · (a, b, c)

respectively

If .X, Y, and Z are all non-empty subsets of a loop G, we denote by .(X, Y ) and (X, Y, Z), respectively, the set of all commutators of the form .(x, y) and all the associators of the form .(x, y, z), where .x ∈ X, y ∈ Y, z ∈ Z.

.

The nuclei of a group is the group itself. The associators of elements of a group are trivial. However, the commutators of elements of a group are not necessarily trivial except the group is abelian. In a group, the identity −1 = (b, a) is true. But this is not necessarily the case an arbitrary loop. .(a, b)

3.1.3 Some Important Groups of a Loop The symmetric group of a loop .(G, ·) is defined and denoted by SY M(G) = {U : G → G | U is a permutation or a bijection}.

.

In what follows, “.≤” will mean “subgroup of”. The set / \ Multλ (G, ·) = {Lx , L−1 : x ∈ G} x

.

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is called the left multiplication group of .(G, ·) and .Multλ (G, ·) ≤ SY M(G). The set / \ −1 .Multρ (G, ·) = {Rx , Rx : x ∈ G} is called the right multiplication group of .(G, ·) and .Multρ (G, ·) ≤ SY M(G). The set / \ −1 −1 .Mult(G, ·) = {Rx , Rx , Lx , Lx : x ∈ G} is called the multiplication group of .(G, ·) and .Mult(G, ·) ≤ SY M(G). In fact, Multλ (G, ·), Multρ (G, ·) ≤ Mult(G, ·) ≤ SY M(G).

.

The product of right (left) translations of elements of a group are right (left) translations. Hence, the right (left) multiplication group of a group is a group of translations.

3.1.3.1

Inner Mappings of a Loop

If .eα = e in a loop G such that .α ∈ Mult(G), then .α is called an inner mapping and they form a group .Inn(G) called the inner mapping group. The right, left and middle inner mappings −1 −1 R(x, y) = Rx Ry Rxy , L(x, y) = Lx Ly L−1 yx and T (x) = Rx Lx

.

respectively generate the left inner mapping group .Innλ (G), right inner mapping group .Innρ (G) and the middle inner mapping .Innμ (G). In fact, it has been shown that / \ .Inn(G) = R(x, y), L(x, y), T (x) | x, y ∈ G . But this was later improved to / \ / \ Inn(G) = L(x, y), T (x) | x, y ∈ G = R(x, y), T (x) | x, y ∈ G .

.

If .Innλ (G) ≤ AU M(G), Innρ (G) ≤ AU M(G), Innμ (G) ≤ AU M(G) and Inn(G) ≤ AU M(G) where .AU M(G) denotes the automorphism group of G. Then G is called a left A-loop(A.λ -loop), right A-loop(A.ρ -loop), middle A-loop(A.μ -loop) and A-loop respectively where A-loop means automorphism loop.

.

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The left and right inner mappings of a group are trivial. The middle inner mapping of a group is trivial if and only if the group is abelian. Hence, the inner mapping group of a group is the group generated by its middle inner mappings. A group is not necessarily an A-loop. An abelian group is an Aloop.

The structure of the groups of loops discussed in this subsection (and some others not mentioned here) have been found to be important for the structure of the loops (see Oyebo et al. [136]). Some works in the direction of structure of A-loops can be found in [19, 90, 122, 123, 153, 165, 170].

3.1.4 Basic Quasigroup and Loop Properties For associative binary systems, the concept of an inverse element is only meaningful if the system has an identity element. For example, in a group .(G, ·) with identity element .e ∈ G, if .x ∈ G then the inverse element for x is the element .x −1 ∈ G such that x · x −1 = x −1 · x = e.

.

In a loop .(G, ·) with identity element e, the left inverse and right inverse elements of x ∈ G are the respective elements .x λ ∈ G and .x ρ ∈ G as defined in Sect. 3.1.1. If −1 = x λ = x ρ . .Jλ = Jρ , then we simply write .J = Jλ = Jρ and as well we write .x Note that .Jλ Jρ = Jρ Jλ = I . In case .(G, ·) is a quasigroup, then .(G, ·) is called a left inverse property quasigroup (LIPQ) if it has the left inverse property (LIP) i.e. if there exists a bijection .

Jλ : x |→ x λ on G such that x λ · xy = y.

.

Similarly, .(G, ·) is called a right inverse property quasigroup (RIPQ) if it has the right inverse property (RIP) i.e. if there exists a bijection Jρ : x |→ x ρ on G such that yx · x ρ = y.

.

A quasigroup that is both a LIPQ and a RIPQ is said to have the inverse property (IP) hence called an inverse property quasigroup (IPQ). The same definitions hold for a loop and such a loop is called a left inverse property loop (LIPL), right inverse property loop (RIPL) and inverse property loop (IPL) accordingly.

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A group has both the left and right inverse properties.

3.1.4.1

Variations of Inverse Properties

There are some classes of loops which do not have the inverse property but have properties which can be considered as variations of the inverse property. A loop .(G, ·) is called a weak inverse property loop (WIPL) if it obeys the identity x(yx)ρ = y ρ

or

.

(xy)λ x = y λ

(3.2)

for all .x, y ∈ G. A loop .(G, ·) is called a cross inverse property loop (CIPL) if it obeys the identity xy · x ρ = y

.

x · yx ρ = y

or

or

x λ · (yx) = y

or

xλy · x = y

(3.3)

for all .x, y, ∈ G. For a WIPL, the four nuclei .N, Nλ , Nρ , Nμ . coincide, i.e. .N = Nλ = Nρ = Nμ . The same is true for a CIPL and an IPL since they are WIPLs.

A loop .(G, ·) is called an automorphic inverse property loop (AIPL) if it obeys the identity (xy)ρ = x ρ y ρ

or

.

(xy)λ = x λ y λ

(3.4)

for all .x, y, ∈ G. A loop .(G, ·) is called an anti-automorphic inverse property loop (AAIPL) if it obeys the identity (xy)ρ = y ρ x ρ

or

.

(xy)λ = y λ x λ

(3.5)

for all .x, y, ∈ G. A loop .(G, ·) is called a semi-automorphic inverse property loop (SAIPL) if it obeys the identity (xy · x)ρ = x ρ y ρ · x ρ

.

for all .x, y, ∈ G.

or

(xy · x)λ = x λ y λ · x λ

(3.6)

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A group has weak inverse property and anti-automorphic inverse property. But a group does not necessarily have the cross inverse property loop, automorphic inverse property, semi-automorphic inverse property. But an abelian group has them.

Jaiyéo.lá and Effiong [93] studied the variation of inverse properties in Basarab loops (cf. Definition 3.11) while Keedwell and Shcherbacov [109–111], Oyebo et al. [134] have studied their generalizations (m-quasigroup and .(r, s, t)-quasigroup). Some of the classes of loops with variation in inverse property (especially cross and weak inverse properties) have been found to be applicable to cryptography by Keedwell [108], Jaiyéo.lá [74, 77, 86, 87], Jaiyéo.lá and Adéníran [89], Jaiyéo.lá and Smarandache [92], Ilemobade et al. [53]. 3.1.4.2

Some Weak-Associative Laws

Quasigroup and loops are known to lack associativity. But some quasigroups and loops obey identities that are called weak-associative laws. Among such identities is the inverse property. Other weak-associative laws shall be introduced under quasigroup and loop varieties and identities. A quasigroup .(G, ·) is called a left alternative property quasigroup (LAPQ) if the left alternative property (LAP), xx · y = x · xy

.

holds for all .x, y ∈ G. A quasigroup .(G, ·) is called a right alternative property quasigroup (RAPQ) if the right alternative property (RAP), y · xx = yx · x

.

holds for all .x, y ∈ G. A quasigroup .(G, ·) is called an alternative property quasigroup (APQ) if both LAP and RAP hold. The same definitions hold for a loop and such a loop is called a left alternative property loop (LAPL), right alternative property loop (RAPL) and alternative property loop (APL) accordingly. A loop .(G, ·) is called a flexible or elastic loop if the flexibility or elasticity property xy · x = x · yx

.

holds for all .x, y ∈ G.

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.(G, ·) is said to be a power associative loop if .< x > is a subgroup for all .x ∈ G and a diassociative loop if .< x, y > is a subgroup for all .x, y ∈ G. A subloop of G is monogenic if it is generated by one element. Below are generalizations of left and right alternative properties.

Definition 3.3 Let .(G, ·) be a loop. 1. .(G, ·) is said to be a left power alternative loop if .Lx n = Lx ◦ Lx ◦ · · · ◦ Lx = ' '' ' n−times

Lnx for all .x ∈ G and .n ∈ Z+ . 2. .(G, ·) is said to be a right power alternative loop if .Rx n = Rx ◦ Rx ◦ · · · ◦ Rx = ' '' ' n−times

Rxn for all .x ∈ G and .n ∈ Z+ . 3. .(G, ·) is said to be a power alternative loop if it is both left and right power alternative.

A group is both power associative loop, dissociative loop and power alternative. A cyclic group is monogenic. A diassociative loop is both power associative and power alternative. But a power associative loop is not necessarily diassociative or power alternative.

Recall that any finite group obeys the

Lagrange’s Theorem The order of any subgroup of a finite group divides the order of the group. This is not necessarily true for a finite loop.

This necessitated the introduction of other forms of the Lagrange property. Definition 3.4 Let G be a loop. 1. If G is finite, then it is said to have the weak Lagrange property (weak monogenic Lagrange property) if the order of any subloop (monogenic subloop) of G divides the order of G. 2. If G is finite, then it is said to have the strong Lagrange property (strong monogenic Lagrange property) if every subloop of G has the weak Lagrange property (weak monogenic Lagrange property). 3. If G is power associative and finite then it is said to have the weak Cauchy property if for any prime p dividing the order of the loop, there is an element of order p.

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3.1.5 Autotopisms of a Quasigroup and Some Associated Groups Let .(G, ·) be a quasigroup. The triple .(U, V , W ) such that .U, V , W ∈ SY M(G) is called an autotopism of G if xU · yV = (x · y)W ∀ x, y ∈ G.

.

Under component-wise multiplication, the set of such autotopisms forms a group called the autotopism group of .(G, ·) and is denoted by .AU T (G, ·). For any .(U, V , W ), .(U1 , V1 , W1 ), .(U2 , V2 , W2 ) ∈ AU T (G, ·), we have (U1 , V1 , W1 )(U2 , V2 , W2 ) = (U1 U2 , V1 V2 , W1 W2 )

.

and (U, V , W )−1 = (U −1 , V −1 , W −1 ).

.

Note that the identity element of this group is .(I, I, I ) where .xI = x for all .x ∈ G. Let .(G, ·) be a loop. A mapping .U ∈ SY M(G) is called a right pseudoautomorphism or left pseudo-automorphism of .(G, ·) with companion c if .(U, U Rc , .U Rc ) ∈ AU T (G, ·) or .(U Lc , U, U Lc ) ∈ AU T (G, ·). The set of such permutations forms a group called the right pseudo-automorphism group or left pseudo-automorphism group of is .(G, ·) and is denoted by .P Sρ (G, ·) or .P Sλ (G, ·). Let .(G, ·) be a loop. If .U ∈ SY M(G) such that .(U, U, U ) ∈ AU T (G, ·), then U is called an automorphism. The group of automorphisms on G is denoted by .AU M(G, ·). Note that .AU M(G, ·) ≤ P Sρ (G, ·), P Sλ (G, ·).

In a group, every right (left) pseudo-automorphism is an automorphism.

Definition 3.5 Let the set BS(G, ·) = {α ∈ SY M(G) : (αRg−1 , αL−1 f , α) ∈ AU T (G, ·) f or some f, g ∈ G}

.

With the composition of maps defined on it, .BS(G, ·) forms a group called the Bryant-Schneider group of the .(G, ·). Definition 3.6 Let .(G, ·) be a loop. The loop .(H, ◦) given by .H = A(G, ·) × G where (α, x) ◦ (β, y) = (αβ, xβ · y)

.

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and .A(G, ·) ≤ AU M(G, ·) for all .(α, x), (β, y) ∈ H is called the .A(G, ·)holomorph of .(G, ·). Definition 3.7 Let .(G, ◦) be a group and .(G, ·) be a loop. .(G, ·) is said to be embeddable in the group .(G, ◦) if there exists a subset .H ⊂ G such that .(G, ·) ∼ = (H, ∗).

3.1.6 Quasigroups and Loops of Bol-Moufang Types Fenyves [42, 43] in the 1960s were the first to classify loops of Bol-Moufang type by showing that there are sixty of them among which thirty are equivalent to the associativity law. In the beginning of this twentyfirst century, Phillips and Vojtˇechovský [140] and [141] generalized and completed the study of Fenyves by showing that there are sixteen varieties of quasigroups and fourteen varieties of loops of Bol-Moufang type. The identities describing the most popular quasigroups and loops of Bol-Moufang are highlighted below. For some particular varieties, the identity or identities named after them are equivalent to each other in quasigroups or in loops or in both quasigroups and loops. (yx · x)z = y(x · xz)

central identity.

(3.7)

(xy · z)x = x(y · zx)

extra identity.

(3.8)

xy · xz = x(yx · z)

extra identity.

(3.9)

yx · zx = (y · xz)x

extra identity.

(3.10)

xx · yz = (x · xy)z

left central identity.

(3.11)

(x · xy)z = x(x · yz)

left central identity.

(3.12)

(xx · y)z = x(x · yz)

left central identity.

(3.13)

(y · xx)z = y(x · xz)

left central identity.

(3.14)

yz · xx = y(zx · x)

right central identity.

(3.15)

(yz · x)x = y(zx · x)

right central identity.

(3.16)

(yz · x)x = y(z · xx)

right central identity.

(3.17)

(yx · x)z = y(xx · z)

right central identity.

(3.18)

xy · zx = (x · yz)x

Moufang identity.

(3.19)

xy · zx = x(yz · x)

Moufang identity.

(3.20)

(xy · x)z = x(y · xz)

Moufang identity.

(3.21)

(yx · z)x = y(x · zx)

Moufang identity.

(3.22)

.

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(x · yx)z = x(y · xz)

left Bol identity.

(3.23)

(yx · z)x = y(xz · x)

right Bol identity

(3.24)

George and Jaiyéo.lá [47] have just reported what they call second Bol-Moufang type identities in loops. These are identities of length five and four of them were found to be new loop identities which individually characterize the Moufang loop. Thus, we now have eight loop identities that characterize Moufang loop.

3.1.7 Bol Loops In any loop, there is equivalence between any two of the identities corresponding to each of the equation numbers in the triple {(3.19), (3.20), (3.21),(3.22)}. Loops that satisfy (3.19) or its equivalent forms are called Moufang loops. Identities (3.23) and (3.24) have no equivalent forms in loops unlike (3.19) and some other identities among (3.7) to (3.24). A loop that satisfies (3.23) and (3.24) is called a left Bol loop and right Bol loop respectively. In a loop, (3.8) implies (3.19), which implies both (3.7) and (3.24).

Left Bol identity and right Bol identity are mirrors of each other. They are usually referred to as duals. Hence, in loops, algebraic properties proven for one usually implies the mirror of such property in the other. Different authors work on either and simply refer to such as a Bol loop.

3.1.8 Bol Loops: Brief History and the Journey So Far The birth of Bol loops can be traced back to Gerrit Bol [21] in 1937 when he established the relationship between Bol loops and Moufang loops, the latter which was discovered by Moufang Ruth [124]. Thereafter, a theory of Bol loops was evolved through the Ph.D. thesis of Robinson [143] in 1964 where he studied the algebraic properties of Bol loops, Moufang loops and Bruck loops, isotopy of Bol loop and some other notions on Bol loops. Some later results on Bol loops and Bruck loops can be found in Bruck [22], Solarin [156], Adéníran and Akinleye [3], Bruck [23], Burn [24], Gerrit Bol [21], Blaschke and Bol [20], Sharma [147, 148], Adéníran and Solarin [8]. In the 1980s, the study and construction of finite Bol loops caught the attention of many researchers among which are Burn [24–26], Solarin [155], Solarin and Sharma [149, 158–160] and others like Chein and Goodaire [28–30], Foguel at. al. [44], Kinyon and Phillips [116, 117] in the present millennium. One of the most important results in the theory of Bol loops is the solution of the open problem on the existence of a simple Bol loop which was finally laid to rest by Nagy [125–127].

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To any right Bol loop or left Bol loop, there corresponds a middle Bol loop and vice versa. Jaiyéo.lá and David [102], Jaiyéo.lá et al. [103, 104], Syrbu and Drapal [167], Syrbu and Grecu [168] and Syrbu [166] have studied the algebraic properties and structure of middle Bol loop. Osoba and Jaiyéo.lá [131] recently announced some algebraic connections between right and middle Bol loops and their cores while Oyebo and Osoba [135] investigated more algebraic properties of middle Bol loops. In 1978, Sharma [149], Sharma and Sabinin [151] introduced and studied the algebraic properties of the notion of half-Bol loops(left B-lops). Thereafter, Adéníran [1], Adéníran and Akinleye [3], Adéníran and Solarin [9] studied the algebraic properties of generalized Bol loops. Also, Ajmal [10] introduced and studied the algebraic properties of generalized Bol loops and their relationship with M-loops. Some study on the holomorph of generalized Bol loops can be found in Adéníran et. al. [5] and Jaiyéo.lá and Popoola [105]. Recently, Adéníran et. al. [6] characterized generalized Bol loops while Jaiyéo.lá et al. [106] investigated the isostrophy Bryant-Schneider Group-invariance of Bol loops.

Every group is trivially a Bol loop and this necessitated the need to establish the existence of strictly Bol loops, that are not associative. This was the subject of earlier study of Bol loops in the late 1970s and early 1980s. Henceforth, by “Bol loop”, we shall mean a right Bol loop.

3.1.9 Algebraic Properties of Bol Loops We shall now discuss some algebraic properties of a Bol loop. Lemma 3.1 If .(G, ·) is a Bol loop, then (i) .(G, ·) satisfies the right inverse property. (ii) .x λ = x ρ for all .x ∈ G. (iii) .(G, ·) has the right alternative property. Proof (i) In (3.24), let .z = y ρ . Then .(xy · y ρ )y = x(yy ρ · y) = xy for all .x, y ∈ G. Hence, .xy · y ρ = x for all .x, y ∈ G. (ii) In (3.24), let .z = y λ Then .(xy · y λ )y = x(yy λ · y) for all .x, y ∈ G. Now using the RIP and the fact that .y = (y λ )ρ , we obtain .xy = x(yy λ ·y) for all .x, y ∈ G. Therefore .yy λ = e and, hence, .y λ = y ρ for all .y ∈ G. Thus, .y λ = y ρ = y −1 for all .y ∈ G. (iii) In (3.24), let .z = e and get .xy · y = x · yy for all .x, y ∈ G. U n

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Power of Elements in a Bol Loop If x is an element of a Bol loop .(G, ·) and .n ∈ Z+ ∪ {0}, define .x n recursively by .x 0 = e and .x n = x n−1 · x for .n > 0. For any .n ∈ Z− , define .x n by n −1 )|n| . .x = (x

Lemma 3.2 If .(G, ·) is a Bol loop, then xy n = xy n−1 · y = xy · y n−1

.

(3.25)

for all .x, y ∈ G and all integers n. In particular, Bol loops are power associative. Proof Clearly (3.25) holds for .n = 0 and for .n = 1. Now assume that, for .k > 1, xy k = xy k−1 · y = xy · y k−1

.

(3.26)

for all .x, y ∈ G (In particular, .y k = y k−1 y = yy k−1 for all .y ∈ G). Then xy k+l = x · y k y = x(yy k−1 · y) = (xy · y k−1 )y = xy k · y for all x, y ∈ G.

.

Then, replacing x by xy in (3.25), we get xy · y k = (xy · y k−1 )y = x(yy k−l · y) = x(y k−l y · y) = x · y k y = xy k+l

.

for all .x, y ∈ G. Thus, (3.25) holds for all integers .n > 0. Now, for all integers n > 0 and all .x, y ∈ G, expression (3.25) applied to x and .y −1 gives

.

x(y −1 )n+1 = x(y −1 )n · y −l = xy −n · y −1

.

and (3.25) applied to xy and .y −l gives .xy · (y −1 )n+1 = (xy · y −1 )(y −1 )n = xy −n . U n Hence, .xy −n = xy −n−1 · y = xy · y −n−1 and the result is proved. Theorem 3.1 If .(G, ·) is a Bol loop, then xy m · y n = xy m+n

.

(3.27)

for all .x, y ∈ G and all integers m and n. In particular, Bol loops are power associative. Proof The desired result clearly holds for .n = 0 and by Lemma 3.2, it holds for n = 1. For any integer .n > 1, assume that (3.27) holds for all integers m and all .x, y ∈ G. Then, by Lemma 3.2,

.

xy m+n+1 = xy m+n · y = (xy m · y n )y = xy m · y n+1

.

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for all .x, y ∈ G and all integers m. So (3.27) holds for all .x, y ∈ G, all integers m, and all nonnegative integers n. (In particular, for use below, .(y n )−1 = y −n for all nonnegative integers n and all .y ∈ G.) Replacing m by .m−n, we have .xy m−n ·y n = xy m and, hence, .xy m−n = xy m · (y n )−1 = xy m · y −n for all integers .n ≥ 0, all integers m, and all .x, y ∈ G. In particular, .y m y n = y m+n for all .y ∈ G and all integers m and n. Consequently, the Bol loop .(G, ·) is power-associative. U n

3.1.10 Characterization and Constructions of Bol Loops This subsection is devoted to characterization for the constructions of some Bol loops of finite orders.

3.1.10.1

Characterization of Bol Loops

Theorem 3.2 If || is the representation of a loop G, then G is a Bol loop if and only if α, β ∈ || implies αβα ∈ ||. Proof By (3.24), (yx · z)x = y(xz · x) ⇒ yRx Rz Rx = yR(xz·x) ⇒ Rx Rz Rx = R(xz·x)

.

for all x, z ∈ G. Thus, if α = Rx ∈ || and β = Rz ∈ ||, then αβα ∈∈ ||. Conversely, let α, β ∈ || implies αβα ∈ ||. Then, take α = Rx , β = Rz so that αβα = R(xz·x) . Then, Rx Rz Rx = R(xz·x) ⇒ yRx Rz Rx = yR(xz·x) ⇒ (yx · z)x = y(xz · x) which is (3.24). U n Theorem 3.3 If || is the representation of a Bol loop and α ∈ ||, then α n ∈ || for any n ∈ Z. Proof Going by Theorem 3.2, for any α, β ∈ ||, we have αβα ∈ ||. Let β = I , then α 2 ∈ ||. Again, with β = α, we get ααα = α 3 ∈ ||. With β = α 2 , we have αα 2 α = α 4 ∈ ||. Let us assume that for any α ∈ || we have α k−1 ∈ || for k ∈ N. Then, with β = α k−1 , we get αα k−1 α = α k . Thus, by induction, α n ∈ || for any n ∈ N+ Recall that by Theorem 3.1, a Bol loop is a right inverse property loop. Hence, β = Rx −1 ∈ ||ρ ⇒ β = Rx−1 ∈ ||ρ for all x ∈ G. By the earlier result, and taking α = Rx , we have β n = (α −1 )n = α −n ∈ ||ρ for all n ∈ Z− . Whence, β n ∈ ||ρ ∀ n ∈ Z. U n Theorem 3.4 If || is a representation of a finite Bol loop and α ∈ ||, then α is a product of disjoint cycles of equal length.

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Corollary 3.1 The order of an element of a finite Bol loop divides the order of the loop. Theorem 3.5 If p is a prime number, then a Bol loop of order 2p is a group. Theorem 3.6 If p is a prime number, then a Bol loop of order p2 is a group. Theorem 3.7 If || is the representation of a Bol loop of order 8, then, ||1 (8) : R(2) = (1234)(5678); R(3) = (13)(24)(57)(68); R(4) = R(2)−1 ;

.

R(5) = (1537)(2648); R(6) = (1638)(2547); R(7) = R(5)−1 ; R(8) = R(6)−1 ; ||2 (8) : R(2) = (1234)(5678); R(3) = (13)(24)(57)(68); R(4) = R(2)−1 ; R(5) = (1537)(2648); R(6) = (16)(25)(38)(47); R(7) = R(5)−1 ; R(8) = (18)(27)(36)(45); ||3 (8) : R(2) = (1234)(5678); R(3) = (13)(24)(57)(68); R(4) = R(2)−1 ; R(5) = (1537)(26846); R(6) = (16)(25)(38)(47); R(7) = R(5)−1 ; R(8) = (18)(27)(36)(45); ||4 (8) : R(2) = (1234)(5678); R(3) = (13)(24)(57)(68); R(4) = R(2)−1 ; R(5) = (15)(26)(37)(48); R(6) = (16)(27)(38)(45); R(7) = (17)(28)(35)(46); R(8) = (18)(25)(36)(47); ||5 (8) : R(2) = (1234)(5678); R(3) = (13)(24)(57)(68); R(4) = R(2)−1 ; R(5) = (15)(26)(37)(48); R(6) = (16)(25)(38)(47; R(7) = (17)(28)(35)(46); R(8) = (18)(27)(36)(45); ||6 (8) : R(2) = (12)(34)(56)(78); R(3) = (13)(25)(47)(68); R(4) = (14)(23)(58)(67); R(5) = (15)(26)(37)(48); R(6) = (16)(25)(38)(47; R(7) = (17)(28)(35)(46); R(8) = (18)(27)(36)(45). The method of proof of Theorem 3.7 is similar to that of orders 12 and 16. We shall illustrate with order 16 below. Theorems 3.5, 3.6 and 3.5 together with the previously known fact that Bol loops of prime order are cyclic groups established that the next smallest orders for which a non-associative Bol loop may occur is 12. At least one non-associative Bol loop of order 12 exists, the smallest Moufang loop [27]. Solarin and Sharma [161] did an exhaustive search for Bol loops of order 12 and established that in addition to the groups there are exactly two non-isomorphic Bol loops, which are not Moufang.

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It is our purpose in this section to prove that (up to Isomorphism) there are exactly 21 non associative Bol loops of order 16, which contain at least one element of order 8 according to Sharma and Solarin [162]. We take || to be the representation of a Bol loop of order 16, || = {R(i)|1 ≤ i ≤ 16} containing no element of order 16. If α ∈ || such that α 8 = I , then throughout this part we assume α = R(2) and α, α 2 , α 3 , α 4 , α 5 , α 6 and α 7 , are as follows: .

R(2) = (1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16).

(3.28)

R(3) = (1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16).

(3.29)

R(4) = (1 4 7 2 5 8 3 6)(9 12 15 10 13 16 11 14).

(3.30)

R(5) = (1 5)(2 6)(3 7)(4 8)(9 13 )(10 14)(11 15)(12 16).

(3.31)

R(6) = (1 6 3 8 5 2 7 4)(9 14 11 16 13 10 15 12).

(3.32)

R(7) = (1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12).

(3.33)

R(8) = (1 8 7 6 5 4 3 2)(9 16 15 14 13 12 11 10)

(3.34)

The main difficulty in classifying Bol loops was in distinguishing which of the loops we found were isomorphic to each other. ln case of Moufang loops, [27] considered the order structure (the number of elements of each order) and of the number of pairs of commuting elements of maximal order in finding isomorphic Moufang loops. But in case of Bol loops, we found that order structure and the number of pairs of commuting elements were not sufficient to classify Bol loops. ln order to illustrate this point, we consider ||1 (16) and ||2 (16) below. lt is easy to see that order structure in ||1 (16) and ||2 (16) are the same. The pairs of commuting elements in ||1 (16) and ||1 (16) are as follows: (a) (b) (c) (d)

2, 3, 4, 6, 7, 8, 10, 12, 14 and 16 commute with each other. 5 commutes with all elements. 9 and 13 commute with each other. 11 and 15 commute with each other.

It is interesting to note that inspite of the same order structures and pairs of commuting elements, ||1 (16) and ||2 (16) are not isomorphic. In order to overcome this difficulty we considered orders of Nλ and Nμ in ||1 (16) and ||2 (16) and found that o(Nλ1 ) = o(Nλ2 ), but o(Nμ1 ) /= o(Nμ2 ). By imposing the additional condition that o(Nλ1 ) = o(Nλ2 ) and o(Nμ1 ) = o(Nμ2 ) we solved the problem of isomorphism. ||1 (16): R(2) to R(8) are given by (3.28)–(3.34) R(9) = (1 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)

.

R(10) = (1 10 7 16 5 14 3 12)(2 11 8 9 6 15 4 13)

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R(11) = (1 11 5 15)(2 14 6 10)(3 9 7 13)(4 12 8 16) R(12) = (1 12 3 14 5 16 7 10)(2 13 4 15 6 9 8 11) R(13) = (1 13 5 9)(2 16 6 12 )(3 11 7 15)(4 14 8 10) R(14) = (1 14 7 12 5 10 3 16)(2 15 8 13 6 11 4 9) R(15) = (1 15 5 11)(2 10 6 14)(3 13 7 9)(4 16 8 12) R(16) = (1 16 3 10 5 12 7 14)(2 9 4 11 6 13 8 15) where Nλ = {1, 5, 9.13}, Nρ = Nμ = {1, 3, 5.7}.

||2 (16): R(2) to R(8) are given by (3.28)–(3.34). R(9) = (1 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)

.

R(10) = (1 10 7 16 5 14 3 12)(2 11 8 9 6 15 4 13) R(11) = (1 11 5 15)(2 10 6 14)(3 9 7 13)(4 16 8 12) R(12) = (1 12 3 14 5 16 7 10)(2 13 4 15 6 9 8 11) R(13) = (1 13 5 9 )(2 16 6 12)(3 11 7 15)(4 14 8 10) R(14) = (1 14 7 12 5 10 3 16)(2 15 8 13 6 11 4 9) R(15) = (1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16) R(16) = (1 16 3 10 5 12 7 14)(2 9 4 11 6 13 8 15) where Nλ = {1, 5, 9, 13}, Nμ = Nρ = {1, 5}. Theorem 3.8 If || is a representation of a Bol loop of order 16 which is not a group or a Moufang loop and α, β, γ , δ, ρ ∈ || such that α 8 = β 4 = γ 4 = δ 4 = ρ 4 = I , and α 4 = β 2 = γ 2 = δ 2 = ρ 2 , then || is given by the only representation:

||3 (16): R(2) to R(8) are given by (3.28)–(3.34) R(9) = (1 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)

.

R(10) = (1 10 5 14)(2 9 6 13)(3 16 7 12)(4 15 8 11) R(11) = (1 11 5 15)(2 14 6 10)(3 9 7 13)(4 12 8 16) R(12) = (1 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13) R(13) = (1 13 5 9)(2 16 6 12)(3 11 7 15)(4 14 8 10) R(14) = (1 14 5 10)(2 13 6 9)(3 12 7 16)(4 11 8 15)

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R(15) = (1 15 5 11)(2 10 6 14)(3 13 7 9)(4 16 8 12) R(16) = (1 16 5 12)(2 15 6 11)(3 14 7 10)(4 13 8 9) where Nλ = {1, 3, 5, 7, 9, 11, 13, 15} and Nμ = Nρ = {1, 3, 5, 7}. Proof Without loss of generality take α = R(2), β = R(9), γ = R(10), δ = R(12) and ρ = R(11), then R(9) to R(16) are of order 4. By hypothesis α 4 = β 2 = γ 2 = δ 2 = ρ 2 = R(5). Thus R(9) = (1 9 5 13)(2 x 6 y)(3 u 7 v)(4 w 8 z), where x ∈ {10, 11, 12, 14, 15, 16}. Take x = 11 in R(9), then y = 15 and R(9)R(7) = R(2)R(13). Thus, 2R(9)R(7) = 2R(2)R(13) ⇒ v = 9, which is impossible. Hence x cannot be equal to 11. Following the same argument we can easily show that x /∈ {10, 11, 14, 15, 16}. Thus x = 12 in R(9) and y = 16. Thus R(9) = (1 9 5 13)(2 12 6 16)(3 u 7 v)(4 w 8 z). But R(9)R(8) = R(2)R(13). Also 2R(9)R(8) = 2R(2)R(13) ⇒ v = 11 and u = 15. 3R(9)R(8) = 3R(2)R(13) ⇒ z = 14 and w = 10. Thus R(9) = (1 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14).

.

(3.35)

Further we have R(10) = (1 10 5 14)(2 x1 6 y1 )(3 u1 7 v1 )(4 w1 8 z1 ), where x1 ∈ {9, 13, 11, 15}. (i) Take x1 = 9 in R(10) then y1 = 13. Thus R(10) = (1 10 5 14)(2 9 6 13) (3 u1 7 v1 ) (4 w1 8 z1 ). But v1 = 12 and u1 = 16. Similarly 3R(10)R(4) = 3R(2)R(14) ⇒ R(10)R(4) = R(2)R(14). Then 2R(10)R(4) = 2R(2)R(14) ⇒ z1 = 11 and w1 = 15. Thus R(10) = (1 10 5 14)(2 9 6 13)(3 16 7 12)(4 15 8 11).

.

(3.36)

R(16) = R(9)R(10)R(9) = (1 16 5 12)(2 15 6 11)(3 14 7 10)(4 13 8 9).

.

R(11) = R(10)R(9)R(10) = (1 11 5 15)(2 14 6 10)(3 9 7 13)(4 12 8 16). Thus we get the representation || which is ||3 (16). (ii) Take x1 = 13 in R(10) then y1 = 9 and R(10) = (1 10 5 14)(2 13 6 9)(3 u1 7 v1 )(4 w1 8 z1 ).

.

But R(10)R(8) = R(2)R(14). Thus 2R(10)R(8) = 2R(2)R(14) ⇒ v1 = 2 and u1 = 6. Also 3R(10)R(8) = 3R(2)R(14) ⇒ z1 = 15 and w1 = 11. Hence R(10) = (1 10 5 14)(2 13 6 9)(3 16 7 12)(4 11 8 15).

.

(3.37)

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R(9)R(10)R(9) = R(12) = (1 12 5 16)(2 15 6 11)(3 10 7 14)(4 13 8 9)

.

R(10)R(9)R(10) = R(15) = (1 15 5 11)(2 10 6 14)(3 9 7 13)(4 16 8 12). Thus we get the representation || of dicyclic group (r3 a3 ). (iii) Take x1 = 11 in R(10), then y1 = 15 and R(10) = (1 10 5 14)(2 11 6 15)(3 u1 7 v1 )(4 w1 8 z1 ).

.

But R(10)R(6) = R(2)R(14). Hence 2R(10)R(6) = 2R(2)R(14) ⇒ v1 = 16 and u1 = 12. Also 3R(10)R(6) = 3R(2)R(14) ⇒ z1 = 9 and w1 = 13. Thus R(10) = (1 10 5 15)(2 11 6 15)(3 12 7 16)(4 13 8 9). But R(2)R(10)R(2) = (1 12)()()()()()()() /∈ || because R(12) is of order 4 by assumption. Hence x1 cannot be equal to 11. Proceeding on the same lines, we can easily show that x1 can not be equal to 15. This completes the proof of the theorem. U n Remark 3.1 The remaining 20 loops were obtained in like manner by Sharma and Solarin [162]. These 21 non-isomorphic Bol loops of order 16 were found to fall into 8 isotopic classes. These 8 classes were used to construct 8 distinct Bol loops of order 4n, n ≥ 2 by Solarin and Sharma [159]. Solarin and Sharma [163] considered Bol loops of order 16 in which 13 elements are of order 2, and 14 elements are of order 4. Recently, Kareem [107] and Aliu [11] constructed some quaternion-type Bol loops of order 2n (n ≥ 3) and some Bol loops of order n2 .

3.1.10.2

Constructions of Bol Loops

Lemma 3.3 Let G(◦) = C2n × C2 and the binary operation is defined as follows: .

( β α ) ( β ) ( β +β α ) x 1, a i ◦ x 2, e = x 1 2, a i {( ) ( β α) ( β ) x β1 +β2 , a αi +1 if x 1, a i ◦ x 2, a = ( ) x 5β1 +β2 , a αi +1 if

β2 ≡ 0 (mod 2) β2 ≡ 1 (mod 2)

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n = 3, 4, 6, 12. ( ) ( ) ( ) Proof Let A = x β1 , e , B = x β2 , e and C = x β3 , e then ( ) ( ) (AB ◦ C)B = x β1 +2β2 +β3 , e and A(BC ◦ B) = x β1 +2β2 +β3 , e .

.

From Table 3.1, it can be seen that (AB ◦C)B = A(BC◦B) in all cases whenever 25 ≡ 1 (mod 2n), where n = 2, 3, 4, 6, 12.

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Table 3.1 Bol identity computation 1 2

A ( β ) x 1, e ( β ) x 1, e

B ( β ) x 2, e ( β ) x 2, e

C ( β ) x 3, e ( β ) x 3, a

3

( β ) x 1, e

(

)

(

4

( β ) x 1, e

(

)

(

5 6

( β ) x 1, a ( β ) x 1, a

( β ) x 2, e ( β ) x 2, e

( β ) x 3, e ( β ) x 3, a

7

( β ) x 1, a

(

)

(

8

( β ) x 1, a

(

)

(

x β2 , a

x β2 , a

x β2 , a

x β2 , a

x β3 , e

x β3 , e

x β3 , e

)

)

)

x β3 , a

)

(AB o C)B ( β +2β +β ) x 1 2 3, e ( β +2β +β ) x 1 2 3, a , β even ( 35β +6β +β ) x 1 2 3, a , β3 odd ( β +2β +β ) x 1 2 3, e , β even ( 225β +6β +5β ) x 1 2 3, e , β2 odd ( β +2β +β ) x 1 2 3, a , β , β even ( 225β3 +6β +5β ) x 1 2 3, a , β odd, β3 even ( 25β +6β +β ) x 1 2 3, a , β even, β odd ( 2125β +26β3 +5β ) 1 2 3, a , x β2 , β3 odd ( β +2β +β ) x 1 2 3, a ( β +2β +β ) x 1 2 3, a , β even ( 35β +6β +β ) x 1 2 3, a , β3 odd ( β +2β +β ) x 1 2 3, a , β even ( 225β +6β +5β ) x 1 2 3, a , β2 odd ( β +2β +β ) x 1 2 3, e , β , β even ( 25β 3+6β +β ) x 1 2 3, e , β even, β3 odd ( 225β +6β +5β ) x 1 2 3, e , β odd, β3 even ( 2125β +26β ) 1 2 +5β3 , e , x β2 , β3 odd

A(BC o B) ( β +2β +β ) x 1 2 3, e ( β +2β +β ) x 1 2 3, a , β even ( 35β +6β +β ) x 1 2 3, a , β3 odd ( β +2β +β ) x 1 2 3, e , β even ( 2β +6β +5β ) x 1 2 3, e , β2 odd ( β +2β +β ) x 1 2 3, e , β , β3 even ( 2β +6β ) x 1 2 +5β3 , a , β odd, β3 even ( 25β +6β +β ) x 1 2 3, a , β even, β3 odd ( 25β +26β +5β ) 2 3, a , x 1 β2 , β3 odd ( β +2β +β ) x 1 2 3, a ( β +2β +β ) x 1 2 3, a , β even ( 35β +6β +β ) x 1 2 3, a , β3 odd ( β +2β +β ) x 1 2 3, a , β even ( 2β +6β +5β ) x 1 2 3, a , β2 odd ( β +2β +β ) x 1 2 3, e , β , β even ( 25β 3+6β +β ) x 1 2 3, e , β even, β3 odd ( 2β +6β +5β ) x 1 2 3, e , β odd, β even ( 25β +26β 3+5β ) 2 3, e , x 1 β2 , β3 odd

( ) Also (e, e) is the two sided identity. Moreover, if A = x β , e , then A−1 = {( ) ( −β ) x −β , a if β is even β −1 . Therefore, the = ( x , e ; if A = (x , a), then A ) −5β x , a if β is odd inverses are defined. ( ) ( ) ( ) Also for non-associativity, let A = x β1 , e ; B = x β2 , e ; C = x β3 , a ; where β2 and β3 are odd integers, then

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( ) ( ) AB ◦ C = x 5β1 +5β2 +β3 , a and A ◦ BC = x β1 +5β2 +β3 , a

.

AB ◦ C /= A ◦ BC whenever 4 /≡ 0 (mod 2n); thus the construction is nonassociative except when n = 2 which gives the group C4 × C2 . Hence it is a Bol loop of order 4n, n = 3, 4, 6, 12. U n Remark 3.2 The construction in Lemma 3.3 gives the Bol loop of order 12 with two generators, when n = 3; ||14 (16) when n = 4; ||1 (24) when n = 6. This construction is typical of constructions of Bol loops of order 4n, n ≥ 2. There are seven similar constructions, Solarin and Sharma [159], and the proofs are similar. Lemma 3.4 Let G(◦) = C2n × C2 and the binary operation is defined as follows: ( β α ) ( β ) ( β +β α ) x 1, a i ◦ x 2, e = x 1 2, a i {( ) ( β α ) ( β α ) x β1 +β2 , a α1 +α2 if 1 1 2 2 x ,a ◦ x ,a = ( ) β −β α +α 2 1 1 2 x if ,a .

β2 ≡ 0 (mod 2) β2 ≡ 1 (mod 2)

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n ≤ 2. G is ||5 (8), when n = 2; the Bol loop of order 12 with two generators, when n = 3; ||5 (16), when n = 4. Lemma 3.5 Let G(◦) = C2n × C2 and the binary operation is defined as follows: ( β ) ( β ) ( β +β ) x 1, e ◦ x 2, e = x 1 2, e ( β ) ( β ) ( β +β ) x 1, a ◦ x 2, e = x 1 2, a {( ) ( β α ) ( β ) x β2 −β1 +nα1 , a α1 +1 if β2 ≡ 1 (mod 2) x 1, a 1 ◦ x 2, a = ( ) 3β +β +nα α +1 1 2 1 1 x if β2 ≡ 0 (mod 2) ,a .

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n is any positive even integer greater than 2. G is D4 when n = 2; ||1 (16) when n = 4. Lemma 3.6 Let G(◦) = C2n × C2 and the binary operation is defined as follows: ( β α ) ( β ) ( β +β α ) x 1, a 1 ◦ x 2, e = x 1 2, a 1 ( β α ) ( β α ) ( β +β α +α ) x 1 , a 1 ◦ x 2 , a 2 = x 1 2 , a 1 2 , if β2 ≡ 0 (mod 2) {( ) x β2 −β1 +nα1 , a α1 +α2 if β2 ≡ 1 (mod 4) = ( ) x 3β1 +β2 , a α1 +α2 if β2 ≡ 3 (mod 4) .

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n is any positive even integer. G is ||5 (8) when n = 2; ||3 (16) when n = 4.

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Lemma 3.7 Let G(◦) = C2n × C2 and the binary operation is defined as follows: ( β α ) ( β ) ( β +β α ) x 1, a 1 ◦ x 2, e = x 1 2, a 1 ( β α ) ( β α ) ( β +β α +α ) x 1 , a 1 ◦ x 2 , a 2 = x 1 2 , a 1 2 , if β1 ≡ 0 (mod 2), β2 ≡ 0 (mod2) ⎧( ) β1 +β2 +nα1 , a α1 +α2 , if β ≡ 1 (mod2), ⎪ ⎪ 1 ⎨ x = β2 ≡ 0 (mod2) ⎪ ⎪ ⎩(x β2 −β1 , a α1 +α2 ) if β ≡ 1 (mod 2) .

2

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n is any positive even integer. G is ||5 (8) when n = 2; ||7 (16) when n = 4. Lemma 3.8 Let G(◦) = C2n × C2 and the binary operation is defined as follows: (

) ( ) ) ( x β1 , a αi ◦ x β2 , e = x β1 +β2 , a αi {( ) ( β α ) ( β α ) x 3β1 +β2 +nα2 , a α1 +α2 if β2 ≡ 0 (mod 2) x 1, a 1 ◦ x 2, a 2 = ( ) 5β +β α +α 1 2 1 2 x if β2 ≡ 1 (mod 2) ,a .

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n is any positive even integer. G is ||2 (8), when n = 2; ||12 (16), when n = 4. Lemma 3.9 Let G(◦) = C2n × C2 and the binary operation is defined as follows: ( β α ) ( β ) ( β +β α ) x 1, a 1 ◦ x 2, e = x 1 2, a 1 ( β α ) ( β α ) ( β +β +nα α +α ) x 1 , a 1 ◦ x 2 , a 2 = x 1 2 1 , a 1 2 , if β2 ≡ 1 (mod 2) ⎧( ) ⎪ x 3β1 +β2 +nα1 , a α1 +α2 ifβ1 ≡ 0 (mod2), ⎪ ⎪ ⎪ ⎨ β2 ≡ 0 (mod2) = ( ) 3β −β +nα α +α ⎪ x 1 2 1, a 1 2 if β1 ≡ 1 (mod2), ⎪ ⎪ ⎪ ⎩ β2 ≡ 0 (mod2) .

For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n is any positive even integer. G is ||2 (8) when n = 2; ||13 (16) when n = 4. Lemma 3.10 Let G(◦) = C2n × C2 and the binary operation is defined as follows: .

( β α ) ( β ) ( β +β α ) x 1, a i ◦ x 2, e = x 1 2, a i {( ) ( β ) ( β ) x 3β1 +β2 , a 1 2 x ,e ◦ x ,a = ( ) x 5β1 +β2 , a

if

β2 ≡ 0 (mod 2)

if

β2 ≡ 1 (mod 2)

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( β ) ( β ) ( 3β −β ) x 1 , a ◦ x 2 , a = x 1 2 , e , if β2 ≡ 0 (mod 2) {( ) x β1 +5β2 , e if β1 ≡ 0 (mod2), β2 ≡ 1 (mod2) = ( ) x β1 +β2 , e if β1 ≡ 1 (mod2), β2 ≡ 1 (mod2) For all 1 ≤ αi ≤ 2, then G(◦) is a Bol loop of order 4n, where n is any positive even integer. G is ||2 (8), when n = 2; ||16 (16), when n = 4. The following Theorem 3.9 constructs a Bol loop from every non-abelian group through a semi-direct product of the group. Theorem 3.9 Let G be a non-abelian group such that X = G × G. Define (X, ◦) as follows: (h1 , g1 ) ◦ (h2 , g2 ) = (h1 h2 , h2 g1 h−1 2 g2 ) for all (h1 , g1 ), (h2 , g2 ) ∈ X.

.

Then (X, ◦) is a Bol loop. Proof Let x, y, z ∈ X. By checking, it is true that x ◦(y ◦z) /= (x ◦y)◦z. Therefore, (X, ◦) is non-associative. X is loop. Let us verify that x ◦ (y ◦ z) /= (x ◦ y) ◦ z. ) ( L.H.S = x ◦ (y ◦ z) =(h1 g1 ) ◦ h2 h3 , h3 g2 h−1 g 3 3 ) ( −1 −1 = h1 h2 h3 , h2 h3 g1 h−1 3 h2 h3 g2 h3 g3 ) ( R.H.S = (x ◦ y) ◦ z = h1 h2 , h2 g1 h−1 2 g2 ◦ (h3 , g3 ) ( ) −1 = h1 h2 h3 , h3 h2 g1 h−1 g h g 2 3 2 3

.

Since L.H.S /= R.H.S, then X is not associative. Let us now verify the Bol identity ((x ◦ y) ◦ z) ◦ y = x ◦ ((y ◦ z) ◦ y).

.

) ( −1 ◦ (h2 , g2 ) L.H.S = ((x ◦ y) ◦ z) ◦ y = h1 h2 h3 , h3 h2 g1 h−1 g h g 2 3 2 3 ( ) −1 −1 = h1 h2 h3 h2 , h2 h3 h2 g1 h−1 g h g h g 2 3 3 2 2 2 ) ( −1 R.H.S = x ◦ ((y ◦ z) ◦ y) =(h1 , g1 ) ◦ h2 h3 h3 , h2 h3 g2 h−1 3 g3 h2 g2 ) ( −1 −1 = h1 h2 h3 h2 , h2 h3 h2 g1 h−1 2 g2 h3 g3 h2 g2 .

.

Therefore, L.H.S = R.H.S. Hence, (X, ◦) is a Bol loop.

U n

3 Some Varieties of Loops

121

When G = S3 , X = S3 × S3 , o(X) = 36. It was found that this Bol loop of order 36 has a normal subloop of order 18, which is not associative, consequently, a Bol loop. This informed our search for other Bol loops of order 18. Using the approach described earlier for order 16, we were able to establish that up to isomorphism, there exist two Bol loops of order 18 (see [150]). This result was eventually extended to Bol loops of 2p2 for all prime p ≥ 3 by Sharma and Solarin [152]. Moreover, going by Theorem 3.9 with G = Dn , X = Dn × Dn is a Bol loop of order 4n2 , n ≥ 3. When G = D6 , we constructed the cayley table for X = D6 × D6 computationally using GAP. The order of X = D6 × D6 is 144. We verified using GAP the non-associativity of the semi-direct product. Thus, it was a non-associative loop. In addition, it was of interest to verify certain loop-theoretical conditions such as the Bol identities since that was the focus of our construction which was shown to be satisfied. We then decided to generate from our construction, all the possible non-associative triples (x, y, z) and investigated the kind of structures that will be generated by some of these triples. There were 736, 128 possible non-associative triples such that x /= y, x /= z and y /= z. We considered the loop generated by one of the triples and discovered using gap that it satisfies Moufang identity and it was a loop of order 36.

Bol Loops of Even Orders These constructions established the status of Bol loops of even orders except for even composite numbers of the form 2pq, where p and q are odd primes. This was resolved with the study of Bol loops of order pq, where p and q are primes, p > q.

Bol Loops of Order 15 Let .|| be the representation of a Bol loop of order 15. We assume .α, β ∈ || such that .α 5 = β 3 = I . Then .||11 is given by: α = R(2) = (12345)(678910)(1112131415), α 2 = R(3), α 3 = R(4), α 4 = R(5)

.

β 3 = R(6) = (1611)(2713)(3815)(4912)(51014), β 2 = R(11). The remaining elements are obtained by .α i β j α i , 1 ≤ i ≤ 5, 1 ≤ j ≤ 3 based on Theorems 3.2 and 3.3. .||12 is given by α = R(2) = (12345)(678910)(1112131415), α2 = R(3), α3 = R(4), α4 = R(5)

.

β3 = R(6) = (1611)(2914)(3712)(41015)(5813), β2 = R(11).

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The remaining elements are obtained by .α i β j α i , 1 ≤ i ≤ 5, 1 ≤ j ≤ 3 based on Theorems 3.2 and 3.3. Solarin and Sharma [164] studied Bol loops of order 3p, where p is a prime (.p > 3). Niederreiter and Robinson [129] studied Bol loop of order pq.

3.2 Central Loops In any loop, there is equivalence between any two of the identities corresponding to each of the equation numbers in each of the triples {(3.8), (3.9), (3.10)}, {(3.11), (3.12), (3.13), (3.14)} and {(3.15), (3.16), (3.17),(3.18)}. These facts are found in [42], [43] and [43] respectively. In fact, in a loop, any one of (3.11), (3.12), (3.13) and any one of (3.15), (3.16), (3.17) together are equivalent to (3.7) and vice versa. Although in a loop, any one of (3.8), (3.9), (3.10) implies (3.7) but the converse is not true. Loops that satisfy (3.11) and (3.15) or their equivalent forms are called left central and right central loops, for short, LC-loops and RC-loops respectively. A loop that obeys (3.7) is called a central loop or C-loop short. At times, the three varieties LC-loops, RC-loops and C-loops shall be sometimes referred to as central loops. Loops that satisfy (3.8) or its equivalent forms are called extra loops. Some types of loops will later be found to be related to central loops. We introduce them in Definition 3.8. Definition 3.8 Let .(G, ·) be a loop. 1. G is called a Steiner loop if .x 2 = e , yx · x = y and .xy = yx for all .x, y ∈ G. 2. A flexible loop G is called a ARIF loop if .(zx) · (yxy) = (z(xyx))y for all .x, y, z ∈ G. 3. A diassociative loop .(G, ·) of order 16 with three generators .a1 , a2 , a3 such that : ai4 = 1; ai2 = aj2 /= 1; ai aj = aj3 ai ; i /= j, ai aj · ak = ai3 · aj ak

.

where .i, j, k are all distinct, is called a Cayley loop.

3.2.1

Central Loops: Brief History and the Journey So Far

Beg [17, 18], Fenyves [43], Solarin and Chiboka [33, 157], Ramamurthi and Solarin [142] Kunen [120], Phillips and Vojtechovsky [139], Adéníran [2], Adéníran and Jaiyéo.lá [4], Jaiyéo.lá and Adéníran [83–85, 88] and Jaiyéo.lá [62, 72, 82], Oyebo [132], Oyebo and Adeniran [133] have made contributions to the study of LC-loops, RC-loops and C-loops in the past.

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In [43], the author worked on their: general properties, relationship with LCloops and RC-loops. He gave an example of a C-loop. The author in [17, 18] furthered study on algebraic study of the LC-loops and RC-loops. In [157], C-loops built on loop of units in a central algebra were found to be conjugacy closed loops while some examples of C-loops that are not conjugacy closed were reported in [139]. Kunen [120] studied quasigroups that satisfy the LC or RC or C identities just like he had studied quasigroups that satisfy each of the Moufang identities or the left Bol identity or the right Bol identity in [119]. In [139], the authors revived the study of C-loops after the work in [43] by considering their: general properties; inverse property, power alternativity, nuclear square, power associativity, diassociativity, flexibility and commutativity, subloop of associators, nucleus; normality and possible order, relationship with; Steiner, ARIF, torsion, loops, analogies to; extra and Moufang loops, quotient loop, order; Lagrange-like and Cauchy-like properties and decomposition. They gave examples of C-loops. In fact, they supplied the smallest non-associative C-loop (of order 10) and the smallest non-commutative non-associative C-loop (of order 12). The authors in [118] worked on the extensions and construction of finite C-loops of particular properties. In [142], the authors investigated finite RC-loops. The articles [33, 157] contain some results on C-loops. In [33], they found a relationship between C-loops and Cayley loops. In [157], they found some C-loops (units in a central algebra) that are both conjugacy closed loops and M-loops. Finite central loops, central loops with central square property and some other properties were considered in [4, 84] while [72, 83, 85, 88] explored the isotopic characterization and universality of central loops. The generalization of central loops was initiated in [82]. A special embedding of C-loop in a group was carried out in [133]. Adeniran et al. [7] studied the holomorph of C-loops and exhibited the form of their Bryant-Schneider group of a C-loop. Let us now discuss some of these results.

3.2.2 Some Characterizations of Central Loops 3.2.2.1

Autotopic Characterizations of Central Loops

Theorem 3.10 A loop G is an LC-loop ⇔ (L2x , I, L2x ) ∈ AU T (G) ∀ x ∈ G. Proof Let G be an LC-loop ⇔ (x · xy)z = (xx)(yz) ⇔ (x · xy)z = x(x · yz) ⇔ (L2x , I, L2x ) ∈ AU T (G) ∀ x ∈ G. U n Theorem 3.11 A loop G is an RC-loop ⇔ (I, Rx2 , Rx2 ) ∈ AU T (G) ∀ x ∈ G. Proof Let G be an RC-loop, then z(yx · x) = zy · xx ⇔ y(yx · x) = (zy · x)x ⇔ (I, Rx2 , Rx2 ) ∈ AU T (G) ∀ x ∈ G. U n

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left Bol loop

Moufang loop

LCC-loop property

extra loop

RCC-loop property

group

CC-loop property

right Bol loop

RC-loop C-loop

LC-loop

Fig. 3.1 Hasse diagram of Bol-Moufang type loops

Theorem 3.12 A loop G is a C-loop ⇔ (Rx2 , L−2 x , I ) ∈ AU T (G) ∀ x ∈ G. Proof Let G be a C-loop then (yx · x)z = y(x · xz) ⇒ yRx2 · z = y · zL2x ⇔ (Rx2 , L−2 U n x , I )AU T (G) ∀ x ∈ G. Theorem 3.13 A loop is a C-loop ⇔ it is both an LC-loop and an RC-loop. Proof The proof of this follows by using the autotopisms (I, Rx2 , Rx2 ), (L2x , I, L2x ) U n and (Rx2 , L−2 x , I ) in Theorems 3.10, 3.11, and 3.12. Remark 3.3 Theorem 3.13 relates LC-loops, RC-loops and C-loops. Hence shall be used to prove results for C-loops by combining those results that are true for LC-loops and RC-loops. This result is depicted in Fig. 3.1.

3.2.2.2

The Representation Sets of Central Loops

Theorem 3.14 Let ||λ (||ρ ) be the left(right) representation of a loop G. G is a LC(RC)-loop ⇔ α, β ∈ ||λ (||ρ ) ⇒ αβ 2 ∈ ||λ (||ρ ). Proof Let G be an LC-loop, then (x · xy)z = x(x · yz) by (3.12). We have Lx·xy = Ly L2x . Replacing x · xy in G and making α = Ly and β = Lx , αβ 2 ∈ ||λ . Conversely, do the reverse of the above. For ||ρ , when G is an RC-loop, z(yx · x) = (zy · x)x by (3.16). The proof goes in the same manner by using Rx . U n

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Theorem 3.15 Let ||λ (||ρ ) be the left(right) representation of a loop G. G is a C-loop⇔ α, β ∈ ||λ (||ρ ) ⇒ αβ 2 , α 2 β ∈ ||λ (||ρ ). Proof Let G be a C-loop, then by (3.7), (yx ·x)z = y(x ·xz) ⇒ Lyx·x = L2x Ly . Let α = Lx , β = Ly , then replacing yx · x in G, α 2 β ∈ ||λ . G is a C-loop ⇔ G is an RC-loop and LC-loop by Theorem 3.13, hence by Theorem 3.14, αβ 2 , α 2 β ∈ ||λ . Conversely let α, β ∈ ||λ ⇒ αβ 2 , α 2 β ∈ ||λ . Take α = Lx , β = Ly then (yx · x)z = y(x · xz) ⇒ G is a C-loop. For ||ρ , the procedure is similar with Rx . U n Theorem 3.16 If ||λ (||ρ ) is the left(right) representation of a LC(RC)-loop and β ∈ ||λ (||ρ ), then β n ∈ ||λ (||ρ ) ∀ n ∈ Z. Proof By Theorem 3.14, α, β ∈ ||λ (||ρ ) ⇒ αβ 2 ∈ ||λ (||ρ ). Using induction, when α = I , β 2 ∈ ||λ (||ρ ), when α = β, β 3 ∈ ||λ (||ρ ) and when α = β k , β k+2 ∈ ||λ (||ρ ) hence, β n ∈ ||λ (||ρ ) ∀ n ∈ Z+ . If G is an LC(RC)-loop then it is a left(right) inverse property loop by Theorem 3.18. −1 Hence, β = Lx −1 (Rx −1 ) ∈ ||λ (||ρ ) ⇒ β = L−1 x (Rx ) ∈ ||λ (||ρ ) ∀ x ∈ G. By n − the earlier result, β ∈ ||λ (||ρ ) ∀ n ∈ Z . Whence, β n ∈ ||λ (||ρ ) ∀ n ∈ Z. U n Corollary 3.2 If ||λ (||ρ ) is the left(right) representation of a C-loop and α ∈ ||λ (||ρ ), then α n ∈ ||λ (||ρ ) ∀ n ∈ Z. Proof G is a C-loop ⇔ G is an RC-loop and LC-loop. The rest of the proof follows from Theorem 3.16. U n

3.2.2.3

Construction of a Finite Central Loop

Let .||ρ be the right representation of the smallest non-commutative non-associative C-loop .(G, ·) of order 12. If .α, β, γ ∈ ||ρ are given by : α = (0 10 1 11 2 9)(3 7 4 8 5 6) = R10 ,

.

β = (0 3)(1 4)(2 5)(6 10)(7 11)(8 9) = R3 , γ = (0 7 2 6 1 8)(3 10 5 9 4 11) = R7 . Then by Theorem 3.15 and Corollary 3.2, .α, β, γ ∈ ||ρ will generate other members of .||ρ by considering appropriate multiplications. [ ]2 ( ) 2 α 2 = R10 = (0 10 1 11 2 9)(3 7 4 8 5 6) = (0 10 1 11 2 9)(3 7 4 8 5 6) .

.

(3.38) = (0 1 2)(3 4 5)(6 7 8)(9 10 11) = R1 ..

(3.39)

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A. R. T. Sòlárìn et al. 2 α 2 β = R10 R3.

(3.40)

= R1 R3. (3.41) ( )( ) = (0 1 2)(3 4 5)(6 7 8)(9 10 11) (0 3)(1 4)(2 5)(6 10)(7 11)(8 9) . (3.42) (

= (0 4 2 3 1 5)(6 11 8 10 7 9) = R4 .. (3.43) ( )( ) α 2 β α 2 = R4 R1 = (0 4 2 3 1 5)(6 11 8 10 7 9) (0 1 2)(3 4 5)(6 7 8)(9 10 11) . )

(3.44) = (0 5 1 3 2 4)(6 9 7 10 8 11) = R5 ..

(3.45)

α 3 = αα 2 = R10 R1. (3.46) ( )( ) = (0 10 1 11 2 9)(3 7 4 8 5 6) (0 1 2)(3 4 5)(6 7 8)(9 10 11) . (3.47) (3.48)

α −1

= (0 11)(1 9)(2 10)(3 8)(4 6)(5 7) = R11 .. ( )−1 −1 = R10 = (0 10 1 11 2 9)(3 7 4 8 5 6) .

(3.50)

α −2

= (0 9 2 11 1 10)(3 6 5 8 4 7) = R9 .. ( )−1 ( )−1 −1 = α2 = R10 = (0 1 2)(3 4 5)(6 7 8)(9 10 11) .

(3.52)

γ −1

= (0 2 1)(3 5 4)(6 8 7)(9 11 10) = R2 .. ( )−1 = R7−1 = (0 7 2 6 1 8)(3 10 5 9 4 11) . = (0 8 1 6 2 7)(3 11 4 9 5 10) = R8 ..

(3.54)

(3.49)

(3.51)

(3.53)

γ 3 = R73 = R7 R7 R7. (3.55) (( )( )) = (0 7 2 6 1 8)(3 10 5 9 4 11) (0 7 2 6 1 8)(3 10 5 9 4 11) (

) (0 7 2 6 1 8)(3 10 5 9 4 11) .

(3.56)

= (0 6)(1 7)(2 8)(3 9)(4 10)(5 11) = R6 .. (3.57) [ ]2 β 2 = R32 = (0 3)(1 4)(2 5)(6 10)(7 11)(8 9) . (3.58) ( )( ) = (0 3)(1 4)(2 5)(6 10)(7 11)(8 9) (0 3)(1 4)(2 5)(6 10)(7 11)(8 9) . (3.59) = (0)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) = R0 .

(3.60)

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Table 3.2 A non-associative C-loop of order 12 .·

0 1 2 3 4 5 6 7 8 9 10 11

0 0 1 2 3 4 5 6 7 8 9 10 11

1 1 2 0 4 5 3 7 8 6 10 11 9

2 2 0 1 5 3 4 8 6 7 11 9 10

3 3 4 5 0 1 2 10 11 9 8 6 7

4 4 5 3 1 2 0 11 9 10 6 7 8

5 5 3 4 2 0 1 9 10 11 7 8 6

6 6 7 8 9 10 11 0 1 2 3 4 5

7

8

9

7 8 6 10 11 9 1 2 0 4 5 3

8 6 7 11 9 10 2 0 1 5 3 4

9 10 11 6 7 8 5 3 4 2 0 1

10 10 11 9 7 8 6 3 4 5 0 1 2

11 11 9 10 8 6 7 4 5 3 1 2 0

Hence, using these results, the bordered multiplication table is shown in Table 3.2.

3.2.3 Some Basic Properties of Central Identities in Quasigroups and Loops Theorem 3.17 Let (G, ·) be a quasigroup. (i) If G satisfies the identity (3.11) or (3.15), then (G, ·) is a loop. (ii) If G satisfies the identity (3.7) or (3.12) or (3.16) or (3.13) or (3.12), then (G, ·) is not a loop. (iii) If G satisfies the identity (3.7), then identities (3.11) and (3.15) or identities (3.12) and (3.16) or identities (3.13) and (3.17) are not true. Proof (i) For a, b ∈ G such that ab = b. Then for any x ∈ G, (x·xa)b = xx·ab = xx·b. Cancelling, xa = a ∀ x ∈ G ⇒ a is a right identity. Putting z = a in the identity implies G is left alternative, hence a is a left identity. A similar prove is true for the second identity. (ii) If p is a prime and 0 < a, b < p, let I (a, b, p) be the structure Zp , with a product operation ◦ defined by : x◦y = ax+by. I (a, b, p) is a quasigroup, and is not a loop unless a = b = 1. I (2, 2, 3) satisfies the first identity, hence G can’t be a loop. On the other hand, for the second and third identities, G is not a loop. I (3, 1, 5) and I (1, 3, 5) satisfy the fourth and fifth identities respectively, hence, G is not a loop in each case. (iii) This follows by (i) and (ii). U n

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Remark 3.4 These results establish the fact that not all quasigroups that satisfy a central identity are loops. Compare these with the left Bol and right Bol identities, in which cases a quasigroup with left (right) Bol has the right (left) identity element. Theorem 3.18 Let (G, ·) be a LC- (RC-) loop. Then : (i) (ii) (iii) (iv) (v) (vi)

(G, ·) is a left (right) alternative loop. (G, ·) is a left (right) inverse property loop. (G, ·) is a left (right) nuclear square loop. (G, ·) is a left (right) power alternative loop. (G, ·) is a middle square loop. (G, ·) is a power associative loop.

Proof Let z = e and y = e in (3.11) and (3.15 respectively, then (i) is true. To prove (ii), let y = x −1 and z = x −1 in (3.12) and (3.16) respectively. By (i), (3.13) and (3.17), (iii) follows. (iv) is proved by (i) and (iii). By (ii), Nμ (G) = Nλ (G)(Nμ (G) = Nρ (G)) hence, (v) holds. (vi) follows from U n (i)–(v). Corollary 3.3 Let (G, ·) be a C-loop. Then: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

(G, ·) is an alternative loop. (G, ·) is an inverse property loop. (G, ·) is a nuclear square loop. (G, ·) is a power alternative loop. the three nuclei coincide. (G, ·) is a power associative loop. if G is flexible, it is diassociative. if G is commutative, it is diassociative.

Proof The proof of (i)–(vi) follows from Theorem 3.18. If G is flexible, it is an ARIF loop which is diassociative. Hence, (vii) is true. A commutative C-loop is flexible, hence by (vii), (viii) holds. U n

We shall frequently make use of these basic properties of central loops in Theorem 3.18 and Corollary 3.3.

3.2.4 Miscellaneous Results on C-Loops 3.2.4.1

Nucleus of a C-Loop

Proposition 3.1 The nucleus N of a C-loop G is a normal subgroup.

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Proof We need only to show that xN = Nx ⇔ x −1 nx ∈ N ∀ x ∈ G, n ∈ N . The nuclei of a C-loop coincide, thus it suffixes only to show that x −1 nx ∈ Nλ ⇔ Lx −1 nx Ly = L(x −1 nx)y ⇔ Lxnx Ly = L(xnx)y , which is true. U n Proposition 3.2 Let G be a C-loop with nucleus N . Then G/N is a Steiner loop. Proof In a C-loop, x 2 ∈ N ∀ x ∈ G. Thus, G/N is an inverse property loop of exponent 2. Whence, G/N is a Steiner loop. n U Lemma 3.11 There is no C-loop G with nucleus of index 2. Proof This is proved by contradiction. Let N, xN be the two cosets of G/N. It can U n be shown that x ∈ N . Remark 3.5 Even though it is tough to find a one line argument to show that |N | /= 2. However, one can do this by hand and computer.

Question Can you find a one line argument to show that |N | /= 2 in Lemma 3.11?

3.2.4.2

Relationships Between C-Loops and Some Other Loops

The following results relate C-loops with some other types of loops. Theorem 3.19 A loop G is a Steiner loop if and only if it is a commutative inverse property loop of exponent 2. Proof If G is a Steiner loop, .x · e = x and thus by symmetric property, .x · x = xx = e .∀ x ∈ G. Thus by definition, G is an inverse property loop of exponent 2. Conversely, a commutative inverse property loop of exponent 2 is totally symmetric : .x = x −1 and .x −1 (xy) = y ⇒ x(xy) = y. U n An improvement of this result is stated in Lemma 3.12. Lemma 3.12 A loop G is a Steiner loop if and only if it is an inverse property loop of exponent two. Proof From the definition of Steiner loop in Definition 3.8, it is an inverse property loop of exponent 2. Conversely, let .z = xy. Then .xz = x(xy) = x −1 (xy) = y, and similarly , .x = yz, .yx = z. Thus G is commutative. As .(yx)x = y by the right inverse property, G is a Steiner loop. U n Lemma 3.13 Every Steiner loop is a C-loop. Proof Note that .(xy)y = x follows from the definition of Steiner loop in Definition 3.8 and .y(yz) = z follows from commutativity. Thus, .x(y(yz)) = xz = U n ((xy)y)z.

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Lemma 3.14 Let .(G, ·) be a Cayley loop, then .(G, ·) is a central loop. Proof Using the definition of a Cayley loop in Definition 3.8, let .a1 , a2 , a3 be the generators of G. Then : .(ai aj · aj )ak = ai aj2 · ak = ai3 ak = ak ai , .ai (aj · aj ak ) = U n ai (aj2 ak ) = ai ak3 = ak ai . Therefore, G is a C-loop.

Going by Lemmas 3.13 and 3.14, Steiner loops and the Cayley loop are examples of central loops.

Corollary 3.4 A loop is a Steiner loop .⇔ it is a C-loop of exponent two. Proof This follows from Lemma 3.12 since a C-loop has the inverse property.

U n

Lemma 3.15 Flexible C-loops are ARIF loops. 2 =R R Proof Every C-loop satisfies .Rxy x y(xy) = R(xy)x Ry . Thence, a C-loop is an ARIF loop. U n

Theorem 3.20 Let .(Q, ·) be a left conjugacy closed (right conjugacy closed, conjugacy closed) loop that is also an LC-(RC-, C-) loop, then .(Q, ·) is a left Bol (right Bol, Moufang) loop. Proof The proof follows by using the autotopisms that are equivalent to the U n definitions of these loops. cf. Definition 3.9.

3.2.4.3

Order of C-Loops

Proposition 3.3 Let G be a non-associative C-loop of order n with nucleus N of order m. Then : (i) n/m ≡ 2 mod 6 or n/m ≡ 4 mod 6, (ii) n is even, (iii) if n = pk for some prime p and positive integer k, then p = 2 and k > 3. Moreover, there is a non-associative non-Steiner C-loop of order 2k for every k > 3. Proof In a C-loop, G/N is a Steiner loop, then since |G/N| = r ⇔ r ≡ 1 mod 6 or r ≡ 3 mod 6 for a Steiner quasigroup G, (i) follows immediately. (ii) and (iii) follow by (i). U n Lemma 3.16 C-loops have the strong monogenic Lagrange property. Proof A finite loop that is left power alternative and has the left inverse property has the strong monogenic Lagrange property. U n Corollary 3.5 Let x be an element of a finite C-loop G. Then the order of x divides the order of G.

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Proof This follows by Lemma 3.16.

U n

Weak Cauchy Property in C-Loops In [139], the authors have been able to show by a counterexample that C-loops do not have the weak Cauchy property.

3.2.4.4

Commutative C-Loops

Lemma 3.17 ([139]) Let G be a commutative C-loop. Then (xy)n = x n y n for all x, y ∈ G and n ∈ Z+ . Proof In a commutative inverse property and alternative loop, this result is true. Thus, it is true for C-loops. U n Proposition 3.4 Let G be a commutative C-loop, and let K = {x ∈ G : x 2 = e}. Then K is a normal subloop of G and G/K is a group. Proof By Lemma 3.17, the map x |→ x 2 is an endomorphism of G, thus its kernel K is a normal subloop of G. U n Corollary 3.6 Let G be a commutative C-loop and let A be the subloop of G generated by all associators (x, y, z), where x, y, z ∈ G. Then A is of exponent 2. Proof This follows from Proposition 3.4.

U n

Lemma 3.18 Let G be a C-loop. Then all associators of G commute with all nuclear elements of G. In particular, associators commute with all squares. Proof In a loop whose nucleus is normal, all associators of G commute with all nuclear elements of G. Hence, the claim follows. n U Lemma 3.19 Let G be a finite commutative C-loop. Let ; U = {x ∈ G : |x| is a power of 2}, V = {x ∈ G : |x| is relatively prime to 2}

.

Then : (i) (ii) (iii) (iv) (v) (vi)

U ≤ G, V ≤ G, V is contained in the nucleus of G ; hence V is a commutative group, V < G, U < G, U V = {uv : u ∈ U, v ∈ V } = G, U ∩ V = {e}.

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Proof This is proved by keeping in mind that since G is commutative and diassociative, (xy)n = x n y n ∀ x, y ∈ G. U n Theorem 3.21 Let G be a finite commutative C-loop. Then, G = U × V where U = {x ∈ G : |x| is a power of 2}, V = {x ∈ G : |x| is odd}

.

Proof This follows from Lemma 3.19 and the fact that in a loop G with normal subloops U, V such that U ∩ V = {e} and U V = {uv : u ∈ U, v ∈ V } = G, G = U ×V. U n

3.2.5 The Bryant Schneider Group and the Holomorph of a C-Loop Adeniran [2] established a condition under which an element of the Bryant Schneider group of a C-loop will form an automorphism. Chiboka [34] got a similar result for extra loops but the condition was necessary and sufficient which is not the case for C-loops in [2]. This latter author went further to show that elements of the Bryant Schneider group of a C-loop can be expressed as a product of pseudoautomorphism and right translations of elements of the nucleus of the loop but Chiboka [34] corresponding result on extra loops was a general form of this and was also possible for left translations. The two authors confirmed that the Bryant Schneider groups of these two varieties of loops are a kind of generalized holomorph of the loops. In fact, this result is true for Bol-loops according to Robinson [146]. Robinson [144] explored the holomorphy of extra loops while Robinson [145] discussed the embeddiment of a Bol loop in a group. Theorem 3.22 Let .(Q, ·) be a C-loop. Then, .θ ∈ BS(Q, ·) implies .θ ∈ AU M(Q, ·) provided .(θ Rg −1 , θ Lf −1 , θ ) ∈ AU T (Q, ·) such that .f, g ∈ N(Q, ·). Proof This is achieved by using .(Rx−2 , L2x , I ) ∈ AU T (Q, ·) of Theorem 3.12 and .(θ, θ, θ ) ∈ AU T (Q, ·). U n Theorem 3.23 Let .(Q, ·) be a C-loop and let .θ ∈ SY M(Q). Then .θ ∈ BS(Q, ·) if there exist a unique .α ∈ P Sρ (Q, ·) and a unique .f ∈ N(Q, ·) such that .θ = αRf . Proof This goes in the same manner as Theorem 3.22.

U n

Theorem 3.24 Let .(Q, ·) be a C-loop. If .x, y ∈ Q, let .o be a binary operation defined on the right pseudo-automorphism group .P Sρ (Q, ·) by α o β = αRx βRy R(xβ·y)−1 ∀ α, β ∈ P Sρ (Q, ·)

.

Let .H = P Sρ (Q, ·) × Q and define a binary operation .◦ on H by

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(α, x) ◦ (β, y) = (α o β, xβ · y).

.

Then, .(H, ◦) is a group and .(H, ◦) ∼ = BS(Q, ·). U n

Proof This is achieved by using Theorem 3.23. Theorem 3.25 Let .(Q, ·) be an extra loop, and .θ ∈ BS(Q, ·) such that α = (θ Rg , θ L−1 g , θ ) ∈ AU T (Q, ·)

.

for some .g ∈ Q. Then, .θ ∈ AU M(Q, ·) ⇔ f ∈ C(Q, ·). Proof We prove this by using the autotopisms which are equivalent to the identities of extra and Moufang loop. U n

3.3 Quasigroups and Loops of Non Bol-Moufang Type Quasigroups and loops that do not meet the criteria of the Bol-Moufang type as discussed in the previous chapters are regarded as quasigroups and loops of non BolMoufang type. Many of such loops fall into some varieties of loops called Osborn loops. Examples are VD-loops, conjugacy closed loops, Basarab loop and universal WIPLs. The identities describing some quasigroups and loops that are not of BolMoufang types are given below. Definition 3.9 A loop .(Q, ·) is called: −1

−1

(a) a VD-loop if .(·)x = (·)Rx Lx and .x (·) = (·)Lx Rx i.e. .Tx = Rx L−1 x ∈ P Sρ (Q, ·) with companion .c = x and .Tx−1 = Lx Rx−1 ∈ P Sλ (Q, ·) with companion .c = x respectively for all .x ∈ Q. These are respectively equivalent to (x\(yx)) · [(x\(yx))x] = (x\(yz · x))x and x(xy/x) · (xz/x) = x((x · yz)/x)

.

(b) a conjugacy closed loop (CC-loop) if the following identities hold in .(Q, ·): .

x · yz = (xy)/x · xz ' '' ' left conjugacy closed loop (LCC-loop)

and

zy · x = zx · x\(yx) ' '' ' right conjugacy closed loop (RCC-loop)

(c) a universal WIPL if the WIP holds in .(Q, ·) and all its isotopes. Remark 3.6 Going by Definition 3.9(b), a loop .(G, ·) is a CC-loop if .(G, ·) is an LCC-loop and an RCC-loop. This can also be expressed in functional form: there exist functions .f, g : G × G → G, then .(G, ·) is an LCC-loop if x · yz = f (x, y) · xz, RCC-loop if zy · x = zx · g(x, y) ∀ x, y, z ∈ G.

.

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VD-Loop, Moufang Loops and CC-Loops Every VD-loop is a G-loop based on a characterization of G-loops. The two identities defining it are equivalent to the fact that middle inner mappings and their inverses are right and pseudo-automorphisms with each element as companions. A loop is a G-loop if and only if each element is a companion of some right and left pseudo-automorphisms. Moufang loops with nuclear fourth powers and CC-loops with nuclear squares are VD-loops.

Definition 3.10 A conjugacy closed quasigroup (CC-quasigroup) is a quasigroup that obeys the identities x · (yz) = {[x · (y · (x\x))]/x} · (xz) and (zy) · x = (zx) · {x\[((x/x) · y) · x]}.

.

Definition 3.11 A loop .(Q, ·) is called: (a) a generalized Moufang loop, if one of the identities .x(yz · x) = (y λ x λ )ρ · zx and .(x · zy)x = xz · (x ρ y ρ )λ holds in Q. (b) a K-loop (or Basarab loop), if the identities .(x · yx ρ ) · xz = x · yz and .(yx)[(x λ · xz) · x] = yz · x hold in Q. Remark 3.7 Note that the generalized Moufang loop in Definition 3.11 is equivalent to both Osborn loop and the WIP.

Kinyon’s Conjecture Kinyon [112] conjectured that “Every CC-quasigroup is isotopic to an Osborn loop”.

3.3.1 Isotopy Consider .(G, ·) and .(H, ◦) been two groupoids (quasigroups, loops). Let .A, B and C be three bijective mappings, that map G onto H . The triple .α = (A, B, C) is called an isotopism of .(G, ·) onto .(H, ◦) if and only if xA ◦ yB = (x · y)C ∀ x, y ∈ G.

.

So, .(H, ◦) is called a groupoid (quasigroup, loop) isotope of .(G, ·). If .C = I is the identity map on G so that .H = G, then the triple .α = (A, B, I ) is called a principal isotopism of .(G, ·) onto .(G, ◦) and .(G, ◦) is called a principal isotope of .(G, ·). Eventually, the equation of relationship now becomes

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x · y = xA ◦ yB ∀ x, y ∈ G

.

which is easier to work with. But if .A = Rg and .B = Lf where .Rx : G → G, the right translation is defined by .yRx = y · x and .Lx : G → G, the left translation is defined by .yLx = x · y for all .x, y ∈ G, for some .f, g ∈ G, the relationship now becomes x · y = xRg ◦ yLf ∀ x, y ∈ G

.

or x ◦ y = xRg−1 · yL−1 f ∀ x, y ∈ G.

.

With this new form, the triple .α = (Rg , Lf , I ) is called an .f, g-principal isotopism of .(G, ·) onto .(G, ◦), f and g are called translation elements of G or at times written in the pair form .(g, f ), while .(G, ◦) is called an .f, g-principal isotope of .(G, ·). The last form of .α above given rises to an important result in the study of loop isotopes of loops. Theorem 3.26 Let .(G, ·) and .(H, ◦) be two distinct isotopic loops. For some .f, g ∈ G, there exists an .f, g-principal isotope .(G, ∗) of .(G, ·) such that .(H, ◦) ∼ = (G, ∗). With this result, to investigate the isotopic invariance of an isomorphic invariant property in loops, one simply needs only to check if the property in consideration is true in all .f, g-principal isotopes of the loop. A property is isotopic invariant if whenever it holds in the domain loop i.e. .(G, ·) then it must hold in the co-domain loop i.e. .(H, ◦) which is an isotope of the formal. In such a situation, the property in consideration is said to be a universal property hence the loop is called a universal loop relative to the property in consideration as often used by Nagy and Strambach [128] in their algebraic and geometric study of the universality of some types of loops. For instance, if every isotope of a “certain” loop is a “certain” loop, then the formal is called a universal “certain” loop. So, we can now restate Theorem 3.26 as: Theorem 3.27 Let .(G, ·) be a “certain” loop where “certain” is an isomorphic invariant property. .(G, ·) is a universal “certain” loop if and only if every .f, gprincipal isotope .(G, ∗) of .(G, ·) has the “certain” loop property.

A Group is a G-Loop A loop is called a G-loop if it is isomorphic to all its loop isotopes. Every group is a G-loop. To every loop isotope K of a group G there exists a principal loop isotope H of G such that .H ∼ = K (cf. Theorem 3.26). Taking ∼ advantage of associativity, .G ∼ H . Hence, . G = = K. An extra loop is a G-loop.

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3.3.2 Osborn Loops A loop .(I, ·) or .I (·) is called an Osborn loop if it obeys the identity: (x λ \y) · zx = x(yz · x)

.

(3.61)

for all .x, y, z ∈ I . The term Osborn loops first appeared in a work of Huthnance Jr [52] on generalized Moufang loops in 1968. However, its definition based on the identity (3.61) is according to Basarab and Belioglo [16] in 1979 after Basarab [12, 13] earlier works on isotopy of WIPLs and Osborn loops.

3.3.2.1

Osborn Loops (1959–1961)

The origin of Osborn loop can be traced to the work of J.M. Osborn [130] which he started in 1960 on weak inverse property loops (WIPLs). He observed that universal WIPL obeys identity: yx · (zEy · y) = (y · xz) · y for all x, y, z ∈ G

.

(3.62)

−1 = L R L−1 R −1 . where .Ey = Ly Ly λ = Ry−1 ρ Ry y y y y Some of the results from the works of J. M. Osborn are highlighted below.

Theorem 3.28 Every universal WIPL is an Osborn loop Theorem 3.29 A universal WIPL which is also an IPL or CIPL or commutative loop is a Moufang loop.

3.3.2.2

Osborn Loops (1962–1968)

Eight years after J.M. Osborn’s work [130] on WIPLs, E.D. Huthnance [52] in 1968 studied the theory of generalized Moufang loops in his Ph.D thesis. He therefore, named a loop that necessarily and sufficiently obeys the identity (3.62) a generalized Moufang loop and later on in the same thesis, he called them M-loops. On the other hand, he called a universal WIPL an Osborn loop. It was this year 1968, that the term Osborn loop was first used, named after J.M. Osborn’s work [130] of 1960. Even at this time, very little was still known about Osborn loops. We could describe it as a baby that has just undergone its naming ceremony, so to say. Time was still needed. However let us consider some results obtained by Huthnance. Theorem 3.30 Let G be an Osborn loop. .Nρ (G) = Nλ (G) = Nμ (G) = N(G) and .N(G)< G Remark 3.8 The result above holds for universal Osborn loops. Such Osborn loops are G-loops.

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Theorem 3.31 Let G be a universal Osborn loop with nucleus N . Then, .G/N has the WIP and so .(G/N)/N(G/N) is a Moufang loop. Thus, if .N2 is the second nucleus of G, then .G/N2 is a Moufang loop.

Osborn Loop, a Generalization of Moufang Loop Every Moufang loop is an Osborn loop but every Osborn is not Moufang. Since a Moufang loop is a RIPL, then comparing the Moufang identity with Osborn’ identity, it can be seen that Moufang loops are Osborn loops. Thus, Osborn loops generalize Moufang loops. No wonder Huthnance called them generalized Moufang loops. However, this is different from the generalized Moufang loop defined by Basarab [15].

Lemma 3.20 An Osborn loop that is flexible or which has the LAP or RAP or LIP or RIP or AAIP is a Moufang loop. But an Osborn loop that is commutative or which has the CIP is a commutative Moufang loop. Lemma 3.21 Let G be a WIP Osborn loop. If .a = x ρ x, then for all .x ∈ G: 2

2

xa = x λ , ax λ = x ρ , x ρ a = x λ , ax = x ρ ,

.

xa −1 = ax, a −1 x λ = x λ a, a −1 x ρ = x ρ a.

.

or equivalently Jλ : x |→ x · x ρ x, Jρ : x |→ x ρ x · x λ ,

.

Jλ : x |→ x ρ · x ρ x, Jρ2 : x |→ x ρ x · x,

.

x(x ρ x)−1 = (x ρ x)x, (x ρ x)−1 x λ = x λ · x ρ x, (x ρ x)−1 x ρ = x ρ (x ρ x).

.

Earliest Constructions of Infinite Osborn Loops Some of the earliest examples of infinite Osborn loops were constructed by Huthnance [52] in 1968. Two of them are given below:

Example 3.2 Let .K = 2Z × Z. Define a binary operation .• on K as : [2i, m] • [2j, n] = [2(i + j ), (m + n) − ij (2j − 1)] ∀ i, j, m, n ∈ Z.

.

(K, •) is an Osborn loop without the WIP.

.

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Example 3.3 Let .H = Z × Z × Z. Define a binary operation .* on H by : [2i, k, m] * [2j, p, q] = [2i + 2j, k + p − ij (2j − 1), q + m − ij (2j − 1)]

.

[2i + 1, k, m] * [2j, p, q] = [2i + 2j + 1, k + p − ij (2j − 1) − j 2 + j, q + m − ij (2j − 1) − j 2 ] [2i, k, m] * [2j + 1, p, q] = [2i + 2j + 1, m + p − ij (2j + 1), q + k − ij (2j + 1)] [2i + 1, k, m] * [2j + 1, p, q] = [2i + 2j + 2, m + p − ij (2j + 1) − j 2 + j, q + k − ij (2j + 1) − j 2 ] ∀ i, j, k, m, p, q ∈ Z. (H, *) is an Osborn loop.

.

3.3.2.3

Osborn Loops (1969–1979)

In 1970, Basarab [12] continued the work of J.M. Osborn [130] of 1961 on universal WIPLs by studying isotopes of WIPLs that are also WIPLs. His studies reveal that not all varieties of Osborn loops are universal WIPLs as sighted by Huthnance [52] in 1968. Basarab, therefore, worked on identity that is all embracing, to capture all varieties of Osborn loops. Therefore, in 1979, he dubbed a loop .(G, ·) satisfying any of the four equivalent identities: (x λ \y) · zx = x(yz · x).

.

OS1

x(yz · x) = (x · yEx ) · zx.

(3.63) (3.64)

(x · yz)x = xy · (zEx−1 · x).

(3.65)

xy · (z/x ) = (x · yz)x

(3.66)

ρ

where .Ex = Rx Rx ρ = (Lx Lλx )−1 = Rx Lx Rx−1 L−1 x for all .x, y, z ∈ G an Osborn loop. It should be observed that this type of Basarab’ Osborn loop is not necessarily a universal WIPL as generally defined by Huthnance [52]. However, there should be no confusion whatsoever, about the two definitions. Universal WIPLs are a variety of Osborn loops (of Basarab). And Osborn loops generalize Moufang loops. Below are some results gotten in the period under review. Theorem 3.32 Every VD-loop is an Osborn loop. Theorem 3.33 An Osborn loop Q in which .x 2 ∈ N(Q) for all .x ∈ Q is a G-loop.

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Theorem 3.34 If in an Osborn loop Q and .x 2 = e for all .x ∈ Q, then Q is an abelian group. Theorem 3.35 An Osborn loop .(Q, ·) in which .x 2 ∈ N(Q) for all .x ∈ Q and .N (Q) /= {e} is an extension of a group by means of an abelian group. Theorem 3.36 Let G be an Osborn loop. .Innρ (G) = Innλ (G).

3.3.2.4

Osborn Loops(1980–1990)

In 1990, Chiboka [32] adopted the Huthnance [52] definition of an Osborn loop. ρ She later deduced some properties of .Ex such as .Ex = Exλ = Ex . 3.3.2.5

Osborn Loops(1991–2005)

In this period, Basarab [14, 15] extended his study to Osborn’s G-loops and generalized Moufang loops. Adopting the Huthnance’s definition would mean working on a variety of Osborn loop. Hence, Kinyon [112] in 2005 re-awaken researchers’s interest in this aspect of loop theory, called Osborn loop that was seemingly dormant for a couple of years after the work of Basarab and Chiboka in 1979 and 1990 respectively. For the first time, Kinyon [112] made a giant effort to classify Osborn loops. He found that, the smallest Osborn that is not associative is of order 16, and there are two of such examples. These are concrete examples and are presented in Tables 3.3 and 3.4. He also posed two open problems.

The Two Open Problems on Osborn Loops by Kinyon The first open problem that was posed was the question “Is every Osborn loop universal?”. If not every Osborn loop is universal, then does there exist a “nice” identity characterizing universal Osborn loops? The second open problem that was posed was “If not every Osborn loop is universal, does there exist a proper Osborn loop with trivial nucleus?”

We now consider some of the results during the period under review. Theorem 3.37 Let Q be an Osborn loop. Then .Multλ (Q) and .Multρ (Q) are normal subgroups of .Mult(Q). Theorem 3.38 Let G be an Osborn loop. .R(x, y) ∈ P Sρ (G) with companion (xy)λ (y λ \x) and .L(x, y) ∈ P Sλ (G) for all .x, y ∈ G. Furthermore, .R(x, y)−1 = −1 λ λ [L−1 y ρ , Rx ] = L(y , x ) for all .x, y ∈ G.

.

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The second part of the Theorem 3.38 is trivial for Moufang loops. For CC-loops, it was first observed by Drápal [39] and then later by Kinyon and Kunen [114]. Corollary 3.7 If G is an Osborn loop that is also an .Aρ -loop and an .Aλ -loop, then G/N is a commutative Moufang loop.

.

Corollary 3.8 For a CC-loop G with nucleaus .N, G/N is a commutative Moufang loop. Proof Since in a CC-loop G, .R(x, y), L(x, y) ∈ AU M(G) then G is both an .Aρ loop and an .Aλ -loop but not an A-loop. So, from Corollary 3.7 the result follows. U n Theorem 3.39 Every CC-loop is an Osborn loop. After the introduction of CC-loops by Goodaire and Robinson [49], their structures have been studied by Goodaire and Robinson [50], Kunen [121], Kinyon and Kunen [113, 114], Kinyon et al. [115], Phillips [138] A recent work on LCCloop was done by George et al. [46].

3.3.2.6

Examples of Osborn Loops

Example 3.4 The smallest order for which proper (non-Moufang and non-CC) Osborn loops with non-trivial nucleus exists is 16. There are two of such loops as earlier mentioned in Sect. 3.3.2.5. • Each of the two is a G-loop. • Each contains as a subgroup, the dihedral group(D4 ) of order 8. • For each loop, the center coincides with the nucleus and has order 2. The quotient by the center is a non-associative CC-loop of order 8. • The second center is Z2 × Z, and the quotient is Z4 . • One loop satisfies L4x = Rx4 = I , the other does not. Their multiplication tables are presented in form of acceptable loops in Tables 3.3 and 3.4. These two Osborn loops are Smarandache loops (that is, a loop that has at least a non-trivial subgroup). Smarandache quasigroup and loop have been studied by Jaiyéo.lá [63–70, 73]. The Smarandache subgroup in each of them is the dihedral group (D4 ) of order 8. Efforts to go higher have been really challenging because Osborn loop is a ’super’ loop. This is not unexpected of a loop that generalizes Moufang loops, VD loops, CC-loops and universal WIPLs. This definitely won’t be a loop of smaller order compared to say a Moufang loop or CC-loop of smallest order; 12 or 16. No wonder the Osborn loop of smallest order is of order 16. Whereas, the least of Bol and Moufang loops are of orders 8 and 12 respectively. So, constructing higher orders of Osborn loops will require great efforts. This makes the construction of proper Osborn of higher orders important.

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Table 3.3 The first Osborn loop of order 16 that is a G-loop ·

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15

3 3 4 1 2 8 7 6 5 11 12 9 10 16 15 14 13

4 4 3 2 1 7 8 5 6 12 11 10 9 15 16 13 14

5 5 6 7 8 1 2 3 4 15 16 13 14 12 11 10 9

6 6 5 8 7 2 1 4 3 16 15 14 13 11 12 9 10

7 7 8 5 6 4 3 2 1 13 14 15 16 9 10 11 12

8 8 7 6 5 3 4 1 2 14 13 16 15 10 9 12 11

9 9 10 11 12 13 14 15 16 5 6 8 7 1 2 4 3

10 10 9 12 11 14 13 16 15 6 5 7 8 2 1 3 4

11 11 12 9 10 16 15 14 13 7 8 6 5 4 3 1 2

12 12 11 10 9 15 16 13 14 8 7 5 6 3 4 2 1

13 13 14 15 16 10 9 12 11 3 4 2 1 7 8 6 5

14 14 13 16 15 9 10 11 12 4 3 1 2 8 7 5 6

15 15 16 13 14 11 12 9 10 1 2 4 3 6 5 7 8

16 16 15 14 13 12 11 10 9 2 1 3 4 5 6 8 7

12 12 11 10 9 15 16 13 14 6 5 7 8 1 2 4 3

13 13 14 15 16 10 9 12 11 2 1 3 4 6 5 7 8

14 14 13 16 15 9 10 11 12 1 2 4 3 5 6 8 7

15 15 16 13 14 11 12 9 10 4 3 1 2 7 8 6 5

16 16 15 14 13 12 11 10 9 3 4 2 1 8 7 5 6

Table 3.4 The second Osborn loop of order 16 that is a G-loop o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3.3.2.7

2 2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15

3 3 4 1 2 8 7 6 5 11 12 9 10 16 15 14 13

4 4 3 2 1 7 8 5 6 12 11 10 9 15 16 13 14

5 5 6 7 8 1 2 3 4 15 16 13 14 12 11 10 9

6 6 5 8 7 2 1 4 3 16 15 14 13 11 12 9 10

7 7 8 5 6 4 3 2 1 13 14 15 16 9 10 11 12

8 8 7 6 5 3 4 1 2 14 13 16 15 10 9 12 11

9 9 10 11 12 13 14 15 16 7 8 6 5 3 4 2 1

10 10 9 12 11 14 13 16 15 8 7 5 6 4 3 1 2

11 11 12 9 10 16 15 14 13 5 6 8 7 2 1 3 4

Osborn Loops (2006 Till Date)

Following the first open problem posed by Kinyon [112] in 2005 on universality of Osborn loops, in a series of research works, Jaiyéo.lá [71, 80, 81], Jaiyéo.lá and Adéníran [55, 58, 95–97, 99, 130], Jaiyéo.lá et al. [100, 101] developed the

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characterization of the universality of Osborn loops in various fashions (identities, permutations, simplicial complexes etc). Thereafter, some of the identities deduced in the studies found application for Osborn loops to cryptography as reported by Jaiyéo.lá and Adéníran [98] and Jaiyéo.lá [75, 79]. Furthermore, Jaiyéo.lá and Effiong [93, 94], Effiong [41] have studied the properties and structure of Basarab loop. Some other identities that equivalently define an Osborn loop (asides (3.63) to (3.66)) exist in literature and were highlighted in Isere et al. [59]. In fact, Drapal and Kinyon [40] rediscovered some of these identities and some additional ones. One of such is OS0

.

x(yx λ · x) · zx = x(yz · x)

(3.67)

In [101], the second open problem “Does there exist a proper Osborn loop with a trivial nucleus?” of Kinyon for finite Osborn loops was expressed in terms of the orders of the nucleus, 2.nd Bryant-Schneider and automorphism groups of the loop. Furthermore, some necessary conditions (based on the orders of the 1.st BryantSchneider, 2.nd Bryant-Schneider and automorphism groups) for the existence of a universal (left, right universal) Osborn loop with trivial nucleus were deduced. Also, some sufficient conditions (based on the orders of the 1.st, 2.nd Bryant-Schneider and automorphism groups) for the non-existence of a universal(left, right universal) Osborn loop with trivial nucleus were deduced. It was observed that these results altogether could be used to peep through the windows of the rooms of solutions to the Phillips’ open problem and Doro’s conjecture.

Phillips’ Problem and Doro’s Conjecture Is there a Moufang loop of odd order with trivial nucleus? Does a Moufang loop with trivial nucleus necessarily have normal commutant?

It is good news to announce that Csörgö [35] recently proved the Phillips’ open problem in the negative. That is, there is no Moufang loop of odd order with trivial nucleus. However, the results in [101] captured beyond Moufang loops, but also other Osborn loops. Hence, there is still the need to investigate Phillips’ open problem and Doro’s conjecture for some other Osborn loops (e.g. CC-loop, VD-loop etc.). The Doro’s conjecture was conceived in Doro [38]. Gagola III [45] stated an answer to this question in affirmation while Grishkov and Zavarnitsine [51] showed that the answer is actually generally negative by constructing two infinite series of Moufang loops of exponent 3 whose commutant is not a normal subloop. Csörgö [36] gave necessary and sufficient conditions in the multiplication group for the conjecture to be true. We shall make remarks about Phillips’ problem and Doro’s conjecture in proper Osborn loops at the end of this section.

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Some of the results obtained in this period by authors cited above are presented below while some others are reserved for Sect. 3.3.3. Theorem 3.40 A loop .(Q, ·, \, /) is a universal Osborn loop if and only if it obeys any of the identities below. .

x · u\{(yz)/v · [u\(xv)]} = (x · u\{[y(u\([(uv)/(u\(xv))]v))] /v · [u\(xv)]})/v · u\[((uz)/v)(u\(xv))]

.

x · u\{(yz)/v · [u\(xv)]} = {x · u\{[y(u\(xv))]/v · [x\(uv)]}} /v · u\[((uz)/v)(u\(xv))].

}

}

= OS'0

= OS'1

Proof Let .Q = (Q, ·, \, /) be an Osborn loop with any arbitrary principal isotope Q = (Q, A, /, /) such that .x A y = xRv−1 · yL−1 u = (x/v) · (u\y) ∀ u, v ∈ Q. If .Q is a universal Osborn loop then, .Q is an Osborn loop. .Q obeys identity OS.0 implies

.

'

x A [(y A z) A x] = {x A [(y A x λ ) A x]} A (z A x)

.

(3.68)

'

where .x λ = xJλ' is the left inverse of x in .Q. The identity element of the loop .Q is uv. So, '

'

λ λ −1 −1 x A y = xRv−1 · yL−1 u implies y A y = y Rv · yLu = uv implies

. '

−1 y λ Rv−1 RyL−1 = uv implies yJλ' = (uv)R −1−1 Rv = (uv)R(u\y) Rv = [(uv)/(u\y)]v. u

.

yLu

Thus, using the fact that .x A y = (x/v) · (u\y), .Q is an Osborn loop if and only if (x/v) · u\{[(y/v) · (u\z)]/v · (u\x)} = ((x/v) · u\{[(y/v)(u\([(uv)/(u\x)]v))]

.

/v · (u\x)})/v · u\[(z/v)(u\x)]. Do the following replacements: x ' = x/v ⇒ x = x ' v, z' = u\z ⇒ z = uz' , y ' = y/v ⇒ y = y ' v we have

.

x ' · u\{(y ' z' )/v · [u\(x ' v)]} = (x ' · u\{[y ' (u\([(uv)/(u\(x ' v))]v))]

.

/v · [u\(x ' v)]})/v · u\[((uz' )/v)(u\(x ' v))]. This is precisely identity OS.'0 by replacing .x ' , .y ' and .z' by x, y and z respectively. The proof of the converse is as follows. Let .Q = (Q, ·, \, /) be an Osborn loop that obeys identity OS.'0 . Doing the reverse process of the proof of the necessary

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part, it will be observed that Eq. (3.68) is true for any arbitrary .u, v-principal isotope Q = (Q, A, /, /) of .Q. So, every .f, g-principal isotope .Q of .Q is an Osborn loop. Following Theorem 3.27, .Q is a universal Osborn loop if and only if .Q is an Osborn loop. The proof for the second identity is done similarly by using identity OS.1 . U n

.

Lemma 3.22 Let Q be a loop with multiplication group .Mult(Q). Q )is a universal ( Osborn loop if and only if the triple . α(x, u, v), β(x, u, v), γ (x, u, v) ∈ AU T (Q) ( ) or the triple . R[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv γ (x, u, v)Rv , β(x, u, v), γ (x, u, v) ∈ AU T (Q) for all .x, u, v ∈ Q where .α(x, u, v) = R(u\([(uv)/(u\(xv))]v)) Rv R[u\(xv)] Lu Lx Rv , .β(x, u, v) = Lu Rv R[u\(xv)] Lu and .γ (x, u, v) = Rv R[u\(xv)] Lu Lx are elements of .Mult(Q). Proof This is obtained from identity OS.'0 or OS.'1 of Theorem 3.40.

U n

Theorem 3.41 Let Q be a loop with ( multiplication group .Mult(Q). If Q is a uni-) versal Osborn loop, then the triple . γ (x, u, v)R(u\[(u/v)(u\(xv))]) , β(x, u, v), γ (x, u, v) ∈ AU T (Q) for all .x, u, v ∈ Q where .β(x, u, v) = Lu Rv R[u\(xv)] Lu and .γ (x, u, v) = Rv R[u\(xv)] Lu Lx are elements of .Mult(Q). Proof Theorem 3.40 will be employed. Let .z = e in identity OS.'0 , then x · u\{y/v · [u\(xv)]} = (x · u\{[y(u\([(uv)/(u\(xv))]v))]

.

/v · [u\(xv)]})/v · u\[(u/v)(u\(xv))]. So, identity OS.'0 can now be written as { x · u\{(yz)/v · [u\(xv)]} = {x · u\[y/v · (u\(xv))]} } /{u\[(u/v)(u\(xv))]} · u\[((uz)/v)(u\(xv))].

.

From where we obtain AU T (Q).

( ) γ (x, u, v)R(u\[(u/v)(u\(xv))]) , β(x, u, v), γ (x, u, v)

.

∈ U n

Lemma 3.23 Let .(Q, ·, \, /) be a universal Osborn loop. The following identities are satisfied: .

y{u\([(uv)/(u\(xv))]v)} = {(y[u\(xv)])/v · [x\(uv)]}/[u\(xv)] · v , ' '' ' OSI01

{(uz)/v · u\({(yv)(u\([(uv)/z]v))}/v · z)}/v · (u\[(u/v)z]) = (uz)/v · u\(yz) ' ' '' OSI01.1

and

3 Some Varieties of Loops .

145

(uz)/v · u\{(yv · z)/v · [((uz)/v)\(uv)]} = [(uz)/v · u\(yz)]/{u\[(u/v)z]} · v . ' '' ' OSI01.2

Furthermore, .

{u\({(uy · u)(u\(uu · u))}/u)}/u · uρ = y , uu · u\(uu · u) = (u · uu)u, ' ' '' OSI01.1.1

v λ · u\{(yv · uρ )/v · [v λ \(uv)]} = [v λ · u\(yuρ )]/{u\[(u/v)uρ ]} · v , ' '' ' OSI01.2.1

v λ (y · v λ \v) = (v λ y)/v λ · v , ' '' ' OSI01.2.2

v λ · (v · v λ \v) = v

λ2

· v = (v λ · vv)v and v(v ρ · v\v ρ ) = v λ · v ρ

are also satisfied. Proof To prove these identities, we shall make use of the three autotopisms in Lemma 3.22 and Theorem 3.41. In a quasigroup, any two components of an autotopism uniquely determine the third. So equating the first components of the three autotopisms, it is easy to see that .α(x, u, v)

= γ (x, u, v)R(u\[(u/v)(u\(xv))]) = R[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv γ (x, u, v)Rv .

The establishment of the identities OSI.01 , OSI.01.1 and OSI.01.2 follows by using the bijections appropriately to map an arbitrary element .y ∈ Q as follows: α(x, u, v) = R[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv γ (x, u, v)Rv implies that

OSI.01

.

R(u\([(uv)/(u\(xv))]v)) Rv R[u\(xv)] Lu Lx Rv = R(u\([(uv)/(u\(xv))]v)) γ (x, u, v)Rv

.

= R[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv γ (x, u, v)Rv which gives R(u\([(uv)/(u\(xv))]v)) = R[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv . So, for any y ∈ Q,

.

yR(u\([(uv)/(u\(xv))]v)) = yR[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv implies that

.

y{u\([(uv)/(u\(xv))]v)} = {(y[u\(xv)])/v · [x\(uv)]}/[u\(xv)] · v.

.

OSI.01.1

Consider .α(x, u, v) = γ (x, u, v)R(u\[(u/v)(u\(xv))]) , then for all y ∈ Q, .yα(x, u, v)

= yR(u\([(uv)/(u\(xv))]v)) Rv R[u\(xv)] Lu Lx Rv = yγ (x, u, v)R(u\[(u/v)(u\(xv))])

yα(x, u, v) = yRv R[u\(xv)] Lu Lx R(u\[(u/v)(u\(xv))]) .

Consequently,

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{x · u\({y(u\([(uv)/(u\(xv))]v))}/v · [u\(xv)])}/v

.

= {x · u\(y/v · [u\(xv)])}/(u\[(u/v)(u\(xv))]). Now replace .y/v by y and post-multiply both sides by .(u\[(u/v)(u\(xv))]) to get {x · u\({(yv)(u\([(uv)/(u\(xv))]v))}/v · [u\(xv)])}/v · (u\[(u/v)(u\(xv))])

.

= x · u\(y · [u\(xv)]). Again, let .z = u\(xv) which implies that .x = (uz)/v and hence we now have {(uz)/v · u\({(yv)(u\([(uv)/z]v))}/v · z)}/v · (u\[(u/v)z]) = (uz)/v · u\(yz).

.

OSI.01.2

Consider

R[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv γ (x, u, v)Rv = γ (x, u, v)R(u\[(u/v)(u\(xv))]) ,

.

then for all .y ∈ Q, yR[u\(xv)] Rv R[x\(uv)] R[u\(xv)] Rv γ (x, u, v)Rv = yγ (x, u, v)R(u\[(u/v)(u\(xv))])

.

results in .

({[(y [u\ (xv)]) /v · [x\ (uv)]] / [u\ (xv)] · v} γ (x, u, v)) /v = (yγ (x, u, v)) / (u\ [(u/v) (u\ (xv))])

which is equivalent to the equation below after substituting the value of γ (x, u, v) and post-multiply both sides by v:

.

[ ] x · u\({ (y[u\(xv)])/v · [x\(uv)] /[u\(xv)] · v}/v · [u\(xv)]) ( ) = [x · u\ y/v · [u\(xv)] ]/(u\[(u/v)(u\(xv))]) · v.

.

Do the replacement .z = u\(xv) ⇒ x = (uz)/v to get [ ] ( (uz)/v · u\ (yz)/v · [(uz)/v\(uv)] = [(uz)/v · u\ y/v · z)]/(u\[(u/v)z]) · v.

.

Now, replace y by yv to get [ ] ( (uz)/v · u\ (yv · z)/v · [(uz)/v\(uv)] = [(uz)/v · u\ yz)]/(u\[(u/v)z]) · v.

.

3 Some Varieties of Loops

147

Identity OSI.01.1.1 is deduced from identity OSI.01.1 while identities OSI.01.2.1 and OSI.01.2.2 are deduced from identity OSI.01.2 . The other identities are gotten from OSI.01.1.1 and OSI.01.2.2 . OSI.01.1.1

Put .u = v in identity OSI.01.1 to get

{(uz)/u · u\({(yu)(u\([(uu)/z]u))}/u · z)}/u · (u\z) = (uz)/u · u\(yz).

.

Now replace z by uz to get {(u · uz)/u · u\({(yu)(u\([(uu)/(uz)]u))}/u · uz)}/u · z = (u · uz)/u · u\(y · uz).

.

Then, substitute .z = uρ and compute to have {u\({(yu)(u\(uu · u))}/u)}/u · uρ = u\y.

.

Replacing y by uy, finally have .{u\({(uy · u)(u\(uu · u))}/u)}/u · uρ = y. OSI.01.2.1 Substitute .z = uρ in identity OSI.01.2 to get [ ] ( v λ · u\ (yv · uρ )/v · [v λ \(uv)] = [v λ · u\ yuρ )]/(u\[(u/v)uρ ]) · v.

.

OSI.01.2.2

Put .u = e in identity OSI.01.2.1 to get .v λ (y · v λ \v) = (v λ y)/v λ · v.

By putting .y = e in identity OSI.01.1.1 , we have .uu · u\(uu · u) = (u · uu)u. Also, 2 substitute .y = v into identity OSI.01.2.2 and use the fact that .x λ = x λ · xx to get 2 λ λ λ · v = (v λ · vv)v and v(v ρ · v\v ρ ) = v λ · v ρ . .v · (v · v \v) = v U n Theorem 3.42 A loop .(Q, ·, \, /) is a left universal Osborn loop if and only if it obeys any of the following identities. .

x · [(y · zv)/v · (xv)] = (x · {[y([v/(xv)]v)]/v · (xv)})/v · [z · xv] ' '' ' OSλ0

x · [(y · zv)/v · (xv)] = {x · [(y · xv)/v · (x\v)]}/v · [z(xv)]. ' '' ' OSλ1

Proof The procedure of the proof of this theorem is similar to the procedure used to prove Theorem 3.40 by just using the arbitrary left principal isotope .Q = (Q, A , /, /) such that .x A y = xRv−1 · y = (x/v) · y ∀ v ∈ Q. U n Lemma 3.24 Let Q be a loop with multiplication(group .Mult(Q). Q is a) left universal Osborn loop if and only if the triple . α(x, v), β(x, ( ) v), γ (x, v) ∈ AU T (Q) or . R[xv] Rv R[x\v] R[xv] Rv γ (x, v)Rv , β(x, v), γ (x, v) ∈ AU T (Q) for all .x, v ∈ Q where .α(x, v) = R([v/(xv)]v) Rv R[xv] Lx Rv , β(x, v) = Rv R[xv] and .γ (x, v) = Rv R[xv] Lx are elements of .Mult(Q).

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Proof This is obtained from identity OS.λ0 or OS.λ1 of Theorem 3.42.

U n

Theorem 3.43 Let Q be a loop with multiplication group .Mult(Q). If Q )is a ( left universal Osborn loop, then the triple . γ (x, v)R[v λ ·xv] , β(x, v), γ (x, v) ∈ AU T (Q) for all .x, v ∈ Q where .β(x, v) = Rv R(xv) and .γ (x, v) = Rv R(xv) Lx are elements of .Mult(Q). Proof This follows by using identity OS.λ0 or OS.λ1 of Theorem 3.42 the way identity OS.'0 or OS.'1 of Theorem 3.40 was used to prove Theorem 3.41. U n Lemma 3.25 Let .(Q, ·, \, /) be a left universal Osborn loop. The following identities are satisfied: .

y{[v/(xv)]v} = {[y(xv)]/v · (x\v)}/(xv) · v , ' '' ' OSIλ01

z{(yv · zv)/v · z\v} = [z(y · zv)]/(v λ · zv) · v ' '' ' OSIλ01.2

and .

{z · {[(yv)(v/(zv) · v)]/v · zv}}/v · v λ (zv) = z · y(vz) . ' ' '' OSIλ01.1

Furtermore, .

{v λ {[(yv)(vv)]/v}}/v · v λ = v λ y , {z{[v(v/(zv) · v)]z}}/v · v λ (zv) = z · zv , ' ' ' '' ' '' OSIλ01.1.1

.

OSIλ01.1.2

v{(yv · vv)/v} = [v(y · vv)]/(v λ · vv) · v , v[(v · vv)/v] = (v · vv)/(v λ · vv) · v , ' ' ' '' ' '' OSIλ01.2.1

.

OSIλ01.2.2

v[(vv · vv)/v] = [v(v · vv)]/(v λ · vv) · v , v λ [y · v λ \v] = (v λ y)/v λ · v , ' ' ' '' '' ' OSIλ01.2.3

OSIλ01.2.4 2

v · vv = v λ \v · v and vv · vv = v λ \(v λ v) · v are also satisfied.

3 Some Varieties of Loops

149

Proof To prove these identities, we shall make use of the three autotopisms in Lemma 3.24 and Theorem 3.43. In a quasigroup, any two components of an autotopism uniquely determine the third. So equating the first components of the three autotopisms, it is easy to see that α(x, v) = γ (x, v)R[v λ ·xv] = R[xv] Rv R[x\v] R[xv] Rv γ (x, v)Rv .

.

The establishment of the identities OSI.λ01 , OSI.λ01.1 and OSI.λ01.2 follows by using the bijections appropriately to map an arbitrary element .y ∈ Q as follows: OSI.λ01 α(x, v) = R[xv] Rv R[x\v] R[xv] Rv γ (x, v)Rv

.

implies that R([v/(xv)]v) Rv R[xv] Lx Rv = R([v/(xv)]v) γ (x, v)Rv

.

R([v/(xv)]v) Rv R[xv] Lx Rv = R[xv] Rv R[x\v] R[xv] Rv γ (x, v)Rv which gives R([v/(xv)]v) = R[xv] Rv R[x\v] R[xv] Rv .

.

So, for any y ∈ Q, yR([v/(xv)]v) = yR[xv] Rv R[x\v] R[xv] Rv implies that

.

y{[v/(xv)]v} = {[y(xv)]/v · (x\v)}/(xv) · v

.

OSI.λ01.1

Consider .α(x, v) = γ (x, v)R[v λ ·xv] , then for all y ∈ Q, yα(x, v) = yR([v/(xv)]v) Rv R[xv] Lx Rv = yγ (x, v)R[v λ ·xv]

.

yα(x, v) = yRv R(xv) Lx R[v λ ·xv] . Consequently, {x · ({y([v/(xv)]v)}/v · xv)}/v = {x · (y/v · xv)}/[v λ · xv].

.

Now replace .y/v by y and post-multiply both sides by .[v λ · xv] to get {x · ({(yv)([v/(xv)]v)}/v · xv)}/v · [v λ · xv] = {x · (y · xv)}.

.

OSI.λ01.2

Consider .R[xv] Rv R[x\v] R[xv] Rv γ (x, v)Rv = γ (x, v)R[v λ ·xv] .

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Then for all y ∈ Q, yR[xv] Rv R[x\v] R[xv] Rv γ (x, v)Rv

.

= yγ (x, v)R[v λ ·xv] results in [ ] ( ) ({ [y(xv)]/v · (x\v) /(xv) · v}γ (x, v))/v = yγ (x, v) /[v λ · xv]

.

which is equivalent to the equation below after substituting the value of .γ (x, v) and post-multiply both sides by v: x{[y(xv)]/v · (x\v)} = (x · [y/v · (xv)])/[v λ · xv] · v.

.

Now, replace y by yv to get x{[(yz)(xv)]/v · (x\v)} = (x[y · (xv)])/[v λ · xv] · v.

.

Identities OSI.λ01.1.1 and OSI.λ01.1.2 are deduced from identity OSI.λ01.1 . Identities OSI.λ01.2.1 and OSI.λ01.2.4 are deduced from identity OSI.λ01.2 while identities OSI.λ01.2.2 and OSI.λ01.2.3 are deduced from identity OSI.λ01.2.1 . The other identities are gotten from OSI.λ01.1.1 . OSI.λ01.1.1 OSI.λ01.1.2 OSI.λ01.2.1 OSI.λ01.2.2 OSI.λ01.2.3 OSI.λ01.2.4

Simply put .z = v λ in identity OSI.λ01.1 to get identity OSI.λ01.1.1 . Simply put .y = e in identity OSI.λ01.1 to get identity OSI.λ01.1.2 . Simply put .z = v in identity OSI.λ01.2 to get identity OSI.λ01.2.1 . Substitute .y = e in identity OSI.λ01.2.1 to get identity OSI.λ01.2.2 . Substitute .y = v in identity OSI.λ01.2.1 to get identity OSI.λ01.2.3 . Simply put .z = v λ in identity OSI.λ01.2 to get identity OSI.λ01.2.4 .

By putting .y = e in identity OSI.λ01.1.1 , we have .{v λ {[v(vv)]/v}}/v · v λ = v λ which implies .v λ {[v(vv)]/v} = v, hence, .v(vv) = (v λ \v) · v. Again, by putting .y = v in identity OSI.λ01.1.1 , we have .{v λ {[(vv)(vv)]/v}}/v·v λ = e 2 2 which implies .v λ {[(vv)(vv)]/v} = v λ v, hence, .vv · vv = v λ \(v λ v) · v. U n Remark 3.9 Some other identities deduced from identities in Theorems 3.40, 3.42, and Lemmas 3.23, 3.25 have been found to be cryptographic in nature and thus applicable to cryptography as reported in [76, 78, 91].

3.3.3 Osborn Loops of Order 4n Isere [54], Isere et al. [55–58, 60, 61] in their work, which they started in 2010, developed methods of construction and classification of finite examples of Osborn loops of order 4n where .n = 4, 6, 9, 12, 18 that are non-universal and the characterizations of these examples were also obtained. Details of some of these works shall be considered in the next section.

3 Some Varieties of Loops

3.3.3.1

151

Construction of Non-Universal Osborn Loops

The binary operations as defined in the constructions below hold between two active(non-arbitrary) variables “a” and “b”. Whereas, the combination “.b + c” or “.a + c” is peculiar and unique to Osborn loops as defined in the construction below. Example 3.5 Let .I (·) = C2n × C2 , I = {(x α , y β ), 0 ≤ α ≤ 2n − 1, 0 ≤ β ≤ 1} such that the binary operation .(·) is defined as follows: (x a , e) · (x b , y β ) = (x a+b , y β )

(3.69)

(x a , y α ) · (x b , e) = (x a+b , y α )

(3.70)

.

.

{ (x a , y α ) · (x b , y β ) =

.

{ (x

.

b+c

, y ) · (x , y ) = δ

a

α

(x a+b , y α+β )

if a ≡ 0(mod 2), b ≡ 0(mod 2)

2 (x a+b+ab , y α+β )

if a ≡ 0(mod 2), b ≡ 1(mod 2) (3.71) if a ≡ 0(mod 2), b ≡ 0(mod 2)

(x a+b+c , y α+δ ) 2

if a ≡ 0(mod 2), b ≡ 1(mod 2) (3.72) Then .I (·) is a non-universal Osborn loop of order 4n, where .n = 4, 6 and 12. (x a+b+c+ab , y α+δ )

Proof We first show that .I (·) satisfies Osborn identity (3.61): (Xλ \Y ) · ZX = X(Y Z · X)

.

(a) Let .X = (x a , e); .Y = (x b , e); .Z = (x c , e), then by direct computations, we have (Xλ \Y ) · ZX = (x 2a+b+c , e)

.

and

X(Y Z · X) = (x 2a+b+c , e)

(b) Let .X = (x a , e); .Y = (x b , e); .Z = (x c , y γ ) (Xλ \Y ) · ZX = (x 2a+b+c , y γ )

.

and

X(Y Z · X) = (x 2a+b+c , y γ )

(c) Let .X = (x a , e); .Y = (x b , y β ); .Z = (x c , e) (Xλ \Y ) · ZX = (x 2a+b+c , y β ); X(Y Z · X) = (x 2a+b+c , y β ) b = even

.

2

2

(Xλ \Y )·ZX = (x 2a+b+c+ab , y β ); X(Y Z·X) = (x 2a+b+c+ab , y β ), b = odd

.

(d) Let .X = (x a , e); Y = (x b , y β ); Z = (x c , y γ ) .(X

λ

\Y ) · ZX = (x 2a+b+c , y β+γ ); X(Y Z · X) = (x 2a+b+c , y β+γ ), b = even 2

2

(Xλ \Y ) · ZX = (x 2a+b+c+ab , y β+γ ); X(Y Z · X) = (x 2a+b+c+ab , y β+γ ) b = odd

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(e) Let .X = (x a , y α ); Y = (x b , e); Z = (x c , e) .(X

λ

\Y ) · ZX = (x 2a+b+c , y 2α ); X(Y Z · X) = (x 2a+b+c , y 2α ) a = even 2

2

(Xλ \Y ) · ZX = (x 2a+b+c+a c , y 2α ); X(Y Z · X) = (x 2a+b+c+(b+c)a , y 2α ) a = odd

(f) Let .X = (x a , y α ); Y = (x b , e); Z = (x c , y γ ) .(X λ \Y ) · ZX = (x 2a+b+c , y 2α+γ ); X(Y Z · X) = (x 2a+b+c , y 2α+γ ) a = even 2 2 (X λ \Y ) · ZX = (x 2a+a c+b+c+ , y 2α+γ ); X(Y Z · X) = (x 2a+b+c+(b+c)a , y 2α+γ ) a = odd

(g) Let .X = (x a , y α ); .Y = (x b , y β ); .Z = (x c , e) (Xλ \Y ) · ZX = (x 2a+b+c , y 2α+β ) a = even, b = even

.

X(Y Z · X) = (x 2a+b+c , y 2α+β ) a = even, b = even

.

(Xλ \Y ) · ZX = (x 2a+ab

.

X(Y Z · X) = (x 2a+ab

.

2 +b+c

2 +b+c+

, y 2α+β ) a = even, b = odd

, y 2α+β ) a = even, b = odd 2

(Xλ \Y ) · ZX = (x 2a+b+c+a c , y β ) a = odd, b = even

.

2

X(Y Z · X) = (x 2a+b+c+(b+c)a , y β ) a = odd, b = even

.

(Xλ \Y ) · ZX = (x 2a+b+c+a

.

X(Y Z · X) = (x 2a+b+c+(b+c)a

.

2 c+ab2

, y 2α+β ) a = odd, b = odd

2 +a 2 c+ab2

, y 2α+β ) a = odd, b = odd

(h) Let .X = (x a , y α ); .Y = (x b , y β ); .Z = (x c , y γ ) (Xλ \Y ) · ZX = (x 2a+b+c , y 2α+β+γ ) a = even, b = even

.

X(Y Z · X) = (x 2a+b+c , y 2α+β+γ ) a = even, b = even

.

2

(Xλ \Y ) · ZX = (x 2a+b+c+ab , y 2α+β+γ ) a = even, b = odd

.

2

X(Y Z · X) = (x 2a+b+c+ab , y 2α+β+γ ) a = even, b = odd

.

2

(Xλ \Y ) · ZX = (x 2a+b+c+a c , y β+γ ) a = odd, b = even

.

2

X(Y Z · X) = (x 2a+b+c+(b+c)a , y β+γ ) a = odd, b = even

.

3 Some Varieties of Loops

153

(Xλ \Y ) · ZX = (x 2a+b+c+a

.

2 c+ab2

X(Y Z · X) = (x 2a+b+c+(b+c)a

.

, y 2α+β+γ ) a = odd, b = odd

2 +ab2

, y 2α+β+γ ) a = odd, b = odd

Since .(Xλ \Y ) · ZX = X(Y Z · X) holds in cases whenever .25 ≡ 1( mod 2n), that is .n = 2, 3, 4, 6 and 12, hence, the example is an Osborn loop of order 4n where .n = 2, 3, 4, 6 and 12. Note that .I (·) is trivial when .n = 2 and 3 but non-trivial when .n = 4, 6 and 12. Also .(e, e) is the two sided identity. Moreover, if .X = (x a , e), then .X−1 = −a (x , e). If .X = (x a , y a ) then X−1 = (x −a , y −α ) if a = even and X−1 = (x −(a+a

.

2 b)

, y −α ) if a=odd.

Therefore, the inverses are defined. Also for non-associativity, let X = (x a , y α ); Y = (x b , y β ); Z = (x c , y γ )

.

where ‘a’ is an even integer and ‘b’ an odd integer. 2

(XY )Z = (x a+b+c+ab , y α+β+γ ) and X(Y Z) = (x a+b+c , y α+β+γ ) b = odd

.

Thus, .(XY )Z /= X(Y Z) whenever 4 and 6 are not congruence to 0 mod 2n. Next, we verify that Example 3.5 is non-universal by checking if the second to the last identity in Lemma 3.25 holds in the Osborn loop .(H, *) or not. That is, the identity .Y * Y Y = Y λ \Y * Y . Let .Y = (x b , y β ), ’b’ being an odd integer. Then, by direct computation, we have: 3

Y * Y Y = (x 3b+b , y β ) b = odd

.

Y λ \Y * Y = (x b , y β )λ \(x b , y β ) · (x b , y β ) = (x 3b+3b

.

3 +b5

, y 3 β) b = odd

Thus, .Y * Y Y /= Y λ \Y * Y . Thus, .I (·) = C2n × C2 is not a universal Osborn loop. U n Remark 3.10 These are non-associative Osborn loops of orders 16, 24 and 48. This is interesting since up to now the smallest Osborn loop constructed is of order 16. Therefore, by Example 3.5, we have a solution to the first open problem of Kinyon [112]; not all Osborn loops are universal. Example 3.6 Let .I (·) = C2n × C2 , I = {(x α , y β ), 0 ≤ α ≤ 2n − 1, 0 ≤ β ≤ 1} such that the binary operation .(·) is defined as follows:

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(x a , e) · (x b , y β ) = (x a+b , y β )

(3.73)

(x a , y α ) · (x b , e) = (x a+b , y α )

(3.74)

.

.

⎧ a+b , y α+β ) ⎪ ⎪ ⎨(x a α b β .(x , y ) · (x , y ) = (x a+3b , y α+β ) ⎪ ⎪ ⎩(x a+3b , y α+3β ) { (x

.

b+c

, y ) · (x , y ) = δ

a

α

(x a+b+c , y α+δ )

if a ≡ 0(mod 2), b ≡ 0(mod 2) if a ≡ 0(mod 2), b ≡ 1(mod 2) if a ≡ 1(mod 2), b ≡ 1(mod 2) (3.75) if a ≡ 0(mod 2), b ≡ 0(mod 2)

if a ≡ 0(mod 2), b ≡ 1(mod 2) (3.76) b+c β+γ .(x ,y ) · (x a , y α ) = (x 3a+3b+c , y α+3β+γ ) if a ≡ 1(mod 2), b ≡ 1(mod 2) (3.77) (x a+3b+c , y α+δ )

Then .I (·) is a non-universal Osborn loop of order 4n, where .n = 6, 9, and 18. Remark 3.11 These examples of Osborn loops are of orders 24, 36 and 72. We have a peak order of 72. Example 3.7 Let .I (·) = C2n × C2 , I = {(x α , y β ), 0 ≤ α ≤ 2n − 1, 0 ≤ β ≤ 1} such that the binary operation .(·) is defined as follows: (x a , e) · (x b , y β ) = (x a+b , y β )

(3.78)

(x a , y α ) · (x b , e) = (x a+b , y α )

(3.79)

.

.

⎧ a+b , y α+β ) ⎪ ⎪ ⎨(x a α b β .(x , y )·(x , y ) = (x a−b , y α+β ) ⎪ ⎪ ⎩(x a−b , y α−β ) { (x

.

b+c

, y ) · (x , y ) = δ

a

α

(x b+c , y β+γ ) · (x a , y α ) =

.

if a ≡ 0(mod 2), b ≡ 0(mod 2) if a ≡ 0(mod 2), b ≡ 1(mod 2)

(3.80)

if a ≡ 1(mod 2), b ≡ 1(mod 2)

(x a+b+c , y α+δ )

if a ≡ 0(mod 2), b ≡ 0(mod 2)

if a ≡ 0(mod 2), b ≡ 1(mod 2) (3.81) { c−a−b α−β+γ (x ,y ) if a ≡ 1(mod 2), b ≡ 1(mod 2) (x a−b+c , y α+δ )

(x b+c−a , y β+γ −α )

if a ≡ 1(mod 2), b ≡ 0(mod 2) (3.82) Then .I (·) is a non-universal Osborn loop of order 4n, where .n = 6, 9 and 18. Remark 3.12 These examples of Osborn loops are of orders 24, 36 and 72. Example 3.8 Let .I (·) = C2n × C2 , I = {(x α , y β ), 0 ≤ α ≤ 2n − 1, 0 ≤ β ≤ 1} such that the binary operation .(·) is defined as follows: (x a , e) · (x b , y β ) = (x a+b , y β )

.

(3.83)

3 Some Varieties of Loops

155

(x a , y α ) · (x b , e) = (x a+b , y α )

.

⎧ a+b , y α+β ) ⎪ ⎪ ⎨(x a α b β .(x , y )·(x , y ) = (x a , y α+β ) ⎪ ⎪ ⎩(x a , y α ) { (x

.

(x

.

b+c

b+c

, y ) · (x , y ) = δ

,y

β+γ

a

α

α

if a ≡ 0(mod 2), b ≡ 0(mod 2) if a ≡ 0(mod 2), b ≡ 1(mod 2)

(3.85)

if a ≡ 1(mod 2), b ≡ 1(mod 2)

(x a+b+c , y α+δ )

if a ≡ 0(mod 2), b ≡ 0(mod 2)

(x a+c , y α+δ )

if a ≡ 0(mod 2), b ≡ 1(mod 2) (3.86) if a ≡ 1(mod 2), b ≡ 1(mod 2)

) · (x , y ) = a

(3.84)

{ (x c , y α+γ ) (x b+c , y β+γ )

if a ≡ 1(mod 2), b ≡ 0(mod 2) (3.87)

Then .I (·) is a non-universal Osborn loop of order 4n, where .n = 4, 8 and 16. The proofs of Examples 3.6–3.8 are similar to that of Example 3.5. Remark 3.13 In all, the constructions in Examples 3.5, 3.6, 3.7, and Example 3.8 are Osborn loops of orders 16, 24, 32, 36, 48, 64 and 72, as against the only Osborn loops of order 16 that were constructed prior to this time (Example 3.4). These four examples are non-isomorphic (cf. [57]). Hence, we have three more solutions to the first problem of Kinyon [112] asides Example 3.5; not all Osborn loops are universal.

3.3.3.2

Non-Universal Osborn Loops with Trivial Nuclei

Let .x = (x a , y α ), y = (x b , y β ), u = (x d , y δ ) ∈ I . Example 3.5: Then considering the definition of left nucleus, by computation, if .b ≡ 1(mod 2), it becomes ) ( ) ( 2 2 ux · y = x a+b+d+(a+d)b , y α+β+δ , u · xy = x a+b+d+ab , y α+β+δ .

.

So, .u ∈ / Nλ (I, ·). Next, by considering the definition of right nucleus, computation, we have: ) ( x · yu = x a+b+d , y α+β+δ , xy· ( ) 2 u = x a+b+d+ab , y α+β+δ , if b ≡ 1(mod 2).

.

Hence, .u ∈ / Nρ (I, ·). Finally, considering the definition of middle nucleus, by computation, it becomes

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) ( ) ( 2 2 x · uy = x a+b+d+b d , y α+β+δ , xu · y = x a+b+(a+d)b , y α+β+δ ,

.

if b ≡ 1(mod 2). Thus, .u ∈ / Nμ (I, ·). Therefore, .u ∈ / N(I, ·). Example 3.6: Let us consider the left nucleus; the right nucleus and, the middle nucleus: ) ( ) ( a+d+3b α+β+δ , u · xy = x a+d+3b , y α+β+δ , if b ≡ 1(mod 2). .ux · y = x ,y Thus, .u ∈ Nλ (I, ·). Let us now consider the right nucleus. Considering the definition of right nucleus, by computation, it becomes ) ( ) ( x · yu = x a+b+d , y α+β+δ , xy · u = x a+d+3b , y α+β+δ , if b ≡ 1(mod 2).

.

Whence, .u ∈ / Nρ (I, ·). Finally, let us consider the middle nucleus. ) ( ) ( xu · y = x a+d+3b , y α+β+δ , x · uy = x a+d+3b , y α+β+δ , if b ≡ 1(mod 2).

.

Thus, .u ∈ Nμ (I, ·). Therefore, .u ∈ / N(I, ·). Example 3.7: First, we check for .Nλ (I, ·). Consider: ) ( ) ( u · xy = x a−b+d , y α+β+δ , ux · y = x a−b+d , y α+β+δ , if b ≡ 1(mod 2).

.

Thus, .u ∈ Nλ (I, ·). Next, we check .Nμ (I, ·). Consider: ) ( x · uy = x a−b+d , y α+β+δ , ux · y = (x a−b+d , y α+β+δ ), if b ≡ 1(mod 2).

.

Thus, .u ∈ Nμ (I, ·). Next, we check .Nρ (I, ·). Consider: ) ( ) ( xy · u = x a−b+d , y α+β+δ , x · yu = x a+b+d , y α+β+δ , if b ≡ 1(mod 2).

.

Thus, .u ∈ / Nρ (I, ·). Thence, .u ∈ / N(I, ·). Example 3.8: By computation we have: ) ( ) ( ux · y = x a+d , y α+β+δ , u · xy = x a+d , y α+β+δ , if b ≡ 1(mod 2).

.

Thus, .u ∈ Nλ (I, ·). Next, we check .Nμ (I, ·). Consider: ) ( ) ( x · uy = x a+d , y α+β+δ , ux · y = x a+d , y α+β+δ , if b ≡ 1(mod 2).

.

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Thus, .u ∈ Nμ (I, ·). Next, we check .Nρ (I, ·). Consider: ) ( ) ( xy · u = x a+d , y α+β+δ , x · yu = x a+b+d , y α+β+δ , if b ≡ 1(mod 2).

.

Thus, .u ∈ / Nρ (I, ·). Therefore, .u ∈ / N(I, ·). From the foregoing, some elements .(I, ·) do not associate with every other elements of .(I, ·) under the same condition. Remark 3.14 The four of Examples 3.5–3.8 are proper Osborn loops that are nonuniversal with trivial nuclei. Therefore, they answer the second part of the open problem posed by Kinyon [112] in 2005 positively; there exist proper Osborn loops with trivial nuclei. Now that Phillips’ Problem (for Moufang loops of odd order) has been shown to have a negative answer, what can be said of it for proper Osborn loops? Are there proper Osborn loops of odd orders? What can be said of Doro’s conjecture for proper Osborn loops since Examples 3.5–3.8 are simple proper Osborn loops with trivial nuclei? Acknowledgments We acknowledge Professor L.V. Sabini and Professor B.L. Sharma who laid the foundation of research in Quasigroups and Loops in Nigeria.

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115. Kinyon, M. K., Kunen, K., Phillips, J. D.: Diassociativity in conjugacy closed loops. Comm. Alg. 32, 767–786 (2004). 116. Kinyon, M. K., J. D. Phillips, J. D.: Commutants of Bol loops of odd order. Proc. Amer. Math. Soc. 132, 617–619 (2004). 117. Kinyon, M. K., Phillips, J. D., Vojtˇechovský P.: When is the commutant of a Bol loop a subloop? Trans. Amer. Math. Soc. 360, 2393–2408 (2008). 118. Kinyon, M. K., Phillips, J. D., Vojtˇechovský, P.: C-loops : Extensions and construction. Journal of Algebra and its Applications 6(1), 1–20 (2007). 119. Kunen, K.: Moufang Quasigroups. J. Alg. 183, 231–234 (1996). 120. Kunen, K.: Quasigroups, Loops and Associative Laws. J. Alg. 185, 194–204 (1996). 121. Kunen, K.: The structure of conjugacy closed loops. Trans. Amer. Math. Soc. 352, 2889–2911 (2000). 122. Kinyon, M. K., Kunen, K., Phillips, J. D. and Vojtˇechovský, P.: Structure of automorphic loops. Trans. Amer. Math. Soc. 368(12), 8901–8927 (2016). 123. Merlini Giuliani, M. L., Dos Anjos, G. S.: On the structure of the automorphism group of certain automorphic loop. Comm. Algebra 45(5), 2245–2255 (2017), 124. Moufang, R.: On Quasigroups, Zur Struktur von Alterntivkorpern, Math. Ann. 110, 416–430 (1935) 125. Nagy, G. P. A class of finite simple Bol loops of exponent 2. Trans. Amer. Math. Soc. 361 , 5331–5343 (2009). 126. Nagy, G. P.: A class of simple proper Bol loop. Manuscripta Mathematica 127(1), 81–88 (2008). 127. Nagy, G. P.: Some remarks on simple Bol loops. Comment. Math. Univ. Carolin. 49(2), 259– 270 (2008). 128. Nagy, T.. Strambach, K.: Loops as invariant sections in groups, and their geometry. Can. J. Math. 46(5), 1027–1056 (1994). 129. Niederreiter, H., Robinson, K.H.: Bol loops of Order pq. Math. Proc. Camb. Phil. Soc. 89, 241–256 (1981). 130. Osborn, J. M. : Loops with the weak inverse property. Pac. J. Math. 10, 295–304 (1961). 131. Osoba, B., Jaiyéo.lá, T. G.: Algebraic Connections between Right and Middle Bol loops and their Cores. Quasigroups and Related Systems 30, 149–160 (2022). 132. Oyebo, Y.T.: On holomorphs and embedding into a group of Central loops. M.Sc. Dissertation, University of Agriculture, Abeokuta (2007). 133. Oyebo, Y. T., Adeniran J. O.: A Special Embedding of a Class of Central Loop in a Group, Journal Of Nigerian Mathematical Society 30, 39–51 (2011). 134. Oyebo, Y.T., Jaiyéo.lá, T. G., Adeniran, J. O.: A study of the holomorphy of (r, s, t)-inverse loops. Journal of Discrete Mathematical Sciences and Cryptography 26(1), 67–86 (2023). https://doi.org/10.1080/09720529.2021.1885810 135. Oyebo, Y. T., Osoba, B.: More results on the algebraic properties of middle Bol loops. Journal of the Nigerian mathematical society 41(2), 129–142 (2022). 136. Oyebo, Y.T., Osoba, B., Jaiyéo.lá, T.G.: Crypto-automorphism Group of some quasigroups. Discussiones Mathematicae - General Algebra and Applications, Available online: 2023-0417. https://doi.org/10.7151/dmgaa.1433 137. Pflugfelder, H. O.: Quasigroups and Loops : Introduction, Sigma series in Pure Math. 7, Heldermann Verlag, Berlin (1990). 138. Phillips, J. D.: A short basis for the variety of WIP PACC-loops. Quasigroups and Related Systems 14(1), 73–80 (2006). 139. Phillips, J. D., Vojtˇechovský, P.: C-loops ; An Introduction. Publ. Math. Debrecen. 68(1–2), 115–137 (2006). 140. Phillips, J. D., Vojtˇechovský, P.: The varieties of loops of Bol-Moufang type. Alg. Univer. 54(3), 259–383 (2005). 141. Phillips, J. D., Vojtˇechovský, P.: The varieties of quasigroups of Bol-Moufang type: An equational approach. J. Alg. 293, 17–33 (2005).

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142. Ramamurthi, V. S., Solarin, A. R. T.: On finite right central loops. Publ. Math. Debrecen 35 260–264 (1988). 143. Robinson, D. A.: Bol loops. Ph.D Thesis, University of Wisconsin, Madison (1964). 144. Robinson, D.A.: Holomorphy Theory of Extra Loops. Publ. Math. Debrecen 18, 59–64 (1971). 145. Robinson, D.A.: A Special Embedding of Bol-Loops in Group. Acta Math. Hungaricae 18, 95–113 (1977). 146. Robinson, D.A.: The Bryant-Schneider group of a loop. Ann. Soc. Sci. Bruxelles Sér. I 94(2– 3), 69–81 (1980). 147. Sharma, B. L.: Left loops which Satisfy the left Bol identity, Proc. Amer. Math. Soc. 61, 189–195 (1976). 148. Sharma, B. L.: Left Loops which satisfy the left Bol identity, (II). Ann. Soc. Sci. Bruxelles Sér. I 91, 69–78 (1977). 149. Sharma B. L. On the Algebraic properties of half Bol Loops. Ann. Soc. Sci. Bruxelles Sér. I 93(4), 227–240 (1979). 150. Sharma, B.L.: Classification of Bol loops of order 18. Acta Univ. Carolin. Math. Phys. 25(1), 37–44 (1984). 151. Sharma, B. L., Sabinin, L. V.: On the existence of Half Bol loops. An. Sti. ¸ Univ. “Al. I. Cuza” Ia¸si Sec¸t. I a Mat. (N.S.) 22(2), 147–148 (1976). 152. Sharma, B. L., Solarin, A. R. T.: On finite Bol loops of order 2p 2 . Simon Stevin Quart Jour. Pure and Applied math. 60(2), 133–156 (1986). 153. Shcherbakov, V. A.: Some properties of the full associated group of an IP-loop (Russian). Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat. Nauk 79(2), 51–52 (1984). 154. Shcherbacov, V. A.: Elements of quasigroup theory and applications, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL. (2017). 155. Solarin, A.R.T.: Characterization of Bol loops of small orders. Ph.D Thesis, University of Ife, Ile-Ife (1986). 156. Solarin A.R.T.: On the Identities of Bol Moufang Type. Kyungpook Math. 28(1), 51–62 (1988). 157. Solarin, A. R. T., Chiboka, V. O.: A Note on G-loops. Collection of Scientific papers of the Faculty of Science, Kragujevac 17, 17–26 (1995). 158. Solarin, A.R.T., Sharma, B. L.: On the Construction of Bol loops. An Siintt. Univ. “Ali Ciza”, Lasi Sect Ia - Mat.(NS), 27(1), 13–17 (1981). 159. Solarin, A.R.T., Sharma, B. L.: Some examples of Bol loops. Acta Carol, Math. and Phys. 25(1), 59–68 (1984). 160. Solarin, A.R.T., Sharma, B. L.: On the Construction of Bol loops II. An Siintt. Univ. “AI.I Ciza”, Lasi Sect. la., Mat (NS) 30(2), 7–14 (1984). 161. Solarin, A.R.T., Sharma, B. L.: On Bol loops of order 12. An. Stiin¸ ¸ t. Univ. Al. I. Cuza Iasi Sect. Ia Mat. 29(2) suppl., 69–80 (1983). 162. Sharma, B.L., Solarin, A.R.T.: On Bol loops of order 16 (I). Kyungpok Math. Jour. 24(1), 69–91 (1984). 163. Solarin, A.R.T., Sharma, B. L.: On Bol loops of order 16 with 13 elements of order 2, and with 14 elements of order 4. Proc. Nat. Acad. Sci. India 57(4), 546–556 (1987). 164. Sharma, B.L., Solarin, A.R.T.: On the classification of Bol loops of order 3p (p > 3). Communications in Algebra 16 (1), 37–55 (1988). 165. Stanovský, D., Vojtˇechovský, P.: Commutator theory for loops. J. Algebra 399, 290–322 (2014). 166. Syrbu, P.: On total multiplication groups. In The XII International Algebraic Conference, Vasyl Stus Donetsk National University, Vinnytsia, Ukraine, 110–111. July, 02–06, (2019) Available via https://jiac.donnu.edu.ua/article/view/7008/7038 167. Syrbu, P., Drapal, A.: On total multiplication groups of loops. In LOOPS 2019 Conference, Budapest University of Technology and Economics, Hungary, July, 7–13 (2019). Available via https://algebra.math.bme.hu/LOOPS19/slides/syrbu.pdf

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168. Syrbu, P., Grecu, I.: On total inner mapping groups of middle Bol loops. In Proceedings of the Fifth Conference of Mathemtical Society of Moldova, IMCS-55, Chisinau, Republic of Moldova, 161–164. September 28 to October 1, (2019) Available via https://ibn.idsi.md/sites/ default/files/imag_file/154-157_15.pdf 169. Vasantha Kandasamy, W. B.: Smarandache loops, Department of Mathematics, Indian Institute of Technology, Madras, India, (2002). 170. Vojtˇechovský, P.: Three lectures on automorphic loops. Quasigroups Related Systems 23(1), 129–163 (2015).

Chapter 4

The Z3 -Graded Extension of the Poincaré Algebra .

Richard Kerner

4.1 Introduction There can be little doubt that of all symmetries displayed by physics of elementary particles and fields, the invariance under the action of discrete groups is by far the best confirmed by the experiment, and also the most fundamental. The simplest discrete group is .S2 , the group of permutations of two objects. These permutations are cyclic, therefore the same group can be interpreted as .Z2 . Such an identification is no more possible for the next permutation group, .S3 , which contains six elements, out of which only the cyclic ones form a three-dimensional subgroup .Z3 . The cyclic group .Z2 plays crucial role in quantum physics of particles and fields. Its two representations in complex plane, implemented as symmetries of arguments of complex wave functions, the trivial one and the faithful one, lead to different fundamental sectors creating the great divide between two quantum statistics characterizing bosons and fermions. In the space of functions depending on two arguments, we can have two different behaviors with respect to permutations; ψ(x, y) = ψ(y, x) for bosons,

.

and ψ(x, y) = −ψ(y, x) for fermions. (4.1)

The relationship between spin and statistics is another illustration of the importance of discrete symmetries. It establishes a one-to-one dependence between the irreducible representations of the Lorentz group and the two possible statistical

R. Kerner (O) Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Sorbonne, Universités CNRS UMR 7600, Tour 23-13, 5-ème étage, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_4

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behaviors defined by (4.1): half-integer spin representations for fermions, and integer spin representations for bosons. The Dirac equation for the electron provides an example of entanglement of two apparently independent .Z2 symmetries. The discovery of dichotomic spin parameter which in the case of the electron can take on only two exclusive values led Pauli to the conclusion that a Schödinger-like equation for the electron should involve a twocomponent wave function: E

.

( 1) ( 1) ( 1) ψ ψ ψ 2 = mc + c σ · p ψ2 ψ2 ψ2

(4.2)

where .σ = (σx , σy , σz ) denotes the three Pauli matrices, which form the basis of .2 × 2 traceless hermitian matrices, .E = −i h∂ ¯ t and .p = (px , py , pz ) is the momentum operator, with .pk = −i h∂ ¯ k . This equation does not satisfy the Lorentzinvariant condition: iterating it leads to wrong relation between energy, momentum and mass, .E 2 = m2 c4 + 2mcp + p2 instead of .E 2 = m2 c4 + c2 p2 , which made Pauli abandon this version introducing the approximate non-relativistic equation for the electron interacting with electromagnetic field [6]. It turns out that relativistic covariance can be restored via introduction of another two-component Pauli spinor, the two similar equations intertwining them with mass terms of opposite sign. Let the two Pauli spinors be denoted by .ψ+ and .ψ− . Then the following system of equations satisfies relativistic dispersion relation, and is Lorentz covariant: Eψ+ = mc2 ψ+ + σ · pψ− ,

.

Eψ− = −mc2 ψ− + σ · pψ+ ,

(4.3)

which is Dirac’s equation in a less usual basis. [7]. By iterating it, we get the relativistic condition satisfied simultaneously by both Pauli spinors: E 2 ψ+ = (m2 c4 + c2 p2 )ψ+ ,

.

E 2 ψ− = (m2 c4 + c2 p2 )ψ− .

In a more familiar manner, the same system is written in a manifestly relativistic form, with the 4-component Dirac spinor composed of two Pauli spinors .ψ+ and .ψ− , and the .4 × 4 Dirac matrices expressed in terms of tensor products of .2 × 2 matrices as follows: γ μ pμ ψ = mψ, where γ 0 = σ3 ⊗1l2 ,

.

γ i = (iσ2 ) ⊗ σ i .

(4.4)

The Dirac equation is invariant with respect to .Z2 × Z2 symmetry. The first .Z2 concerns the spin of the electron, which can have two projections on the momentum; the second .Z2 group, imposed by the requirement of Lorentz invariance, concerns the particle-antiparticle symmetry. Recently in [1–4] a generalization of the Dirac equation for quarks was proposed, incorporating the color degrees of freedom via extending the discrete symmetry of the system to the .Z2 × Z2 × Z3 group. The cyclic group .Z3 is generated by the third

4 The .Z3 -Graded Extension of the Poincaré Algebra 2π i

167 4π i

root of unity, denoted by .j = e 3 , with .j 2 = e 3 , .j 3 = 1, and .1 + j + j 2 = 0. Just as taking into account the dichotomic half-integer spin variable, the introduction of color degrees of freedom requires additional .Z3 symmetry acting on a new discrete variable taking three possible (and exclusive) values, named symbolically “red”, “blue” and “green”. The .3 × 3 matrices had to be introduced, all representing third roots of the .3 × 3 unit matrix. Six Pauli spinors represent three colors and three anti-colors: ( .ϕ+

=

( 1) ( 1) ( 1) ( 1) ( 1) ) 1 ϕ+ χ+ ψ+ ϕ− χ− ψ− , χ+ = , ψ+ = , ϕ− = , χ− = , ψ− = 2 2 2 2 2 2 , ϕ+ χ+ ψ+ ϕ− χ− ψ− (4.5)

on which Pauli sigma-matrices act in a natural way. By analogy with the pair of Eq. (4.3) in which multiplying the mass by .−1 led to the anti-particle appearance, now the mass term is multiplied by the generator of the .Z3 group, j , each time the colour changes. This yields the following set of what may be called the “colour Dirac equation”: E ϕ+ = mc2 ϕ+ + c σ · p χ− ,

.

E χ+ = j mc2 χ+ + c σ · p ψ− ,

.

E ψ+ = j 2 mc2 ψ+ + c σ · p ϕ− ,

.

E ϕ− = −mc2 ϕ− + c σ · p χ+ E χ− = −j mc2 χ− + c σ · p ψ+ E ψ− = −j 2 mc2 ψ− + c σ · p ϕ+

(4.6)

In an appropriate basis, the system (4.6) can be represented in a Dirac-like form as follows: r μ pμ ψ = mc1l12 ψ,

(4.7)

.

where .ψ is the generalized 12-component spinor made of6 Pauli spinors (4.5), and the generalized .12 × 12 Dirac matrices .r μ are constructed as follows: r 0 = B † ⊗ σ3 ⊗1l2 ,

.

r i = Q2 ⊗ (iσ2 ) ⊗ σ i ,

(4.8)

where ⎛

1 0 † 2 .B = ⎝0 j 0 0

⎞ 0 0⎠ , j

⎛ ⎞ 10 0 B = ⎝0 j 0 ⎠ , 0 0 j2

⎛

⎞ 01 0 Q2 = ⎝0 0 j 2 ⎠ , j 0 0

(4.9)

The two traceless matrices B and .Q2 are both cubic roots of the unit .3 × 3 matrix. They generate the entire Lie algebra of the .SU (3) group. The system (4.7) becomes diagonal only after sixth iteration, yielding the dispersion relation of sixth order: (r μ pμ )6 = (p06 − p6 )1l2 = m6 c6 1l2

.

(4.10)

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This expression is not manifestly relativistic invariant, but it represents a unique light cone multiplied by a positive form-factor: (p06 − p6 ) = (p02 − p2 )(jp02 − p2 )(j 2 p02 − p2 ) = (p02 − p2 )(p04 + p02 p2 + | p |4 . (4.11)

.

Such field theories of higher order were considered by T.D.Lee and G.Wick [8] and were recently an object of renewed interest [9]. The colour Dirac matrices .r μ defined in (4.8) do not span the usual Clifford algebra, and do not transform as relativistic 4-vectors under ordinary Lorentz transformations. In order to implement Lorentz covariance, the set of .r-matrices must be extended up to six different realizations, forming doublets transforming under the extension of the Lorentz algebra, containing the usual Lorentz algebra, (0)

(1)

(2)

and two conjugate replicas forming a .Z3 -graded algebra .L = L ⊕ L ⊕ L, acting on the generalized multi-spinors formed by six 12-dimensional colour Dirac spinors, the total dimension of the representation space being .6 × 12 = 72 (see the details in (r) (s)

((r+s)|3 )

[10]. The multiplication rules in L are .Z3 -graded, i.e. one has . L · L ⊂ L . The aim of the present article is to define a similar .Z3 -graded extension of the Poincaré algebra realized in terms of differential operators acting on an extended Minkowskian space-time.

4.2 The Z3 × Z2 Symmetry Let us recall briefly the properties of the cyclic (.Z3 ) and the permutation (.S3 ) groups of three elements. Their representation in terms of rotations and reflections in the complex plane are shown in the following Fig. 4.1: Let us denote by j and .j 2 the two complex third roots of unity, given by j =e

.

2π i 3

√ √ 4π i 1 i 3 1 i 3 2 , j =e 3 =− − =− + 2 2 2 2

(4.12)

√ satisfying obvious identities .1 + j + j 2 = 0, so that .j + j 2 = −1, j − j 2 = i 3, The six .S3 symmetry transformations contain the identity, two rotations, one by ◦ ◦ .120 , another one by .240 , and three reflections, in the x-axis, in the j -axis and in 2 the .j -axis. The .Z3 subgroup contains only the three rotations. Odd permutations must be represented by idempotents, i.e. by operations whose square is the identity operation. We can make the following choice: ( ) ( ) ( ) ABC ABC ABC . → (z → z¯ ), → (z → zˆ ), → (z → z∗ ), CBA BAC CBA (4.13) Here the bar .(z → z¯ ) denotes the complex conjugation, i.e. the reflection in the real line, the hat .z → zˆ denotes the reflection in the root .j 2 , and the star .z → z∗ the

4 The .Z3 -Graded Extension of the Poincaré Algebra

169

Fig. 4.1 Rotations (.Z3 -group) and reflections added (.S3 group)

reflection in the root j . The six operations close in a non-abelian group with six elements, which are represented as rotation and reflexion operations in the complex plane, as shown in (4.1) above. In what follows, we shall use the .Z3 group for grading of linear spaces and matrix algebras [1, 11, 12]. The .Z3 -graded algebras are composed of three vector subspaces, one of which (of .Z3 -grade zero) constitutes a subalgebra in the ordinary sense: A = A0 ⊕ A1 ⊕ A2

.

(4.14)

The multiplication in the graded algebra (4.14) obeys the following scheme: (r) (s) .

A·A ⊂

(r+s)|3

A , with r, s, .. = 0, 1, 2, (r + s) |3 = (r + s) modulo 3.

(4.15)

The .Z3 symmetry can be combined with the .Z2 symmetry; 3 and 2 being prime numbers, the Cartesian product of the two is isomorphoic with another cyclic group, .Z3 × Z2 = Z6 . The generalized Dirac equation is invariant under the discrete group .Z3 × Z2 × Z2 = Z6 × Z2 (which is not isomorphic with .Z12 because 6 is not a prime number, being divisible by 2 and by 3). The cyclic group .Z6 is represented in the complex plane by its generator .q = πi 2π i e 6 = e 3 , and its powers from 1 to 6. In terms of the .Z3 group generated by j and .Z2 group generated by .−1, we have q = −j 2 , q 2 = j, q 3 = −1, q 4 = j 2 , q 5 = −j, q 6 = 1,

.

as shown in the Fig. 4.2 below. In analogy with colours labeling quark fields, if the “white” combination is represented by 0, then we have two linear colourless sums of three powers of q, namely .1 + q 2 + q 4 = 0 and .q + q 3 + q 5 = 0, and three white combinations of colour with its anti-colour, .q + q 4 = 0, q 2 + q 5 = 0, q 3 + q 6 = 0, just like a fermion and its antiparticle, or three bosons (like e.g. mesons .π 0 , π + and − .π ).

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Fig. 4.2 The six complex numbers .q k can be put into correspondence with three colours and three anti-colours

A .Z3 -graded analog of Pauli’s exclusion principle was introduced and its algebraic and physical consequences investigated in [2, 5].

4.3 The Z3 -Extended Minkowskian Spacetime Let us denote by .M4 the standard four-dimensional Minkowskian spacetime, a 4-dimensional real vector space endowed with pseudo-Euclidean (Minkowskian) metric .ημν = diag[+, −, −, −]. A spacetime vector is given by its coordinates in a chosen orthonormal frame: k μ = [k 0 , k] = [k 0 , k x , k y , k z ]

(4.16)

.

often replaced by a more practical notation with small Greek indices running from 0 to 3: k μ = [k 0 , k] = [k 0 , k 1 , k 2 , k 3 ]

(4.17)

.

The three replicas of a 4-vector .k μ will be labeled with the superscripts relative to the elements of the .Z3 -group as follows: (0) μ

.

k

(0)

(0)

= ( k 0 , k ),

(1) μ

k

(1)

(1)

= ( k 0 , k ),

(2) μ

k

(2)

(2)

= ( k 0 , k ).

(4.18)

In each of the three sectors the specific quadratic form is given, defining the group of transformations keeping it invariant: (0)

(0)

( k 0 )2 − ( k )2 = m 2 ,

.

(1)

(1)

( k 0 )2 − j ( k )2 = j m 2 ,

(2)

(2)

( k 0 )2 − j 2 ( k )2 = j 2 m 2 , (4.19)

4 The .Z3 -Graded Extension of the Poincaré Algebra

171 (r)

(r)

which leads to the following explicit expressions of .( k 0 ) as functions of . k and m (.r = 0, 1, 2): / (0) 0

.

k =±

/ (0) k2

+ m2 ,

(1) 0

/ (1) k2

k = ±j

+ m2 ,

(2) 0

(2)

k = ±j 2 k2 + m2 ,

(4.20)

Let us denote the three quadratic forms, one real and two mutually complex conjugate, by the following three tensors

.

(0) η μν

= diag[+1, −1, −1, −1],

(1) η μν

= diag[+1, −j, −j, −j ],

(2)

η μν = diag[+1, −j 2 , −j 2 , −j 2 ]

(4.21)

defined on each of the subspaces of the generalized Minkowskian space (Z3 ) M 12

.

(0)

(1)

(2)

= M4 ⊕ M4 ⊕ M4

(4.22)

The superscripts .(r) = (0), (1), (2) refer to the .Z3 -grades attributed to each of the three subspaces. These grades will play an important role in defining the .Z3 -graded extension of the Poincaré algebra acting on the extended Minkowskian (Z3 )

space-time . M 12 . We should underline here that the three “replicas” are to be treated as really independent components of the resulting 12-dimensional manifold. For convenience, we shall use the same letters designing three types of space-time components, labeling them with an extra index as follows: μ

μ

μ

xrμ = (x0 , x1 , x2 ) = [τ0 , x0 , y0 , z0 ; τ1 , x1 , y1 , z1 ; τ2 , x2 , y2 , z2 ].

.

(4.23)

Idempotent operators projecting on one of the three subspaces of the generalized (Z3 )

Minkowskian space-time . M 12 can be constructed using the .3 × 3 matrices B and † .B introduced in (4.9) as follows. Let us define two .12 × 12 matrices acting on (Z3 ) .

M

12 :

B = B ⊗1l4 , B† = B † ⊗1l4 ,

.

Then the following three projection operators can be formed: (0) .

|| =

(1)

|| =

1 (l 112 + B + B† ), 3 1 (l 112 + j 2 B + j B† ), 3

(2)

|| =

1 (l 112 + j B + j 2 B† , 3

(4.24)

172

R. Kerner (r)

(r)

(r)(s)

One checks easily that .[ ||]2 = ||, r = 0, 1, 2 and . || || = 0 for .r /= s. Interesting higher-dimensional and complex extensions of Minkowskian space-time were investigated in [13, 14], albeit without introducing the .Z3 grading.

4.4 The Z3 -Graded Lorentz Group The quadratic Minkowskian square of the 4 vector .k μ , .(k 0 )2 − k2 is invariant under the transformations of the Lorentz group. The space rotations touching only the 3-dimensional vector .k leave all the three quadratic expressions invariant, because they depend only on its 3-dimensional Euclidean square .k2 ; therefore we can fix our attention at the Lorentzian boosts. As we can always align the relative velocity along one of the orthonormal axes of the chosen inertial frame, say 0x, those boosts can be considered only between the time and the x coordinates. Here are the three .2 × 2 matrices representing the same Lorentz boost (with real parameter u equal to v .tanh ) leaving invariant one of the three quadratic invariants given in (4.19): c (0) . L00 (0) L11

( ) cosh u sinh u = , sinh u cosh u ( ) cosh u j 2 sinh u = , j sinh u cosh u

(0) L22

( =

) cosh u j sinh u , j 2 sinh u cosh u

(4.25)

The three matrices are self-adjoint: (0)† L00

.

(0)

= L00 ,

(0)† L11

(0)

= L11 ,

(0)† L22

(0)

= L22 ,

(4.26)

The above matrices transform each of the three sectors of the .Z3 -Minkowski space into itself, which founds its reflection in the lower indices is quite transparent: .L00 transforms a vector belonging to the 0-th sector of the .Z3 -graded Minkowskian space into a 4-vector belonging to the same sector, and similarly for the matrix operators .L11 and .L22 . It is also easy to prove that each set is a representation of a one-parameter subgroup representing a particular Lorentz boost, here between the time variable (hereafter always represented by .τ = ct) and one cartesian coordinate, say x. For example, the product of two Lorentz boosts acting on the sector .(1), is a boost of the same type: (0)

(0)

(0)

L11 (u) · L11 (v) = L11 (u + v),

.

and similarly for a product of two boosts acting on the sector .(2),

(4.27)

4 The .Z3 -Graded Extension of the Poincaré Algebra (0)

(0)

173

(0)

L22 (u) · L22 (v) = L22 (u + v),

(4.28)

.

The full set of three independent “classical” (i.e. belonging to the subgroup denoted (0)

by .L00 ) Lorentz boosts is given by three .4×4 matrices, with independent parameters .u, v, w: ⎞ cosh u sinh u 0 0 ⎜ sinh u cosh u 0 0⎟ ⎟, .⎜ ⎝ 0 0 1 0⎠ 0 0 01 ⎛

⎛

cosh v ⎜ 0 ⎜ ⎝ sinh v 0

0 sinh v 1 0 0 cosh v 0 0

⎞ 0 0⎟ ⎟, 0⎠ 1

⎛

cosh w ⎜ 0 ⎜ ⎝ 0 sinh w

⎞ 0 0 sinh w 10 0 ⎟ ⎟ 01 0 ⎠ 0 0 cosh w

(4.29)

To make the extension of the Lorentz boosts complete we need also two sets of complementary matrix operators transforming one sector into another. There are two types of such operators, one raising the .Z3 index of each subspace, another lowering the .Z3 index by 1. It is quite easy to find out their matrix representation. The matrices lowering the .Z3 index by 1 are: (1) . L01 (1) L12

) j cosh u sinh u , = j sinh u cosh u ) ( j cosh u j 2 sinh u , = 2 j sinh u cosh u (

(1) L20

) j cosh u j sinh u , = sinh u cosh u (

(4.30)

The determinant of each of these matrices is equal to j . The matrices raising the .Z3 index by one (or decreasing it by 2, which is equivalent from the point of view of the .Z3 -grading) are: (2) L10

.

(2) L21

) j 2 cosh u j 2 sinh u , sinh u cosh u ) (2) ) ( 2 ( 2 j cosh u j sinh u j cosh u sinh u , L02 = = j sinh u cosh u j 2 sinh u cosh u (

=

(4.31)

The determinant of each of these matrices is equal to .j 2 . The above sets of three matrices each, decreasing and raising the .Z3 index, are mutually hermitian adjoint: (1)† L01

.

(2)

= L10 ,

(1)† L12

(2)

= L21 ,

(1)† L20

(2)

= L02 ,

(4.32)

Here again, the logic of the lower indices is quite transparent: a matrix labeled .L12 transforms a 4-vector belonging to the sector .(2) into a 4-vector belonging to the sector .(1), and so forth, e.g.:

174

R. Kerner (0)

(1)

,

L01 k μ = k μ ,

.

(0)

(2)

(1)

(2)

,

L20 k μ = k μ ,

,

L12 k μ = k μ ,

etc.

(4.33)

The matrices raising or lowering the .Z3 -grade of the particular type of the 4-vector they are acting on do not form a group, because most of the products of two such matrices produce new matrices not belonging to the set defined above. However, inside each of one-parameter families corresponding to a given choice of the single space direction concerned by the Lorentz boost, .0x, 0y or 0z displays the group property if the products are taken according to the chain rule, with second index of the first factor equal to the first index of the second factor, like in the following examples: .

(1) L12 (τ, x;

u) L20 (τ, x; v) = L10 (τ, x; (u + v)),

(1)

(2)

(2) L21 (τ, y;

u) L12 (τ, y; v) = L22 (τ, y; (u + v)), etc.

(1)

(0)

(4.34)

The above .2 × 2 matrices represent a reduced version of Lorentz boosts with relative velocity aligned on the unique axis Ox. As in the previous case, the full .4 × 4 versions are given by the following three matrices corresponding to the three independent Lorentz boosts. The boosts of the increasing type, transforming 4vectors from sector 2 to 0, from sector .1 to 2 and from sector 0 to 1, respectively, are as follows: (1)

(1)

(1)

– the three boosts . L20 (τ, x), L20 (τ, y), L20 (τ, z) are given by: ⎛

⎞ j cosh u j sinh u 0 0 ⎜ sinh u cosh u 0 0⎟ ⎟, .⎜ ⎝ 0 0 1 0⎠ 0 0 01 ⎛ ⎞ j cosh w 0 0 j sinh w ⎜ 0 10 0 ⎟ ⎜ ⎟ ⎝ 0 01 0 ⎠

⎛

j cosh v ⎜ 0 ⎜ ⎝ sinh v 0

0 j sinh v 1 0 0 cosh v 0 0

⎞ 0 0⎟ ⎟, 0⎠ 1

(4.35)

sinh w 0 0 cosh w (1)

(1)

(1)

– the three boosts . L12 (τ, x), L12 (τ, y), L12 (τ, z) are given by: ⎛

j cosh u j 2 sinh u ⎜j 2 sinh u cosh u ⎜ . ⎝ 0 0 0 0

⎞ 00 0 0⎟ ⎟, 1 0⎠ 01

⎛

j cosh v ⎜ 0 ⎜ ⎝j 2 sinh v 0

0 j 2 sinh v 1 0 0 cosh v 0 0

⎞ 0 0⎟ ⎟, 0⎠ 1

⎛

j cosh w ⎜ 0 ⎜ ⎝ 0 j 2 sinh w

0 1 0 0

⎞ 0 j 2 sinh w ⎟ 0 0 ⎟ ⎠ 1 0 0 cosh w (4.36)

4 The .Z3 -Graded Extension of the Poincaré Algebra (1)

(1)

175 (1)

and the three boosts . L01 (τ, x), L01 (τ, y), L01 (τ, z) are given by: ⎛

⎞ j cosh u sinh u 0 0 ⎜ j sinh u cosh u 0 0⎟ ⎟, .⎜ ⎝ 0 0 1 0⎠ 0 0 01

⎛

j cosh v ⎜ 0 ⎜ ⎝ j sinh v 0

⎞ 0 sinh v 0 1 0 0⎟ ⎟, 0 cosh v 0⎠ 0 0 1

⎛ j cosh w ⎜ 0 ⎜ ⎝ 0 j sinh w

0 1 0 0

⎞ 0 sinh w 0 0 ⎟ ⎟ 1 0 ⎠ 0 cosh w (4.37)

The boosts of the decreasing type, transforming 4-vectors from sector 1 to 0, from sector .2 to 1 and from sector 0 to 2, respectively, are as follows: (2)

(2)

(2)

– the three boosts . L10 (τ, x), L10 (τ, y), L10 (τ, z) are given by: ⎛

⎞ j 2 cosh u j 2 sinh u 0 0 ⎜ sinh u cosh u 0 0⎟ ⎟, .⎜ ⎝ 0 0 1 0⎠ 0 0 01 ⎛ 2 ⎞ j cosh w 0 0 j 2 sinh w ⎜ ⎟ 0 10 0 ⎜ ⎟ ⎝ ⎠ 0 01 0

⎛

j 2 cosh v ⎜ 0 ⎜ ⎝ sinh v 0

0 j 2 sinh v 1 0 0 cosh v 0 0

⎞ 0 0⎟ ⎟, 0⎠ 1

(4.38)

sinh w 0 0 cosh w (2)

(2)

(2)

– the three boosts . L21 (τ, x), L21 (τ, y), L21 (τ, z) are given by: ⎛

⎞ j 2 cosh u j sinh u 0 0 ⎜ j sinh u cosh u 0 0⎟ ⎟, .⎜ ⎝ 0 0 1 0⎠ 0 0 01 ⎛ 2 ⎞ j cosh w 0 0 j sinh w ⎜ 0 10 0 ⎟ ⎜ ⎟ ⎝ 0 01 0 ⎠ j sinh w 0 0 cosh w (2)

(2)

⎛

j 2 cosh v ⎜ 0 ⎜ ⎝ j sinh v 0

0 j sinh v 1 0 0 cosh v 0 0

⎞ 0 0⎟ ⎟, 0⎠ 1

(4.39)

(2)

and the three boosts . L02 (τ, x), L02 (τ, y), L02 (τ, z) are given by: ⎛

j 2 cosh u sinh u ⎜ j 2 sinh u cosh u .⎜ ⎝ 0 0 0 0

⎞ 00 0 0⎟ ⎟, 1 0⎠ 01

⎛ 2 j cosh v ⎜ 0 ⎜ ⎝ j 2 sinh v 0

⎞ 0 sinh v 0 1 0 0⎟ ⎟, 0 cosh v 0⎠ 0 0 1

⎛

j 2 cosh w ⎜ 0 ⎜ ⎝ 0 j 2 sinh w

0 1 0 0

⎞ 0 sinh w 0 0 ⎟ ⎟ 1 0 ⎠ 0 cosh w (4.40)

176

R. Kerner (r)

The nine .4×4 matrices . Lst , .r, s, t = 0, 1, 2 act on the .Z3 -extended Minkowskian vector in a specifically ordered way. Let us write a .Z3 -extended vector as a column with 12 entries, composed of three 4-vectors belonging each to one of the .Z3 -graded sectors: (0)

(1)

(2)

(kμ , kμ , kμ )

(4.41)

.

⎛(0) ⎞ ⎜ L00 0 0 ⎟ (0) (0) ⎜ ⎟ .A = ⎜ ⎟ ⎝ 0 L11 0 ⎠ (0)

0

⎛ ⎜ 0 ⎜ A=⎜ 0 ⎝

(1) L01

(1)

0

(1) L20

0 L22

0

⎛

⎞ 0 ⎟

(1) ⎟ L12 ⎟ ⎠

⎞

⎜ 0 0 ⎜(2) A = ⎜ L10 0 ⎝

(2)

(2) L02 ⎟

(2)

⎟ 0 ⎟ ⎠

0 L21 0

0

(4.42) It is easy to see that the so defined matrices display not only the group property, but also the .Z3 grading in the following sense: (0) (0) .

(0)

(0) (1)

(1)

(0) (2)

(2)

(2)

(2) (2)

(1)

(1) (2)

(2) (1)

A · A ⊂ A,

(1) (1)

A · A ⊂ A,

A · A ⊂ A, A · A ⊂ A,

A · A ⊂ A, (0)

A · A = A · A ⊂ A.

(4.43)

In other words, the elements of three subsets of the .Z3 -graded group of boosts behave under associative matrix multiplication as follows: (r) (s) .

A·A ⊂

(r+s)|3

A , with r, s, .. = 0, 1, 2, (r + s) |3 = (r + s) modulo 3.

(4.44)

The three sets of matrices ordered in the particular blocks (4.42) form a threeparameter family which can be considered as the extension of the set of three independent Lorentz boosts. In order to obtain the extension of the entire Lorentz group including the 3-parameter subgroup of space rotations we shall first investigate the .Z3 -graded infinitesimal generators of the Lorentz boosts, and then, taking their commutators, define the .Z3 -graded extension of the space rotations.

4.5 The Z3 -Graded Lorentz Algebra The .Z3 -graded matrix Lie algebra corresponding to the .Z3 -graded Lie group defined above is easily obtained by taking the differentials of corresponding families of generators of 1-parameter abelian subgroups in the vicinity of the unit element (corresponding to the 0 value of the parameter u, .v, etc.). It is sufficient to develop all terms in a Taylor series of powers of the parameter u and keep only linear terms in the formulae for the matrices of the Lie group defined above, which means that

4 The .Z3 -Graded Extension of the Poincaré Algebra

177

the terms like 1 or .cosh u will be suppressed, and the terms with .sinh u will be replaced by 1. Thus, we define: – the full set of three independent “classical” generators (i.e. belonging to the (0)

subgroup . A acting in the first sector of the .Z3 -extended Mikowski space and (0)

(0)

(0)

which we shall denote .K00 (τ, x), K00 (τ, y) and .K00 (τ, z), the three .4 × 4 matrices, defining the boosts between the variables .(τ, x), (τ, y) and .(τ, z), respectively: acting in the first sector of the .Z3 -extended Mikowski space ⎛

0 (0) ⎜1 .K00 (τ, x) = ⎜ ⎝0 0

1 0 0 0

0 0 0 0

⎞ ⎞ ⎞ ⎛ ⎛ 0 0010 0001 (0) ⎟ (0) ⎟ ⎜ ⎜ 0⎟ ⎟ , K00 (τ, y) = ⎜0 0 0 0⎟ , K00 (τ, z) = ⎜0 0 0 0⎟ ⎠ ⎠ ⎝ ⎝ 0 1000 0 0 0 0⎠ 0 0000 1000 (4.45) (0)

(0)

as well as two similar sets of .4 × 4 matrices, denoted respectively .K11 and .K22 , acting in the sectors 1 and 2 of the .Z3 -extended Minkowskian space transforming them onto themselves: ⎛

0 (0) ⎜j .K11 (τ, x) = ⎜ ⎝0 0 ⎛ 0 (0) ⎜0 K11 (τ, z) = ⎜ ⎝0 j

⎞ 00 0 0⎟ ⎟, 0 0⎠ 00 ⎞ 0 0 j2 00 0⎟ ⎟ 00 0⎠

j2 0 0 0

⎛

0 (0) ⎜0 K11 (τ, y) = ⎜ ⎝j 0

0 0 0 0

j2 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

(4.46)

00 0

transforming the sector 1 onto itself, and ⎛

0 (0) ⎜j 2 .K22 (τ, x) = ⎜ ⎝0 0 ⎛ 0 (0) ⎜0 K22 (τ, z) = ⎜ ⎝0 j2

j 0 0 0

⎞ 00 0 0⎟ ⎟, 0 0⎠

00 ⎞ 00j 0 0 0⎟ ⎟ 0 0 0⎠ 000

⎛

0 (0) ⎜0 K22 (τ, y) = ⎜ ⎝j 2 0

0 0 0 0

j 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

(4.47)

transforming sector 2 onto itself. There are also the two sets of complementary matrix operators transforming sectors into one another. There are two types of such infinitesimal generators,

178

R. Kerner

one raising the .Z3 index of each subspace, another decreasing the .Z3 index by 1. Their matrix representation is as follows: (1)

(1)

(1)

– the infinitesimal generators of three boosts .K20 (τ, x), K20 (τ, y), K20 (τ, z) are given by: ⎛ 0 (1) ⎜1 .K20 (τ, x) = ⎜ ⎝0 0

j 0 0 0

⎛ ⎞ 00 00 (1) ⎜0 0 0 0⎟ ⎟ , K20 (τ, y) = ⎜ ⎝1 0 0 0⎠ 00 00

j 0 0 0

⎛ 0 (1) ⎜0 K20 (τ, z) = ⎜ ⎝0 1

⎞ 0 0⎟ ⎟, 0⎠ 0

(1)

(1)

⎞ 00j 0 0 0⎟ ⎟ 0 0 0⎠ 000 (4.48)

(1)

– the three infinitesimal generators of boosts .K12 (τ, x), K12 (τ, y), K12 (τ, z) are given by: ⎛

0 (1) ⎜j 2 .K12 (τ, x) = ⎜ ⎝0 0 ⎛ 0 (1) ⎜0 K12 (τ, z) = ⎜ ⎝0 j2

j2 0 0 0

0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0 2⎞

⎛

0 (1) ⎜0 K12 (τ, y) = ⎜ ⎝j 2 0

0 0 0 0

j2 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

00j 00 0⎟ ⎟ 00 0⎠ 00 0

(1)

(4.49)

(1)

(1)

and the three boosts .K01 (τ, x), K01 (τ, y), K01 (τ, z) are given by: ⎛

0 (1) ⎜j .K01 (τ, x) = ⎜ ⎝0 0

⎛ ⎞ 100 0 (1) ⎜ ⎟ 0 0 0⎟ 0 , K01 (τ, y) = ⎜ ⎝j 0 0 0⎠ 000 0

0 0 0 0

1 0 0 0

⎛ ⎞ 0 0 (1) ⎜ ⎟ 0⎟ 0 , K01 (τ, z) = ⎜ ⎝0 0⎠ 0 j

⎞ 001 0 0 0⎟ ⎟ 0 0 0⎠ 000 (4.50)

The boosts of the increasing type, transforming 4-vectors from sector 0 to 1, from sector .1 to 2 and from sector 2 to 0, respectively, are as follows: (2)

(2)

(2)

– the three infinitesimal boosts .K10 (τ, x), K10 (τ, y), K10 (τ, z) are given by: ⎛

0 j2 (2) ⎜1 0 .K10 (τ, x) = ⎜ ⎝0 0 0 0

0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

⎛ 0 (2) ⎜0 K10 (τ, y) = ⎜ ⎝1 0

0 j2 0 0 0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

4 The .Z3 -Graded Extension of the Poincaré Algebra

⎛

000 (2) ⎜0 0 0 K10 (τ, z) = ⎜ ⎝0 0 0 100 (2)

179

⎞ j2 0⎟ ⎟ 0⎠ 0

(2)

(4.51)

(2)

– the three boosts .K21 (τ, x), K21 (τ, y), K21 (τ, z) are given by: ⎛

0 (2) ⎜j .K21 (τ, x) = ⎜ ⎝0 0 ⎛ 0 (2) ⎜0 K21 (τ, z) = ⎜ ⎝0 j (2)

⎞ 0 0⎟ ⎟, 0⎠ 0 ⎞ 00j 0 0 0⎟ ⎟ 0 0 0⎠ j 0 0 0

0 0 0 0

⎛

0 (2) ⎜0 K21 (τ, y) = ⎜ ⎝j 0

0 0 0 0

j 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

(4.52)

000 (2)

(2)

and the three boosts .K02 (τ, x), K02 (τ, y), K02 (τ, z) are given by: ⎛

0 (2) ⎜j 2 .K02 (τ, x) = ⎜ ⎝0 0 ⎛ 0 (2) ⎜0 K02 (τ, z) = ⎜ ⎝0 j2

⎞ 100 0 0 0⎟ ⎟, 0 0 0⎠ 000 ⎞ 001 0 0 0⎟ ⎟ 0 0 0⎠ 000

⎛

0 (2) ⎜0 K02 (τ, y) = ⎜ ⎝j 2 0

⎞ 010 0 0 0⎟ ⎟, 0 0 0⎠ 000

(4.53)

The so defined infinitesimal generators keep the symmetry properties of the Lie group matrices, i.e. they close under the commutator product, provided that the two factors satisfy the chain rule, with the second index of the first matrix coinciding with the first index of the second matrix, like in the following examples: .

[(2) ] [(1) ] (2) (2) K02 (τ, y), K21 (τ, z) , K20 (τ, z), K02 (τ, x) , etc.

(4.54)

The 27 generators form three groups containing three matrices each, belonging to raising, lowering or neutral type with respect to the .Z3 -grade of the Minkowskian 4-vector

180

R. Kerner

⎛(0) ⎞ K 0 0 00 ⎜ ⎟ (0) (0) ⎜ ⎟ .K = ⎜ ⎟ 0 K 0 11 ⎝ ⎠

⎛

(0)

0

⎛

⎞ (1) 0 K 0 01 ⎜ ⎟ (1) (1) ⎟ ⎜ K = ⎜ 0 0 K12 ⎟ ⎝ ⎠

⎞ (2) 0 0 K 02 ⎜ ⎟ (2) ⎜(2) ⎟ K = ⎜K10 0 0 ⎟ ⎝ ⎠ (2)

(1)

K20 0

0 K22

0

0 K21 0 (4.55) (p)

Each of the three big .12×12 matrices composed of three blocks of .4×4 matrices .Krs (.p, r, s = 0, 1, 2) appears in three different versions corresponding to the choice of one of the three elementary Lorentz boosts in .(τ, x), (τ, y) or .(τ, z) 2-dimensional (r)

spacetime planes. Let us denote them by .Ki , i = 1, 2, 3, corresponding to the (1)

respective choice of the space direction .x, y or z. For example, for .Ky we shall get explicitly ⎛ ⎜ (1) ⎜ Ky = ⎜ ⎝

⎞

(1)

0

K01 (τ, y)

0

0

K20 (τ, y)

0

.

(1)

0

⎟

(1) ⎟ , K12 (τ, y) ⎟ ⎠

(4.56)

0

and so forth. The spatial rotations around the axes .0x, 0y and 0z are represented in the usual 4-dimensional Minkowskian space as follows: ⎛ 0 ⎜0 .Jx = ⎜ ⎝0 0

0 0 0 0

0 0 0 1

⎛ ⎞ 0 0 0 ⎜0 0 0⎟ ⎟ , Jy = ⎜ ⎝0 0 −1⎠ 0 0 −1

0 0 0 0

⎞ 0 1⎟ ⎟, 0⎠ 0

⎛ 0 ⎜0 Jz = ⎜ ⎝0 0

0 0 0 −1 1 0 0 0

⎞ 0 0⎟ ⎟. 0⎠

(4.57)

0

The full set of .12 × 12 matrices representing three independent spatial rotations acting on the twelve-dimensional .Z3 -graded Minkowskian spacetime is as follows: ⎞ Ji 0 0 = ⎝ 0 Ji 0 ⎠ , 0 0 Ji ⎛

(0) .Ji

⎛

(1) Ji

⎞ 0 Ji 0 = ⎝ 0 0 Ji ⎠ , Ji 0 0

⎛

(2) Ji

⎞ 0 0 Ji = ⎝Ji 0 0 ⎠ , 0 Ji 0

(4.58)

They also form a .Z3 graded Lie algebra with respect to the ordinary Lie bracket (the commutator of matrices). Therefore we get the full set of .Z3 -graded relations defining the algebra (.r, s, r + s are modulo 3), conformally with the structure of the .Z3 -graded Lorentz algebra introduced in [10]. [ .

(r) (s) Ki , Kk

] =

(r+s) −εikl Jl ,

[

(r) (s) Ji , Kk

] =

(r+s) εikl Kl ,

[

(r) (s) Ji , Jk

]

(r+s)

= εikl Jl . (4.59)

4 The .Z3 -Graded Extension of the Poincaré Algebra

181

4.6 Z3 -Extended Poincaré Algebra and the Casimir Operators The standard Poincaré algebra is the semi-direct product of the Lorentz algebra and the 4-dimensional abelian algebra of translations .Pμ , satisfying the well-known commutation relations:

.

[ ] Mμν , Mλρ = ημρ Mνλ − ηνλ Mμρ + ημλ Mνρ − ηνρ Mμλ , [

.

] Pμ , Pν = 0,

[ ] Mμν , Pλ = ημλ Pν − ηνλ Pμ ,

In terms of six generators .Ki = M0i and .Jm = standard commutation relations .

[Ji , Jk ] = εikl Jl ,

[Ji , Kk ] = εikl Kl ,

1 2

(4.60)

εikm Mik , .i, k, .. = 1, 2, 3, the

[Ki , Kk ] = −εikl Jl .

(4.61)

must be complemented by the following extra commutation relations with .Pμ = (P0 , Pi ): .

[Ki , P0 ] = Pi ,

[

] Ki , Pj = −δij P0 ,

[Ji , P0 ] = 0, [Ji , Pk ] = εikm Pm . (4.62)

The most appropriate realization of the totality of commutation relations given by (4.59) and (4.62) is via differential operators, with the generators .Pμ identified with partial derivations .∂μ . These operators can be produced from the standard matrix representation by the following well-known procedure. Let us take for example the .4 × 4 matrix representation of 3-dimensional rotations given by formulae (4.57). The differential operators corresponding to .Jx , Jy and .Jz are obtained by taking formally the scalar product of the space-time 4-covector .[τ, x, y, z] with the 4gradient .∂μ transformed by the corresponding matrix .Ji . Take for example the matrix .Jx : ⎛ 0 ( ) ⎜0 . τ, x, y, z ⎜ ⎝0 0

0 0 0 0

⎞⎛ ⎞ ∂τ 0 0 ⎟ ⎜ 0 0 ⎟ ⎜ ∂x ⎟ ⎟ = z∂y − y∂z . 0 −1⎠ ⎝∂y ⎠ 1 0 ∂z

(4.63)

Jz → y∂x − x∂y . Similarly we get .Jy → x∂z − z∂x , Our next aim is to extend the standard Poincaré algebra so as to include the .Z3 -graded Lorentz algebra defined by the set of commutation relations (4.59) (r)

complemented by the set of three types of translation generators, denoted by .Pμ , .r = 0, 1, 2 and .μ, ν, .. = 0, 1, 2, 3. Let us separate time and space components; we shall write then

182

R. Kerner (r)

(r) (r)

Pμ = [P0 , Pi ].

(4.64)

.

We expect the following .Z3 -graded generalization of standard commutation relations between the Lorentz and translation generators: (r)

(s)

[ P0 , Pk ] = 0;

.

(r)

(s)

[ Jk , P0 ] = 0;

.

(r)

(s)

(r+s)

[ Ki , P0 ] = Pi ,

.

(r)

(s)

[ Pi , Pj ] = 0, (r)

(s)

(4.65)

(r+s)

[ Ji , Pk ] = εikl Pl , (r)

(s)

(4.66)

(r+s)

[ Ki , Pk ] = −δik P0 .

(4.67)

In all the above relations the grades .r, s = 0, 1, 2 add up modulo 3. The construction of differential operators providing faithful representation of the .Z3 -graded Poincaré algebra (4.67) shall follow the prescription given by (4.63) with .12 × 12 matrices introduced in previous section, and 12-component generalizations of Minkowskian 4-vectors and co-vectors. Let us introduce the following notation for generalized vectors in triple Minkowskian space-time: .

[τ0 , x0 , y0 , z0 ; τ1 , x1 , y1 , z1 ; τ2 , x2 , y2 , z2 ] ,

(4.68)

The notations are obvious: the lower index “0” refers to the standard Minkowskian component (graded 0), while the indices “1” and “2” refer to two complex extensions, mutually conjugate, of .Z3 grades 1 and 2, respectively. For the moment we leave aside the definition of metrics in the so extended triple Minkowskian space-time. Partial derivatives take, with respect to these variables are represented by the following 12-component column vector (written here as a horizontal co-vector transposed, in order to spare the space): [ .

∂τ0 , ∂x0 , ∂y0 , ∂z0 ; ∂τ1 , ∂x1 , ∂y1 , ∂z1 ; ∂τ2 , ∂x2 , ∂y2 , ∂z2 ;

]T

(4.69)

What is left now is to compute patiently the results of contraction of the co-vector (4.68) with the 12-component generator of generalized translations (4.69) with one of the eighteen .12 × 12 matrices representing the generalized Lorentz algebra (4.59) sandwiched in between. This will produce the 18 generators of the .Z3 -graded Poincaré algebra represented in form of linear differential operators. With twelve translations (4.64) we shall get the 30-dimensional .Z3 -graded covering extension of the Poincaré algebra, of which the usual 10-dimensional subalgebra is the standard Poincaré algebra. The results are a bit cumbersome, but their construction and symmetry properties are quite clear.

4 The .Z3 -Graded Extension of the Poincaré Algebra

183 (r)

Let us start with the nine generalized Lorentz boosts . K i . We have explicitly: .

(0) Kx

= (τ0 ∂x0 + x0 ∂τ0 ) + (j 2 τ1 ∂x1 + j x1 ∂τ1 ) + (j τ2 ∂x2 + j 2 x2 ∂τ2 ),

(0) Ky

= (τ0 ∂y0 + y0 ∂τ0 ) + (j 2 τ1 ∂y1 + j y1 ∂τ1 ) + (j τ2 ∂y2 + j 2 y2 ∂τ2 ),

(0) Kz

= (τ0 ∂z0 + z0 ∂τ0 ) + (j 2 τ1 ∂z1 + j z1 ∂τ1 ) + (j τ2 ∂z2 + j 2 z2 ∂τ2 ); .

(1) Kx

= (τ0 ∂x1 + j x0 ∂τ1 ) + (j 2 τ1 ∂x2 + j 2 x1 ∂τ2 ) + (j τ2 ∂x0 + x2 ∂τ0 ),

(1) Ky

= (τ0 ∂y1 + j y0 ∂τ1 ) + (j 2 τ1 ∂y2 + j 2 y1 ∂τ2 ) + (j τ2 ∂y0 + y2 ∂τ0 ),

(1) Kz

= (τ0 ∂z1 + j z0 ∂τ1 ) + (j 2 τ1 ∂z2 + j 2 z1 ∂τ2 ) + (j τ2 ∂z0 + z2 ∂τ0 ); .

(2) Kx

= (τ0 ∂x2 + j 2 x0 ∂τ2 ) + (j τ2 ∂x1 + j x2 ∂τ1 ) + (j 2 τ1 ∂x0 + x1 ∂τ0 ),

(2) Ky

= (τ0 ∂y2 + j 2 y0 ∂τ2 ) + (j τ2 ∂y1 + j y2 ∂τ1 ) + (j 2 τ1 ∂y0 + y1 ∂τ0 ),

(2) Kz

= (τ0 ∂z2 + j 2 z0 ∂τ2 ) + (j τ2 ∂z1 + j z2 ∂τ1 ) + (j 2 τ1 ∂z0 + z1 ∂τ0 ).

(4.70)

(4.71)

(4.72)

The .Z3 -graded generalized differential operators representing the Lorentz boosts (0)

display remarkable symmetry properties. The “diagonal” generators . K i are hermitian: they are invariant under the simultaneous complex conjugation, replacing j by 2 .j and vice versa, and switching the indices .1 → 2, 2 → 1. (1)

(2)

Under the same hermitian symmetry operation the .Z3 -graded boosts . K i and . K i transform into each other, so that we have (1)† Ki

.

(2)

= Ki,

(2)† Ki

(1)

= Ki.

The commutation relations between the generalized Lorentz boosts given by (4.70), (4.71) and (4.72) define the differential representation of .Z3 -graded exten(r)

sion of pure rotations, .Jk , with .r = 0, 1, 2 and .i, j, .. = 1, 2, 3. By tedious (but not too sophisticated) calculation we can check that the commutation relations between the .Z3 -graded Lorentz boosts imposed as hypothesis in (4.59): (r)

(s)

(r+s)

[ Ki , Kk ] = −εikl Jl ,

.

(s)

lead indeed to the following expressions for spatial rotations .Ji :

184

R. Kerner (0)

Jx = (z0 ∂y0 − y0 ∂z0 ) + (z1 ∂y1 − y1 ∂z1 ) + (z2 ∂y2 − y2 ∂z2 ),

.

(0)

Jy = (x0 ∂z0 − z0 ∂x0 ) + (x1 ∂z1 − z1 ∂x1 ) + (x2 ∂z2 − z2 ∂x2 ), (0)

Jz = (y0 ∂x0 − x0 ∂y0 ) + (y1 ∂x1 − x1 ∂y1 ) + (y2 ∂x2 − x2 ∂y2 ),

(4.73)

Note that the above generators are sums of classical expressions for .Jk , each of them acting in its own sector of the .Z3 -graded extension of Minkowskian spacetime. (1)

The grade 1 generators of rotations .Ji have the same form, but mix up coordinates with derivatives from different sectors, in cyclical order, symbolically .0 → 1, .1 → 2, .2 → 0: (1)

Jx = (z0 ∂y1 − y0 ∂z1 ) + (z1 ∂y2 − y1 ∂z2 ) + (z2 ∂y0 − y2 ∂z0 ),

.

(1)

Jy = (x0 ∂z1 − z0 ∂x1 ) + (x1 ∂z2 − z1 ∂x2 ) + (x2 ∂z0 − z2 ∂x0 ), (1)

Jz = (y0 ∂x1 − x0 ∂y1 ) + (y1 ∂x2 − x1 ∂y2 ) + (y2 ∂x0 − x2 ∂y0 ),

(4.74)

(2)

Finally, the grade 2 generators of spatial rotations, .Ji , repeat the same scheme, but in reverse (anti-cyclic) order, i.e. .0 → 2, 1 → 0, 2 → 1: (2)

Jx = (z0 ∂y2 − y0 ∂z2 ) + (z1 ∂y0 − y1 ∂z0 ) + (z2 ∂y1 − y2 ∂z1 ),

.

(2)

Jy = (x0 ∂z2 − z0 ∂x2 ) + (x1 ∂z0 − z1 ∂x0 ) + (x2 ∂z1 − z2 ∂x1 ), (2)

Jz = (y0 ∂x2 − x0 ∂y2 ) + (y1 ∂x0 − x1 ∂y0 ) + (y2 ∂x1 − x2 ∂y1 ),

(4.75)

It can easily be checked that these differential operators correspond to what we would get by direct construction using the matrix representation given in (4.58). The 18 differential operators acting on the .Z3 -graded extension of Minkowskian space(r)

time; the 9 generalized Lorentz boosts .Ki and the 9 generalized space rotations (s)

Jk , with . r, s = 0, 1, 2 and .i, j = 1, 2, 3, define the faithful representation of the .Z3 -graded generalization of the Lorentz group. In order to introduce the .Z3 -graded extension of full Poincaré group we have to add extra two 4-component generators of translations, each one acting on its own sector of the generalized .Z3 -graded Minkowskian space-time. It turns out that in order to satisfy the .Z3 -graded set of standard commutation relations given by (4.67), the three differential operators .

4 The .Z3 -Graded Extension of the Poincaré Algebra (0)

(1)

Pμ ,

Pμ ,

.

185 (2)

Pμ

must be defined as follows: (0) [ ] Pμ = ∂τ0 , −∂x0 , −∂y0 , −∂z0 .

(4.76)

.

(1) ] [ Pμ = j ∂τ1 , −∂x1 , −∂y1 , −∂z1 . (2)

Pμ =

[

j 2 ∂τ2 , −∂x2 , −∂y2 , −∂z2

(4.77)

] (4.78) (r)

It can be checked by direct computation that the eighteen generators .Ki and (s)

Jk together with the twelve generalized .Z3 -graded translations defined above by (4.76), (4.77, (4.78) satisfy the full set of .Z3 -graded extension of the Poincaré algebra.1 Its total dimension is 30, three times ten, corresponding to three replicas of the classical Poincaré group, one “diagonal”, acting on three components of the .Z3 graded Minkowskian space-time without mixing them, and two other replicas acting on all three components transforming them into one another. The commutations relations are given by the set defined in (4.64), (4.65), (4.66) and (4.67). Classical Poincaré algebra admits two Casimir operators which commute with all generators. These are the 4-square of the translation 4-vector .Pμ P μ , and the 4-square of the Pauli-Lubanski 4-vector .Wμ W μ , where

.

Wμ =

.

1 μνλρ Jνλ Pρ , ε 2

J0i = Ki , Jik = εikl J l .

(4.79)

In terms of more familiar generators .Ki and .Jl the Pauli-Lubanski vector takes on the following form: W0 = Ji P i = J · P,

.

Wi = P0 Ji − εij k P j K k , or W = P 0 J − P ∧ K. (4.80)

The following relations are easily verified: Wμ P μ = 0,

.

[ μ λ] W , P = 0,

[

] J μλ , W ρ = ηλρ W μ − ημρ W λ .

(4.81)

The eigenvalues of these two Casimir operators, corresponding to the mass and orbital spin of a given particle state, define the irreducible representations of the Poincaré group,

(1)

(2)

alternative choice is possible, too, which amounts to multiplying .Pμ by .j 2 and .Pμ by j , making all three time-like coordinates real, and the 3-vectors complexified.

1 An

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R. Kerner

Pμ P μ = m2 ,

.

Wμ W μ = L(L + 1)

(4.82)

In the case of the .Z3 -graded extension the corresponding Casimir operators must be invariant under permutations imposed by the .Z3 symmetry. That is to say, the three types of generators should contribute equally to the generalized Casimir operator. The expression generalizing the mass operator .Pμ P μ should contain not only the (0) (0)

obvious term .Pμ Pμ , but also other contributions of all possible grades, like e.g. (1) (2)

another grade 0 term: .Pμ Pμ , as well as other similar terms of grades 1 and 2. The symmetric and real combination imitating the first Casimir operator in (4.82) is as follows: (0) (0)

(1) (1)

(2) (2)

(0) (1)

(1) (2)

(2) (0)

P2 = Pμ Pμ + Pμ Pμ + Pμ Pμ + Pμ Pμ + Pμ Pμ + Pμ Pμ ,

(4.83)

.

The Pauli-Lubanski 4-vector also possesses its .Z3 -graded extensions. They are of the following form: (0)

(0) (0) (1) (2) (2) (1) 1 εμνλρ (Jνλ Pρ + Jνλ Pρ + Jνλ Pρ ), 2

(1)

(2) (2) (1) (0) (0) (1) 1 εμνλρ (Jνλ Pρ + Jνλ Pρ + Jνλ Pρ ), 2

(2)

(1) (1) (2) (0) (0) (2) 1 εμνλρ (Jνλ Pρ + Jνλ Pρ + Jνλ Pρ ). 2

Wμ =

.

Wμ = Wμ =

(4.84)

With these three graded Pauli-Lubanski vectors we can produce a .Z3 -invariant extended Casimir operator of orbital spin: (0)

(0)

(1)

(1)

(2)

(2)

(0)

(1)

(1)

(2)

(2)

(0)

W2 = Wμ Wμ + Wμ Wμ + Wμ Wμ + Wμ Wμ + Wμ Wμ + Wμ Wμ ,

.

(4.85)

The analysis of eigenvalues of the generalized Casimir operators and the classification of irreducible representations of .Z3 -graded extension of the Poincaré algebra presented here will be the subject of the forthcoming publications. Acknowledgments The author is greatly indebted to Jerzy Lukierski for countless discussions, enlightening remarks and lots of very useful suggestions.

References 1. R. Kerner, O. Suzuki, The discrete quantum origin of the Lorentz group and the Z3-graded ternary algebras, Proceedings of the RIMS Conference on Mathematical Physics, Kyoto 2013, pp. 54–72 (2014) see also: https://ci.nii.ac.jp/naid/110009863886

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2. R. Kerner, Ternary generalization of Pauli’s principle and the Z6 -graded algebras, Physics of Atomic Nuclei, 80 (3), pp. 529–531 (2017). also: arXiv:1111.0518, arXiv:0901.3961 3. R. Kerner, in Mathematical Structures and Applications, Springer, pp. 311–357 (2018) 4. R. Kerner, Ternary Z2 × Z3 graded algebras and ternary Dirac equation, Physics of Atomic Nuclei 81 (6), pp. 871–889 (2018), also: arXiv:1801.01403 5. R. Kerner, The Quantum nature of Lorentz invariance, Universe, 5 (1), p.1, (2019). https://doi. org/10.3390/universe5010001 (2019). 6. W. Pauli, Zeitschrift für Physik, 26 (5), pp. 336–363 (1926). 7. P.A.M. Dirac, The Quantum Theory of the Electron, Proc. Royal Soc. A, 117 (778), pp. 610– 624; ibid 118 (779) pp. 351–361 (1928) 8. T.D. Lee and G.C. Wick,Finite Theory of Quantum Electrodynamics, Phys. Rev. D, 2 p. 1033 (1970). 9. D. Anselmi and M. Piva, Perturbative Unitarity of Lee-Wick Quantum Field Theory, Phys. Rev. D 96 045009 (2017). 10. R. Kerner and J. Lukierski, Internal quark symmetries and colour SU (3) entangled with Z 3 graded Lorentz algebra, Nuclear Physics B, 972, p.115529 (2021) Z3 -graded colour Dirac equation for quarks, confinementt and generalized Lorentz symmetries, Phys. Letters B, Vol. 792, pp. 233–237 (2019), also: arXiv:1901.10936 [hep-th] 11. R. Kerner, Graduation Z3 et la racine cubique de l’équation de Dirac, Comptes Rendus Acad. Sci. Paris, 312, ser. II, pp. 191–195 (1991) 12. V. Abramov, R. Kerner, B. Le Roy, Hypersymmetry: a Z3 -graded generalization of supersymmetry, Journal of Math.Phys. 38 (3), 1650–1669 (1997). 13. D. Finkelstein, Hyperspin and Hyperspace, Phys. Rev. Lett. 56 p.p. 1532–1533, (1986). 14. D.C. Brody, L.P. Hughston, Theory of Quantum Space-Time, Proc. Roy. Soc. A461, pp. 2679– 2699 (2005)

Chapter 5

Ternary Leibniz Color Algebras and Beyond Ibrahima Bakayoko and Ismail Laraiedh

5.1 Introduction An n-ary algebra consists of a vector space A together with a multilinear map .μ on A × A × · · · × A (n times) with values in A. Whenever .n = 3, we say that we have ternary algebras. In other words, ternary algebras are vector space equipped with (at least) a multiplication with three items instead of two, as in classical algebraic structures and satisfying some identities. They originate from the work of Jacobson in 1949 in the study of associative algebra .(A, .) that are closed relative to the ternary operation .[[a, b], c], where .[a, b] = ab−ba. According to some conditions satisfied by the multiplication .μ, we dispose of ternary (partial or total) associative algebras, ternary Leibniz algebras, 3-Lie algebras [50], ternary Leibniz-Poisson algebras, ternary Hopf algebras, ternary Heap algebras, Comstrans algebras, Akivis algebras, Lie-Yamaguti algebras, Lie triple systems [28–30, 51], ternary Jordan algebras [34], Jordan-Lie triple systems [41], Jordan triple and so on. Some of these algebras have either their n- ary generalization, their Hom-version their (Hom)-super version or their color version, among which one can cite n-Lie algebras [37], Hom-Lie triple systems [52], Super 3-Lie Algebras [1–3], n-ary HomNambu [53], Ternary Hom-Nambu-Lie algebras [5], etc. Generalized derivations of (color) n-ary algebras [35].

.

I. Bakayoko (O) Département de Mathématiques, Université de N’Zérékoré, N’Zérékoré, Republic of Guinea I. Laraiedh Department of Mathematics, Faculty of Sciences, Sfax University, Sfax, Tunisia Department of Mathematics, College of Sciences and Humanities - Kowaiyia, Shaqra University, Shaqra, Kingdom of Saudi Arabia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_5

189

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The notion of n-Lie algebras were introduced by Filippov [47] in 1985 as a natural generalization of Lie algebras. More precisely, n-Lie algebras are vector spaces V equipped with n-ary operation which is skew symmetric for any pair of variables and satisfies the following identity: .

[[x1 , x2 , . . . , xn ], y1 , y2 , . . . , yn−2 , yn−1 ] =

n E [x1 , x2 , . . . , xi−1 , [xi , y1 , y2 , . . . , yn−2 , yn−1 ], xi+1 , . . . , xn ]. (5.1) i=1

For .n = 3, it reads [[x, y, z], t, u] = [x, y, [z, t, u]] + [x, [y, t, u], z] + [[x, t, u], y, z].

.

Whenever the identity (5.1) is satisfied and the bracket fails to be totally skew symmetric we obtain n-Leibniz algebras [21]. Moreover, when the bracket .[−, −, −] is skew-symmetric with respect to the last two variables, .(V , [−, −, −]) is said to be quasi-Lie 3-algebras [20]. If in addition, [x, y, z] + [y, z, x] + [z, x, y] = 0

.

is satisfied for any .x, y, z ∈ V , .(V , [−, −, −]) is called a Lie triple system [20]. The n-Lie algebras [13, 15, 23] found their applications in many fields of mathematics and Physics. For instance, Takhtajan has developed the foundations of the theory of Nambu-Poisson manifolds [48]. The general cohomology theory for n-Lie algebras and Leibniz n-algebras was established in [42]. The structure and classification theory of finite dimensional n-Lie algebras was given by Ling [38] and many other authors. For more details of the theory and applications of n-Lie algebras, see [7] and references therein. The concept of 3-Lie algebras [4, 6] are extended to the graded case by Zhang T. in [54], in which he studied the cohomology and deformations of 3-Lie colour algebras, as well as the abelian extensions of 3-Lie colour algebras. The 3-Lie algebras [11, 12, 26, 46] are applied to the study of the gauge symmetry and supersymmetry of multiple coincident M2-branes in [8]. The authors in [39] studied non-commutative ternary Nambu-Poisson algebras and their Hom-type version. They provided construction results dealing with tensor product and direct sums of two (non-commutative) ternary (Hom-) Nambu-Poisson algebras. Examples and a 3-dimensional classification of non-commutative ternary Nambu-Poisson algebras were given. Comstrans algebras were introduced in [55] as an answer of a problem from differential geometry: finding of the algebraic structure in the tangent bundle corresponding to the coordinate n-ary loop of an .(n + 1)-web [25]. The role played by Comstrans algebras is analogous to the role played by the Lie algebra of a Lie group. They arise in many contexts [31–33, 45]. Some examples (on Lie algebras, spaces with bilinear forms, spaces of rectangular matrices, on Minkowski space-

5 Ternary Leibniz Color Algebras and Beyond

191

time that extends the vector triple product on .R3 , on spaces of Hermitian operators, the transposed comtrans algebra), Representation theory and some applications in quantum mechanics are given in [46]. It is well-known that mathematical objects are often understood through studying operators defined on them. For instance, in Gallois theory a field is studied by its automorphisms, in analysis functions are studied through their derivations, and in geometry manifolds are studied through their vector fields. Fifty years ago, several operators have been found from studies in analysis, probability and combinatorics. Among these operators, one can cite, element of centroid [14], averaging operator, Reynolds operator, Leroux’s TD operator, Nijenhuis operator and Rota-Baxter operator [27, 40]. The Rota-Baxter operator originated from the work of G. Baxter [17, 44] on Spitzer’s identity[47] in fluctuation theory. For example, on the polynomial algebra, the indefinite integral f R(f )(x) =

x

f (t)dt

.

0

and the inverse of any bijective derivation are Rota-Baxter operators. Rota-Baxter algebras (associative algebra with Rota-Baxter operator) are used in many fields of mathematics and mathematical Physics. In mathematics, they are used in algebra, number theory, operads and combinatorics [9, 10, 19, 39, 43]. In mathematical physics they appear as the operator form of the classical Yang Baxter equation [9] or as the fundamental algebraic structure in the normalisation of quantum field theory of Connes and Kreimer [22]. In non-associative algebra, the Rota-Baxter operators are used in order to produce another one of the same type or not from the previous one. The Nijenhuis operator on an associative algebra was introduced in [18] to study quantum bi-Hamiltonian systems while the notion Nijenhuis operator on a Lie algebra originated from the concept of Nijenhuis tensor that was introduced by Nijenhuis in the study of pseudo-complex manifolds and was related to the well known concepts of Schouten-Nijenhuis bracket, the Frolicher-Nijenhuis bracket [24], and the Nijenhuis-Richardson bracket. The associative analog of the Nijenhuis relation may be regarded as the homogeneous version of Rota-Baxter relation [36]. We know that some algebraic structures admit left and right version such as right symmetric algebras (left symmetric algebras), right Leibniz algebras (left Leibniz algebras), right BiHom-Lie algebras (right BiHom-Lie algebras) and so on. In current paper, we deal with right Leibniz color algebras and introduce right ternary Leibniz color algebras, right color Lie triple systems, right Comstrans color algebras. We study their properties and the relationship among them. The paper is organized as follows. In section two, we recall basic notions concerning, graded vector spaces, bicharacter and associative color algebras. In section three, we introduce Leibniz color algebras and Leibniz-Poisson color algebras. We investigate some properties of Leibniz color algebras and give some constructions dealing with averaging operator, element of centroid, Nijenhuis operator, Rota-Baxter operator

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and Reynolds operators. We introduce action of Leibniz color algebra onto another one and define the semidirect sum of Leibniz color algebras. In section four, we introduce ternary Leibniz color algebras and give some constructions from Leibniz color algebra, element of centroid, averaging operator, Rota-Baxter operator and Reynolds operator. We also prove that the tensor product of two ternary Leibniz color algebra is a Leibniz color algebra. Moreover, we show that the tensor product of a commutative associative color algebra and a ternary Leibniz color algebra is also a ternary Leibniz color algebra. Moreover, we give some methods of constructing bimodules over ternary Leibniz color algebras. Next, we introduce and give some constructions of ternary Leibniz-Poisson color algebras. As subclass of ternary Leibniz color algebras, constructions of color Lie triple systems are provided. Their connection with Jordan Lie triple systems are given. Finally, we introduce and give some constructions of Comstrans color algebras. Throughout this paper, all graded vector spaces are assumed to be over a field .K of characteristic different from 2.

5.2 Preliminaries In this section, we give the definitions of associative color algebras, Lie color algebras, averaging operators on associative and Lie color algebras, and constructions of Leibniz color algebras. Definition 5.1 (1) Let G be an abelian group. A linear space V is said to be a G-graded if, there exists a family .(Va )a∈G of vector subspaces of V such that V =

O

.

Va .

a∈G

(2) An element .x ∈ V is said to be homogeneous of degree .a ∈ G if .x ∈ Va . We denote .H(V ) the set of all homogeneous elements in V . Example 5.1 Let .V = ⊕a∈G Va and .V , = ⊕a∈G Va, be two G-graded vector spaces. We have the following graded vector spaces: (1) The direct sum V ⊕V, =

.

O (Va ⊕ Va, ),

(5.2)

a∈G

(2) The direct product V ×V, =

.

O (Va × Va, ), a∈G

(5.3)

5 Ternary Leibniz Color Algebras and Beyond

193

(3) The tensor product V ⊗V, =

O O

.

Va ⊗ Vb, .

(5.4)

c∈G a+b=c

Definition 5.2 Let .V = ⊕a∈G Va and .V , = ⊕a∈G Va, be two G-graded vector spaces. A linear map .f : V → V , is said to be homogeneous of degree b if , f (Va ) ⊆ Va+b , ∀a ∈ G.

.

If, f is homogeneous of degree zero i.e. .f (Va ) ⊆ Va, holds for any .a ∈ G, then f is said to be even. Definition 5.3 (1) An algebra .(A, O ·) is said to be G-graded if its underlying vector space is Ggraded i.e. .A = a∈G Aa , and if furthermore .Aa ·Ab ⊆ Aa+b , for all .a, b ∈ G. Let .A, be another G-graded algebra. (2) A morphism .f : A → A, of G-graded algebras is by definition an algebra morphism from A to .A, which is, in addition an even map. Definition 5.4 Let G be an abelian group. A map .ε : G × G → K∗ is called a skew-symmetric bicharacter on G if the following identities hold, (i) .ε(a, b)ε(b, a) = 1, (ii) .ε(a, b + c) = ε(a, b)ε(a, c), (iii) .ε(a + b, c) = ε(a, c)ε(b, c), a, b, c ∈ G,

.

Example 5.2 Some standard examples of skew-symmetric bicharacters are: (i) .G = Z2 ,

ε(i, j ) = (−1)ij , or more generally G = Zn2 = {(α1 , . . . , αn )|αi ∈ Z2 , i = 1, 2, ..., n},

.

ε((α1 , . . . , αn ), (β1 , . . . , βn )) := (−1)α1 β1 +···+αn βn , (ii) .G = Z × Z, ε((i1 , i2 ), (j1 , j2 )) = (−1)(i1 +i2 )(j1 +j2 ) , (iii) .G = {−1, +1}, ε(i, j ) = (−1)(i−1)(j −1)/4 . Example 5.3 Let .σ : G × G → K∗ be any mapping such that σ (x, y + z)σ (y, z) = σ (x, y)σ (x + y, z), ∀x, y, z ∈ G.

.

(5.5)

Then, .δ(x, y) = σ (x, y)σ (y, x)−1 is a bicharacter on G. In this case, .σ is called a multiplier on G, and .δ the bicharacter associated with .σ . For instance, let us define the mapping .σ : G × G → R by

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σ ((i1 , i2 ), (j1 , j2 )) = (−1)i1 j2 , ∀ik , jk ∈ Z2 , k = 1, 2.

.

It is easy to verify that .σ is a multiplier on G and δ((i1 , i2 ), (j1 , j2 )) = (−1)i1 j2 −i2 j1 , ∀ik , jk ∈ Z2 , i = 1, 2.

.

is a bicharacter on G. If x and y are two homogeneous elements of degree a and b respectively and ε is a skew-symmetric bicharacter, then we shorten the notation by writing .ε(x, y) instead of .ε(a, b).

.

Definition 5.5 A color algebra is a G-graded algebra .(A, ·) together with a bicharacter .ε. Definition 5.6 An associative color algebra is a G-graded algebra .(A, ·) together with a bicharacter .ε : G × G → K∗ such that (x · y) · z = x · (y · z)

.

(associativity)

(5.6)

for all .x, y, z ∈ H(A). Example 5.4 Let .A = A0 ⊕ A1 =< a2 , a3 > ⊕ < a1 > and .B = B0 ⊕ B1 =< b3 > ⊕ < b1 , b2 > be two three dimensional vector superspace. The multiplications .a1 a2 = a1 , a2 a2 = a2 , a3 a1 = a1 , a3 a3 = a3 and .b1 b3 = b2 , b2 b3 = b2 , b3 b3 = b3 make A and B into associative superalgebras respectively. Example 5.5 Let .G = Z2 × Z2 and A = A(0,0) ⊕ A(0,1) ⊕ A(1,0) =< e3 > ⊕ < e1 > ⊕ < e2 >

.

be a three dimensional G-graded vector space with the bicharacter ε((α1 , α2 ), (β1 , β2 )) := (−1)α1 β1 +α2 β2 .

.

Then, A is an associative color algebra with the multiplication e2 e3 = e2 ,

.

e3 e1 = e1 ,

e3 e3 = e3 .

5.3 Leibniz Color Algebras One of the main result of this paper is based on the fact that one may associate a ternary Leibniz color algebra to Leibniz color algebra. To this end and to give

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195

various examples of this construction, we develop, in this section, several properties and constructions of Leibniz color algebras.

5.3.1 Generalities In this subsection, we give some results on Leibniz color algebras. They will provide examples of ternary Leibniz color algebras from other algebraic structures. Definition 5.7 ([16]) A Leibniz color algebra is a G-graded vector space L together with an even bilinear map .[−, −] : L ⊗ L → L and a bicharacter .ε : G ⊗ G → K∗ such that [[x, y], z] = [x, [y, z]] + ε(y, z)[[x, z], y]

(5.7)

.

holds, for all .x, y, z ∈ H(L). Remark 5.1 (i) When the bracket .[−, −] is skew-symmetric, L is said to be a Lie color algebra. (ii) When .ε(x, y) ≡ 1, we get a Leibniz algebra. (iii) When .ε(x, y) = (−1)|x||y| , we obtain a Leibniz superalgebra. More precisely, Definition 5.8 A Lie color algebra is a triple .(L, [−, −], ε) in which .(L, [−, −]) is a G-graded vector space and .ε is a bicharacter such that [x, y] = −ε(x, y)[y, x]

(ε-skew- symmetry). (5.8)

.

ε(z, x)[x, [y, z]] + ε(x, y)[y, [z, x]] + ε(x, y)[y, [z, x]] = 0, (ε-Jacobi identity) (5.9) for any .x, y, z ∈ H(L). Example 5.6 Let .G = Z2 × Z2 × Z2 and L = L(1,1,0) ⊕ L(1,0,1) ⊕ L(0,1,1) =< e3 , e4 > ⊕ < e2 , e5 > ⊕ < e1 >

.

be a G-graded five dimensional vector space. The triple .(L, [−, −], ε) is a Lie color algebra with the multiplication [e1 , e4 ] = e2 ,

.

and the bicharacter

[e1 , e5 ] = e3

and

[e4 , e5 ] = e1 ,

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ε((α1 , α2 , α3 ), (β1 , β2 , β3 )) := (−1)α1 β1 +α2 β2 +α3 β3 .

.

Example 5.7 Let .A = A0 ⊕ A1 =< e1 , e2 > ⊕ < e3 > be a two-dimensional superspace. The multiplications [e1 , e1 ] = [e1 , e2 ] = [e2 , e1 ] = 0 and [e2 , e2 ] = e1 ,

.

make L into a Leibniz superalgebra. Lemma 5.1 Let .(L, [−, −], ε) be a Leibniz color algebra. Then R[x,x] = 0,

.

for any .x ∈ L. Proof Taking .y = z in (5.7), we have for any .x, y ∈ L, [x, [y, y]] = [[x, y], y] − ε(y, y)[[x, y], y] = 0,

.

u n

and the conclusion follows. Definition 5.9 Let .(L, [−, −], ε) be a Leibniz color algebra. Then the subset Cl (L) := {c ∈ L| Lc = 0} = {c ∈ L| [c, x] = 0, ∀x ∈ L}

.

(5.10)

is called the left center of L. Cr (L) := {c ∈ L| Rc = 0} = {c ∈ L| [x, c] = 0, ∀x ∈ L}

.

(5.11)

is called the right center of L. C(L) = Cl (L) ∩ Cr (L)

.

(5.12)

is called the center of L. Proposition 5.1 Let L be a Leibniz color algebra. Then [Cr (L), L] ⊆ Cr (L) and

.

[L, Cr (L)] = 0.

In particular, .Cr (L) is an abelian ideal of L. Proof By (5.7), for any .x, y ∈ H(L), c ∈ H(Cr (L)), [x, [c, y]] = [[x, c], y] − ε(c, y)[[x, y], c] = [Rc (x), y] − ε(c, y)Rc ([x, y]) = 0,

.

which means that .[c, y] ∈ Cr (L). The second assertion comes from Definition 5.9. u n

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Definition 5.10 Let L be a Leibniz color algebra. The Leibniz kernel of L is defined as Leib(L) := {[x, x], x ∈ L}

.

(5.13)

Remark 5.2 The Leibniz kernel measures how much a Leibniz algebra deviates from being a Lie algebra. In particular, a Leibniz color algebra is a Lie color algebra if and only if its Leibniz kernel vanishes. Proposition 5.2 Let L be a Leibniz algebra. Then (i) .[Leib(L), L] ⊆ Leib(L) and Leib(L) ⊆ Cr (L). (ii) .Leib(L) is an abelian color ideal of L. Moreover, if .L /= 0, then .Leib(L) /= L. Proof (i) For any .[x, x] ∈ Leib(L), y ∈ L, [[x, x] + y, [x, x] + y] = [[x, x], [x, x]] + [[x, x], y] + [y, [x, x]] + [y, y]

.

= [[x, x], y] + [y, y]

(By Lemma 5.1).

That is .[[x, x], y] = [[x, x] + y, [x, x] + y] − [y, y] ∈ Leib(L). The second statement follows from Lemma 5.1. (ii) Lemma 5.1 and Proposition 5.1 (b) mean that .Leib(L) is a color ideal of L. Moreover, we have .[Leib(L), Leib(L)] = 0. For the second part, suppose that .Leib(L) = L. Then, .[L, L] = 0. In particular, every square of L is zero. u n Therefore, .L = Leib(L) = 0.

5.3.2 Constructions In this subsection, we recall definitions of special even linear operator and give constructions using these maps. Definition 5.11 Let .(A, ·, ε) be a color algebra. Then, an even linear map .ϕ : A → A is said to be: (i) An averaging operator if ϕ(ϕ(x) · y) = ϕ(x) · ϕ(y) = ϕ(x · ϕ(y)),

.

(5.14)

(ii) An element of centroid ϕ(x · y) = ϕ(x) · y = x · ϕ(y),

.

for all .x, y ∈ H(A).

(5.15)

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Proposition 5.3 Let .(L, [−, −], ε) be a Leibniz color algebra and .α : L → L be an injective averaging operator. Then L is also Leibniz color algebra with respect to the bracket [x, y]α = [α(x), y]

.

for all .x, y ∈ H(L). Proof For any .x, y, z ∈ H(L), [[x, y]α , z]α = [α[α(x), y], z]

.

= [[α(x), α(y)], z] = [α(x), [α(y), z]] + ε(y, z)[[α(x), z], α(y)]. As α([[α(x), z], α(y)]) = [α([α(x), z]), α(y)]

.

= α([α([α(x), z]), y]) = α([[x, z]α , y]α ), it follows that, ( ) .α [[x, y]α , z]α ) − [α(x), [α(y), z]]) − ε(y, z)[[α(x), z], α(y)] ) ( = α [[x, y]α , z]α ) − [x, [y, z]α ]α ) − ε(y, z)[[x, z]α , y]α = 0. The conclusion comes from injectivity.

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Proposition 5.4 Let .(L, [−, −], ε) be a Leibniz color algebra and .η : L → L an element of centroid. Then L is also Leibniz color algebra with respect to the bracket [x, y]η = [η(x), y]

.

for all .x, y ∈ H(L). Proof For any .x, y, z ∈ H(L), [[x, y]η , z]η = [η[η(x), y], z]

.

= [[η(x), η(y)], z] = [η(x), [η(y), z]] + ε(y, z)[[η(x), z], η(y)] = [η(x), [η(y), z]] + ε(y, z)[η[η(x), z], y] = [x, [y, z]η ]η ) + ε(y, z)[[x, z]η , y]η . This ends the proof.

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To continue to give other construction from special linear maps we give the below definition. Definition 5.12 Let .(A, ·, ε) be a color algebra. Then, an even linear map .ϕ : A → A is said to be (i) a Nijenhuis operator if ( ) ϕ(x) · ϕ(y) = ϕ ϕ(x) · y + x · ϕ(y) − ϕ(x · y) ,

.

(5.16)

(ii) a Reynolds operator if ( ) ϕ(x) · ϕ(y) = ϕ ϕ(x) · y + x · ϕ(y) − ϕ(x) · ϕ(y) ,

.

(5.17)

(iii) a Rota-Baxter operator (of weight .λ ∈ K) if ( ) ϕ(x) · ϕ(y) = ϕ ϕ(x) · y + x · ϕ(y) + λx · y ,

.

(5.18)

for all .x, y ∈ H(A). Proposition 5.5 Let .(L, [−, −], ε) be a Leibniz color algebra and .P : L → L be a Reynolds operator. Then L is also Leibniz color algebra with respect to the bracket [x, y]P = [P (x), y] + [x, P (y)] − [P (x), P (y)]

.

for all .x, y ∈ H(L). Moreover, P is a morphism of .(L, [−, −]P , ε) onto (L, [−, −], ε).

.

Proof For any .x, y ∈ H(L), [[x, y]P , z]P = [[P (x), y] + [x, P (y)] − [P (x), P (y)], z]P

.

= [[P (x), P (y)], z] + [[P (x), y], P (z)] + [[x, P (y)], P (z)] − [[P (x), P (y)], P (z)] − [[P (x), P (y)], P (z)] = [P (x), [P (y), z]] + [P (x), [y, P (z)] + [x, [P (y), P (z)]] − [P (x), [P (y), P (z)]]

( − [P (x), [P (y), P (z)]] + ε(y, z) [[P (x), z], P (y)] + [[P (x), P (z)], y] ) + [[x, [P (z)], P (y)] − [[P (x), P (z)], P (y)] − [[P (x), P (z)], P (y)]

= [P (x), [P (y), z] + [y, P (z)] − [P (y), P (z)]] + [x, [P (y), P (z)]] ( − [P (x), [P (y), P (z)]] + ε(y, z) [[P (x), P (z)], y] + [[P (x), z] + [x, P (z)]

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− [P (x), P (z)], P (y)] − [[P (x), P (z)], P (y)]

)

= [P (x), [y, z]P ] + [x, P ([y, z]P )] − [P (x), P ([y, z]P )] ) ( + ε(y, z) [P ([x, z]P ), y] + [[x, z]P , P (y)] − [P ([x, z]P ), P (y)] = [x, [y, z]P ]P + ε(y, z)[[x, z]P , y]P . This completes the proof.

u n

Proposition 5.6 Let .(L, [−, −], ε) be a Leibniz color algebra and .R : L → L a Rota-Baxter operator of weight .λ. Then L is also Leibniz color algebra with respect to the bracket [x, y]R = [R(x), y] + [x, R(y)] + λ[x, y]

.

for all .x, y ∈ H(L). Moreover, R is a morphism of .(L, [−, −]R , ε) onto (L, [−, −], ε).

.

Proof For any .x, y ∈ H(L), [[x, y]R , z]R = [[R(x), y] + [x, R(y)] + λ[x, y], z]R

.

= [[R(x), R(y)], z] + [[R(x), y] + [x, R(y)] + λ[x, y], R(z)] + λ[[R(x), y] + [x, R(y)] + λ[x, y], z] = [[R(x), R(y)], z] + [[R(x), y], R(z)] + [[x, R(y)], R(z)] + λ[[x, y], R(z)] + λ[[R(x), y], z] + λ[[x, R(y)], z] + λ2 [[x, y], z]. By Leibniz rule (5.7), [[x, y]R , z]R = [R(x), [R(y), z]] + ε(y, z)[[R(x), z], R(y)] + [R(x), [y, R(z)]]

.

+ ε(y, z)[[R(x), R(z)], y] + [x, [R(y), R(z)]] + ε(y, z)[[x, R(z)], R(y)] + λ[x, [y, R(z)]] + λε(y, z)[[x, R(z)], y] + λ[R(x), [y, z]] + λε(y, z)[[R(x), z], y] + λ[x, [R(y), z]] + λε(y, z)[[x, z], R(y)] + λ2 [x, [y, z]] + λ2 ε(y, z)[[x, z], y]. Furthermore, [x, [y, z]R ]R = [x, [R(y), z] + [y, R(z)] + λ[y, z]]R

.

= [R(x), [R(y), z]] + [R(x), [y, R(z)]] + λ[x, [y, z]] + [x, [R(y), R(z)]] + λ[x, [R(y), z]] + λ[x, [y, R(z)]] + λ2 [x, [y, z]],

5 Ternary Leibniz Color Algebras and Beyond

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and [[x, z]R , y] = [[R(x), z] + [x, R(z)] + λ[x, z], y]R

.

= [[R(x), R(z)], y] + [[R(x), z], R(y)] + [[x, R(z)], R(y)] + λ[[x, z], R(y)] + λ[[R(x), z], y] + λ[[x, R(z)], y] + λ2 [[x, z], y]. It is easy to observe that [[x, y]R , z]R = [x, [y, z]] + ε(y, z)[[x, z], y].

.

u n Proposition 5.7 Let .(L, [−, −], ε) be a Leibniz color algebra and .N : L → L a Nijenhuis operator. Then L is also Leibniz color algebra with respect to the bracket [x, y]N = [N(x), y] + [x, N(y)] − N([x, y])

.

for all .x, y ∈ H(L). Moreover, N is a morphism of .(L, [−, −]N , ε) onto (L, [−, −], ε).

.

Proof For any .x, y ∈ H(L), [[x, y]N , z]N = [N[x, y]N , z] + [[x, y]N , N(z)] − N[[x, y]N , z]

.

= [[N(x), N (y)], z] + [[N(x), y], N(z)] + [[x, N(y)], N(z)] − [N[x, y], N(z)] − N[[N (x), y], z] − N[[x, N(y)], z] + N[N[x, y], z] = [[N(x), N (y)], z] + [[N(x), y], N(z)] + [[x, N(y)], N(z)] − N[N[x, y], z] − N[[x, y], N(z)] + N 2 [[x, y], z] − N[[N(x), y], z] − N[[x, N(y)], z] + N[N[x, y], z]. By exchanging the role of y and z, we get [[x, z]N , y]N = [[N(x), N(z)], y] + [[N(x), z], N(y)] + [[x, N(z)], N(y)]

.

− N[N[x, z], y] − N[[x, z], N(y)] + N 2 [[x, z], y] − N[[N(x), z], y] − N[[x, N(z)], y] + N[N[x, z], y]. Then, [x, [y, z]N ]N = [N(x), [y, z]N ] + [x, N[y, z]N ] − N[x, [y, z]N ]

.

= [N (x), [N(y), z]] + [N(x), [y, N(z)]] − [N(x), N[y, z]] + [x, [N(y), N(z)]] − N[x, [N(y), z]] − N[x, [y, N(z)]] + N[x, N[y, z]]

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= [N (x), [N (y), z]] + [N(x), [y, N(z)]] − N [N(x), [y, z]] − N[x, N[y, z]] + N 2 [x, [y, z]] + [x, [N(y), N(z)]] − N[x, [N(y), z]] − N [x, [y, N(z)]] + N[x, N[y, z]]. We can observe, by using Leibniz rule, that the right hand side of the first identity is equal to the sum of the right hand side of the third identity plus the right hand side of the second identity multiplied by .ε(y, z). u n Now, we introduce action of Leibniz color algebra on another one. Definition 5.13 Let L and .L be two Leibniz color algebras. A color action of .L on L consists of a pair of bilinear maps, .L × L → L, (x, a) |→ [x, a] and .L × L → L, (a, x) |→ [a, x], such that .

[[x, a] , b] = [x, [a, b]] + ε(a, b) [[x, b] , a] .

(5.19)

[[a, x] , b] = [a, [x, b]] + ε(x, b) [[a, b] , x] .

(5.20)

[[a, b] , x] = [a, [b, x]] + ε(b, x) [[a, x] , b] .

(5.21)

[[a, x] , y] = [a, [x, y]] + ε(x, y) [[a, y] , x] .

(5.22)

[[x, a] , y] = [x, [a, y]] + ε(a, y) [[x, y] , a] .

(5.23)

[[x, y] , a] = [x, [y, a]] + ε(y, a) [[x, a] , y]

(5.24)

for all .x, y ∈ L, a, b ∈ L. Remark 5.3 (i) Whenever, .L is just a color vector space (i.e. has not the structure of Leibniz color algebra), axioms (5.19)–(5.21) disappear, and axioms (5.22)–(5.24) mean that .L is a bimodule over L. (ii) Any Leibniz color algebra or any Leibniz superalgebras [49] is a bimodule over itself. Proposition 5.8 Let L and .L be two Leibniz color algebras. Given a Leibniz color action of .L on .L, we can consider the semidirect sum Leibniz color algebra .L x L, which consists of color vector space .L ⊕ L [(x, a), (y, b)] = ([a, b] + [x, b] + [a, y], [x, y]),

.

(5.25)

for all .(x, a), (y, b) ∈ H(L × L). Proof It uses axioms in Definition 5.13.

u n

Corollary 5.1 Let M be a color bimodule over a Leibniz color algebra L. Then M ⊕ L is a Leibniz color algebra with the multiplication

.

[(x, a), (y, b)] = ([x, b] + [a, y], [x, y]),

.

for all .(x, a), (y, b) ∈ H(L × L).

(5.26)

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5.3.3 Associative Color Trialgebras We now introduce color trialgebras and establish their relationship with ternary Leibniz color algebras. Definition 5.14 An associative color trialgebra is a G-graded vector space A equipped with a bicharacter .ε : G ⊗ G → K∗ and three even binary associative operations .|, T, |: A ⊗ A → A (called left, middle and right respectively), satisfying the following relations: .

(x | y) | z = x | (y | z) = x | (y T z).

(5.27)

(x | y) | z = x | (y | z).

(5.28)

(x | y) | z = x | (y | z) = (x T y) | z.

(5.29)

(x T y) | z = x T (y | z).

(5.30)

(x | y) T z = x T (y | z).

(5.31)

(x | y) T z = x | (y T z)

(5.32)

for all .x, y, z ∈ H(A). Example 5.8 Any associative color algebra .(A, ·, ε) is a color trialgebra with · =|=T=|.

.

Example 5.9 Any associative color dialgebra is a color trialgebra with trivial middle product. Example 5.10 If .(A, |, T, |, ε) is a color trialgebra, then so is .(A, |, , T, , |, , ε), where x |, y := y | x,

.

x T, y := y T x,

x |, y := y | x.

Now, let us recall the definition of Leibniz-Poisson color algebras. Definition 5.15 A Leibniz-Poisson color algebra is a G-graded vector space P together with two even bilinear maps .[−, −] : P ⊗ P → P and .· : P ⊗ P → P and a bicharacter .ε : G ⊗ G → K∗ such that (1) .(P , ·, ε) is an associative color algebra, (2) .(P , [−, −], ε) is a Leibniz color algebra, (3) and the following right Leibniz rule: [x · y, z] = x · [y, z] + ε(y, z)[x, z] · y

.

holds, for all .x, y, z ∈ H(P ).

(5.33)

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Remark 5.4 (1) When the color associative product .· is .ε-commutative i.e. .x · y = ε(x, y)y · x, then .(P , ·, [−, −], ε) is said to be a commutative Leibniz-Poisson color algebra. (2) A non-commutative Leibniz-Poisson color algebra in which the associative product is non-commutative and the bracket .[−, −] is .ε-skew symmetric, is called a non-commutative Poisson color algebra [16]. (3) Whenever the color associative product .· is .ε-commutative and the bracket .[−, −] is .ε-skew-symmetric, then .(P , ·, [−, −], ε) is named a commutative Poisson color algebra. Example 5.11 ([16]) Let .(D, |, |, ε) be an associative color dialgebra. Then (D, |, [−, −], ε) is a non-commutative Leibniz-Poisson color algebra with the bracket

.

[x, y] := x | y − ε(x, y)y | x,

.

for all .x, y ∈ H(D). The following proposition connects color trialgebras to Leibniz-Poisson color algebras. It will gives a construction of ternary Leibniz-Poisson color algebras form Leibniz-Poisson color algebras. Proposition 5.9 Let .(A, |, T, |, ε) be a color trialgebra. Then .(A, ·, [−, −], ε) is a Leibniz-Poisson color algebra with respect to the operations x · y := x T y

.

[x, y] = x | y − ε(x, y)y | x,

for all .x, y, z ∈ H(A). Proof From the definition of associative color trialgebra, the multiplication “.·” is associative. Let us show that the bracket .[−, −] endow A with a structure of Leibniz color algebra. For this, let us write down all the twelve terms involved in the HomLeibniz identity: [[x, y], z] = (x | y − ε(x, y)y | x) | z − ε(x, y + z)z | (x | y − ε(x, y)y | x)

.

[[x, z], y] = (x | z − ε(x, z)z | x) | y − ε(x + z, y)y | (x | z − ε(x, z)z | x) [x, [y, z]] = x | (y | z − ε(y, z)z | y) − ε(x, y + z)(y | z − ε(y, z)z | y) | x Using some axioms in Definition 5.14, it is immediate to see that (5.7) holds. Now, let us prove the right Leibniz identity [x T y, z] − x T [y, z] − ε(y, z)[x, z] T y = (x T y) | z − ε(x + y, z)z | (x T y)

.

− x T (y | z − ε(y, z)z | y) − ε(y, z)(x | z − ε(x, z)z | x) T y

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= (x T y) | z − ε(x + y, z)z | (x T y) − x T (y | z) + ε(y, z)x T (z | y) − ε(y, z)(x | z) T y + ε(x + y, z)(z | x) T y. The right hand side vanishes by axioms (5.30)–(5.32).

u n

5.4 Ternary Color Algebras This section is devoted to the construction of some structures of ternary color algebras.

5.4.1 Ternary Leibniz Color Algebras Definition 5.16 A ternary color algebra is a G-graded vector space A together with an even ternary bracket [−, −, −] : A × A × A → A (i.e. [x, y, z] ⊆ Ax+y+z whenever x, y, z ∈ H(A)) and a bicharacter ε : G × G → K∗ . Definition 5.17 A ternary color algebra A is said to be a ternary Leibniz color algebra if the bracket satisfies the following identity: [[x, y, z], t, u] = [x, y, [z, t, u]] + ε(z, t + u)[x, [y, t, u], z]

.

+ ε(y + z, t + u)[[x, t, u], y, z]

(5.34)

for any x, y, z, t, u ∈ H(A). Whenever the bracket [−, −, −] is ε-skew-symmetric for any pair of variables, then (A, [−, −, −], ε) is said to be a ternary Lie color algebra or 3-Lie color algebra [54]. Example 5.12 Let L be a ternary Leibniz color algebra and put L, = K[t, t −1 ]⊗L; L, can be considered as a vector space over Laurent polynomials with coefficient in the ternary Leibniz color algebra L. Considering element of K[t, t −1 ] as degree 0, L and L, has the same graduation. Taking, for any f (t), g(t), h(t) ∈ K[t, t −1 ] and x, y, z ∈ H(L), [f (t) ⊗ x, g(t) ⊗ y, h(t) ⊗ z], = f (t)g(t)h(t) ⊗ [x, y, z],

.

we endow L, with a structure of ternary color algebra

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Proposition 5.10 Let (L, [−, −, −], ε) be a ternary Leibniz color algebra, ξ ∈ A0 such that [ξ, x, ξ ] = 0, for any x ∈ L. Then, L is a Leibniz color algebra with the bracket {x, y} = [x, y, ξ ],

.

for any x, y ∈ H(L). Proof One has, for any x, y, z ∈ H(L), {{x, y}, z} = [[x, y, ξ ], z, ξ ]

.

= [x, y, [ξ, z, ξ ]] + ε(ξ, z + ξ )[x, [y, z, ξ ], ξ ] + ε(ξ + y, z + ξ )[[x, z, ξ ], y, ξ ]. By assumption, {{x, y}, z} = [[x, y, ξ ], z, ξ ] = [x, [y, z, ξ ], ξ ] + ε(y, z)[[x, z, ξ ], y, ξ ]

.

= [x, {y, z}, ξ ] + ε(y, z)[{x, z}, y, ξ ] = {x, {y, z}} + ε(y, z){{x, z}, y}. u n

This ends the proof.

5.4.1.1

Constructions

It is proved in [21, Proposition 3.2] that any Leibniz algebra is also a Leibniz nalgebra. We prove the analog for color case and .n = 3. That is one can get ternary Leibniz color algebras from Leibniz color algebras. Theorem 5.1 Let .(L, [−, −], ε) be a Leibniz color algebra. Then L is a ternary Leibniz color algebra with respect to the bracket {x, y, z} := [x, [y, z]],

.

for any .x, y, z ∈ H(L). Proof Applying twice relation (5.7), for any .x, y, z ∈ H(L), we have {{x, y, z}, t, u} = {[x, [y, z]], t, u} = [[x, [y, z]], [t, u]]

.

= [x, [[y, z], [t, u]]] + ε(y + z, t + u)[[x, [t, u]]], [y, z]] = [x, [y, [z, [t, u]]]] + ε(z, t + u)[x, [[y, [t, u]], z]] + ε(y + z, t + u)[[x, [t, u]]], [y, z]] = {x, y, [z, [t, u]]} + ε(z, t + u)[x, [{y, t, u}, z]] + ε(y + z, t + u)[{x, t, u}, [y, z]]

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= {x, y, {z, t, u}} + ε(z, t + u){x, {y, t, u}, z} + ε(y + z, t + u){{x, t, u}, y, z}. This completes the proof.

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The next assertion connects color trialgebras to ternary Leibniz color algebras. Proposition 5.11 Let .(A, |, T, |, ε) be a color trialgebra. Then A is a ternary Leibniz color algebra with respect to the bracket [x, y, z] := x | (y T z − ε(y, z)z T y) − ε(x, y + z)(y T z − ε(y, z)z T y) | x,

.

for all .x, y, z ∈ H(A). Proof For all .x, y, z, t, u ∈ H(A), one has [[x, y, z], t, u] = [x, y, z] | (t T u − ε(t, u)u T t)

.

− ε(x + y + z, t + u)(t T u − ε(t, u)u T t) | [x, y, z] ( = x | (y T z − ε(y, z)z T y) − ε(x, y + z)(y T z ) − ε(y, z)z T y) | x | (t T u − ε(t, u)u T t) ( − ε(x + y + z, t + u)(t T u − ε(t, u)u T t) | x | (y T z ) − ε(y, z)z T y) − ε(x, y + z)(y T z − ε(y, z)z T y) | x = (x | (y T z)) | (t T u) − ε(t, u)(x | (y T z)) | (u T t) − ε(y, z)(x | (z T y)) | (t T u) + ε(y, z)ε(t, u)(x | (z T y)) | (u T t) − ε(x, y + z)((y T z) | x) | (t T u) + ε(x, y + z)ε(t, u)((y T z) | x) | (u T t) + ε(x, y + z)ε(y, z)((z T y) | x) | (t T u) − ε(x, y + z)ε(y, z)ε(t, u)((z T y) | x) | (u T t) − ε(x + y + z, t + u)(t T u) | (x | (y T z)) − ε(x + y + z, t + u)ε(y, z)(t T u) | (x | (z T y)) + ε(x + y + z, t + u)ε(x, y + z)(t T u) | ((y T z) | x) − ε(x + y + z, t + u)ε(x, y + z)ε(y, z)(t T u) | ((z T y) | x) + ε(x + y + z, t + u)ε(t, u)(u T t) | (x | (y T z)) − ε(x + y + z, t + u)ε(t, u)ε(y, z)(u T t) | (x | (z T y))

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− ε(x + y + z, t + u)ε(t, u)ε(x, y + z)(u T t) | ((y T z) | x) + ε(x + y + z, t + u)ε(t, u)ε(x, y + z)ε(y, z)(u T t) | ((z T y) | x). By exchanging the role of y and t, z and u, we get [[x, t, u], y, z] = [x, t, u] | (y T z − ε(y, z)z T y)

.

− ε(x + t + u, y + z)(y T z − ε(y, z)z T y) | [x, t, u] ( = x | (t T u − ε(t, u)u T t) − ε(x, t + u)(t T u ) − ε(t, u)u T t) | x | (y T z − ε(y, z)z T y) ( − ε(x + t + u, y + z)(y T z − ε(y, z)z T y) | x | (t T u ) − ε(t, u)u T t) − ε(x, t + u)(t T u − ε(t, u)u T t) | x = (x | (t T u)) | (y T z) − ε(y, z)(x | (t T u)) | (z T y) − ε(t, u)(x | (u T t)) | (y T z) + ε(t, u)ε(y, z)(x | (u T t)) | (z T y) − ε(x, t + u)((t T u) | x) | (y T z) + ε(x, t + u)ε(y, z)((t T u) | x) | (z T y) + ε(x, t + u)ε(t, u)((u T t) | x) | (y T z) − ε(x, t + u)ε(t, u)ε(y, z)((u T t) | x) | (z T y) − ε(x + t + u, y + z)(y T z) | (x | (t T u)) − ε(x + t + u, y + z)ε(t, u)(y T z) | (x | (u T t)) + ε(x + t + u, y + z)ε(x, t + u)(y T z) | ((t T u) | x) − ε(x + t + u, y + z)ε(x, t + u)ε(t, u)(y T z) | ((u T t) | x) + ε(x + t + u, y + z)ε(y, z)(z T y) | (x | (t T u)) − ε(x + t + u, y + z)ε(y, z)ε(t, u)(z T y) | (x | (u T t)) − ε(x + t + u, y + z)ε(y, z)ε(x, t + u)(z T y) | ((t T u) | x) + ε(x + t + u, y + z)ε(y, z)ε(x, t + u)ε(t, u)(z T y) | ((u T t) | x).

Next,

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[x, y, [z, t, u]] = x | (y T [z, t, u] − ε(y, z + t + u)[z, t, u] T y)

.

− ε(x, y + z + t + u)(y T [z, t, u] − ε(y, z + t + u)[z, t, u] T y) |x

[ ( = x | y T z | (t T u − ε(t, u)u T t) − ε(z, t + u)(t T u ) ( − ε(t, u)u T t) | z − ε(y, z + t + u) z | (t T u − ε(t, u)u T t) ) ] − ε(z, t + u)(t T u − ε(t, u)u T t) | z T y [ ( − ε(x, y + z + t + u) y T z | (t T u − ε(t, u)u T t) ) − ε(z, t + u)(t T u − ε(t, u)u T t) | z ( − ε(y, z + t + u) z | (t T u − ε(t, u)u T t) ) ] − ε(z, t + u)(t T u − ε(t, u)u T t) | z T y | x = x | (y T (z | (t T u))) − ε(t, u)x | (y T (z | (u T t))) − ε(z, t + u)x | ((y T (t T u)) | z) + ε(z, t + u)ε(t, u)x | (y T ((u T t)) T y) − ε(y, z + t + u)x | ((z | (t T u)) T y) + ε(y, z + t + u)ε(t, u)x | ((z | (u T t)) T y) + ε(y, z + t + u)ε(z, t + u)x | (((t T u) | z) T y) − ε(y, z + t + u)ε(z, t + u)ε(t, u)x | (((u T t) T z) T y) − ε(x, y + z + t + u)(y T (z | (t T u))) | x + ε(x, y + z + t + u)ε(t, u)(y T (z | (u T t))) | x + ε(x, y + z + t + u)ε(z, t + u)((y T (t T u)) | z) | x − ε(x, y + z + t + u)ε(z, t + u)ε(t, u)((y T (u T t)) | z) | x − ε(x, y + z + t + u)ε(y, z + t + u)((z | (t T u)) T y) | x − ε(x, y + z + t + u)ε(y, z + t + u)ε(t, u)((z | (u T t)) T y) | x − ε(x, y + z + t + u)ε(y, z + t + u)ε(z, t + u)(((t T u) | z) T y) |x + ε(x, y + z + t + u)ε(y, z + t + u)ε(z, t + u)ε(t, u) (((u T t) | z) T y) | x.

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Lastly, [x, [y, t, u], z] = x | ([y, t, u] T z − ε(y + t + u, z)z T [y, t, u])

.

− ε(x, y + z + t + u)([y, t, u] T z − ε(y + t + u, z)z T [y, t, u]) | x [( = x | y | ((t T u − ε(t, u)u T t) − ε(y, t + u)(t T u ) ( − ε(t, u)u T t) | y T z − ε(y + t + u, z)z T y | ) (t T u − ε(t, u)u T t) − ε(y, t + u)(t T u − ε(t, u)u T t) | y T ] [( z ε(x, y + z + t + u) y | (t T u − ε(t, u)u T t) ) − ε(y, t + u)(t T u − ε(t, u)u T t) | y T z − ε(y + t + u, z)z ( T y | (t T u − ε(t, u)u T t) )] − ε(y, t + u)(t T u − ε(t, u)u T t) | y | x = x | ((y | (t T u)) T z − ε(t, u)x | ((y | (u T t)) T z) − ε(y, t + u)x | (((t T u) | y) T z) + ε(y, t + u)ε(t, u)x | (((u T t) | y) T z) − ε(y + t + u, z)x | (z T (y | (t T u))) + ε(y + t + u, z)ε(t, u)x | (z T (y | (u T t))) + ε(y + t + u, z)ε(y, t + u)x | (z T ((t T u) | y)) − ε(y + t + u, z)ε(y, t + u)ε(t, u)x | (z T ((u T t) | y)) − ε(x, y + z + t + u)((y | (t T u)) T z) | x + ε(x, y + z + t + u)ε(t, u)((y | (u T t)) T z) | x + ε(x, y + z + t + u)ε(y, t + u)(((t T u) | y) T z) | x − ε(x, y + z + t + u)ε(y, t + u)ε(t, u)(((u T t) | y) T y) | x − ε(y + t + u, z)(z T (y | (t T u))) | x + ε(y + t + u, z)ε(t, u)(z T (y | (u T t))) | x + ε(y + t + u, z)ε(y, t + u)(z T ((t T u) | y)) | x − ε(y + t + u, z)ε(y, t + u)ε(t, u)(z T ((u T t) | y)) | x.

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By uses of axioms in Definition 5.14, we can observe that relation (5.34) holds.

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Corollary 5.2 Let .(A, ·, ε) be a non-commutative associative color algebra. Then (A, [−, −, −], ε) is a ternary Leibniz color algebra, where

.

[x, y, z] := x · (y · z − ε(y, z)z · y) − ε(x, y + z)(y · z − ε(y, z)z · y) · x.

.

Remark 5.5 Observe that this ternary Leibniz color algebra is nothing but the one constructed from the commutator of the associative product. Proposition 5.12 Let .(A, [−, −, −], ε) and .(A, , [−, −, −], , ε) be two ternary Leibniz color algebras. Then .A ⊕ A, is also a ternary Leibniz color algebra with respect to the operations: {x ⊕ x , , y ⊕ y , , z ⊕ z, } := [x, y, z] ⊕ [x , , y , , z, ], ,

.

for any .x + x , , y + y , , z + z, ∈ H(A ⊕ A, ). Proof We have to verify axiom (5.34) for all .x + x , , y + y , , z + z, , t + t , , u + u, ∈ H(A ⊕ A, ), {{x + x , , y + y , , z + z, }, t + t , , u + u, } = [[x, y, z], t, u] + [[x , , y , , z, ], t, u],

.

= [x, y, [z, t, u]] + ε(z, t + u)[x, [y, t, u], z] + ε(y + z, t + u)[[x, t, u], y, z] + [x , , y , , [z, , t , , u, ]] + ε(z, , t , + u, )[x , , [y , , t , , u, ], z, ] + ε(y , + z, , t , + u, )[[x , , t , , u, ], y , , z, ] = ([x, y, [z, t, u]] + [x , , y , , [z, , t , , u, ]]) + ε(z, t + u)([x, [y, t, u], z] + [x , , [y , , t , , u, ]) + ε(y + z, t + u)([[x, t, u], y, z] + [[x , , t , , u, ], y , , z, ]) = {x + x , , y + y , , {z + z, , t + t , , u + u, }} + ε(z, t + u){x + x , , {y + y , , t + t , , u + u, }, z + z, } + ε(y + z, t + u){{x + x , , t + t , , u + u, }, y + y , , z + z, }. This achieves the proof.

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Corollary 5.3 Let = 1, 2, . . . , n be ternary color algebras. Then .A = On i is also a ternary Leibniz color algebra. A i=1 i .A , i

Proposition 5.13 Let .(A, [−, −, −], ε) be a ternary Leibniz color algebra. Then A ⊗ A has a Leibniz color algebra structure for the structure maps defined by

.

{x ⊗ y, x , ⊗ y , } := x ⊗ [y, x , , y , ] + ε(y, x , + y , )[x, x , , y , ] ⊗ y.

.

Proof For all .x, y, z ∈ H(L), we have successively,

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{{x ⊗ y, x , ⊗ y , }, x ,, ⊗ y ,, } = {x ⊗ [y, x , , y , ] + ε(y, x , + y , )[x, x , , y , ]

.

⊗ y, x ,, ⊗ y ,, } = {x ⊗ [y, x , , y , ], x ,, ⊗ y ,, } + ε(y, x , + y , ){[x, x , , y , ] ⊗ y, x ,, ⊗ y ,, } = x ⊗ [[y, x , , y , ], x ,, , y ,, ] + ε(y + x , + y , , x ,, + y ,, )[x, x ,, , z,, ] ⊗ [y, x , , y , ] + ε(y, x , + y , )[x, x , , y , ] ⊗ [y, x ,, , y ,, ] + ε(y, x , + y , )ε(y, x ,, + y ,, )[[x, x , , y , ], x ,, , y ,, ] ⊗ y. {x ⊗ y, {x , ⊗ y , , x ,, ⊗ y ,, }} = {x ⊗ y, x , ⊗ [y , , x ,, , y ,, ]

.

+ ε(y , , x ,, + y ,, )[x , , x ,, , y ,, ] ⊗ y , } = x ⊗ [y, x , , [y , , x ,, , y ,, ]] + ε(y, x , + x ,, + y , + y ,, )[x, x , , [y , , x ,, , y ,, ]] ⊗ y + ε(y , , x ,, + y ,, )x ⊗ [y, [x , , x ,, , y ,, ], y , ] + ε(y , , x ,, + y ,, )ε(y, x , + x ,, + y , + y ,, )[x, [x , , x ,, , y ,, ], y , ] ⊗ y. {{x ⊗ y, x ,, ⊗ y ,, }, x , ⊗ y , } = x ⊗ [[y, x ,, , y ,, ], x , , y , ]

.

+ ε(y + x ,, + y ,, , x , + y , )[x, x , , y , ] ⊗ [y, x ,, , y ,, ] + ε(y, x ,, + y ,, )[x, x ,, , y ,, ] ⊗ [y, x , , y , ] + ε(y, x ,, + y ,, )ε(y, x , + y , )[[x, x ,, , y ,, ], x , , y , ] ⊗ y.

It is an easy fact, to see that {{x ⊗ y, x , ⊗ y , }, x ,, ⊗ y ,, } = {x ⊗ y, {x , ⊗ y , , x ,, ⊗ y ,, }}

.

+ ε(x , + y , , x ,, + y ,, ){{x ⊗ y, x ,, ⊗ y ,, }, x , ⊗ y , }. Which ends the proof.

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Corollary 5.4 Let .(A, [−, −]) be a Leibniz color algebra. Then .A ⊗ A is also a Leibniz color algebra with respect to the operation {x ⊗ y, x , ⊗ y , } := x ⊗ [y, [x , , y , ]] + ε(y, x , + y , )[x, [x , , y , ]] ⊗ y.

.

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Proof It follows from Proposition 5.13 and Theorem 5.1.

Remark 5.6 The above bracket fails to be .ε-skew.symmetric; thus it can not define a Lie color algebra bracket. Definition 5.18 An even linear map .θ : L → L is said to be an element of centroid of L if, for any .x, y, z ∈ H(L), θ [x, y, z] = [θ (x), y, z] = [x, θ (y), z] = [x, y, θ (z)].

.

(5.35)

Example 5.13 Any homothety is an element of centroid of L. In deed, putting θ (x) = λx, x ∈ L, λ ∈ K,

.

we have θ [x, y, z] = λ[x, y, z] = [λx, y, z] = [θ (x), y, z]

.

= [x, λy, z] = [x, θ (y), z] = [x, y, λz] = [x, y, θ (z)]. Proposition 5.14 Let L be a ternary Leibniz color algebra and .θ : L → L an element of centroid of L. Then L becomes a ternary Leibniz color algebra in each of the following cases: (a) .{x, y, z} = [θ (x), y, z], (b) .{x, y, z} = [θ (x), θ (y), z], (c) .{x, y, z} = [θ (x), θ (y), θ (z)], for any .x, y, z ∈ H(L). Proof After applying axiom (5.34), at each step we use one of the equality of relation (5.35). One has: (a) For any .x, y, z, t, u ∈ H(L), {{x, y, z}, t, u} = [θ [θ (x), y, z], t, u]

.

= [[θ (x), y, z], θ (t), u] = [θ (x), y, [z, θ (t), u]] + ε(z, t + u)[θ (x), [y, θ (t), u], z] + ε(y + z, t + u)[[θ (x), θ (t), u], y, z] = [θ (x), y, [θ (z), t, u]] + ε(z, t + u)[θ (x), [θ (y), t, u], z] + ε(y + z, t + u)[θ [θ (x), t, u], y, z] = {x, y, {z, t, u}} + ε(z, t + u){x, {y, t, u}, z} + ε(y + z, t + u){{x, t, u}, y, z}. (b) For any .x, y, z, t, u ∈ H(L),

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{{x, y, z}, t, u} = [θ [θ (x), θ (y), z], θ (t), u]

.

= [[θ (x), θ (y), θ (z)], θ (t), u] = [θ (x), θ (y), [θ (z), θ (t), u]] + ε(z, t + u)[θ (x), [θ (y), θ (t), u], θ (z)] + ε(y + z, t + u)[[θ (x), θ (t), u], θ (y), θ (z)] = [θ (x), θ (y), [θ (z), θ (t), u]] + ε(z, t + u)θ [θ (x), [θ (y), θ (t), u], z] + ε(y + z, t + u)[θ [θ (x), θ (t), u], θ (y), z] = {x, y, {z, t, u}} + ε(z, t + u){x, {y, t, u}, z} + ε(y + z, t + u){{x, t, u}, y, z}. (c) For any .x, y, z, t, u ∈ H(L), {{x, y, z}, t, u} = [θ [θ (x), θ (y), θ (z)], θ (t), θ (u)]

.

= [[θ 2 (x), θ (y), θ (z)], θ (t), θ (u)] = [θ 2 (x), θ (y), [θ (z), θ (t), θ (u)]] + ε(z, t + u)[θ 2 (x), [θ (y), θ (t), θ (u)], θ (z)] + ε(y + z, t + u)[[θ 2 (x), θ (t), θ (u)], θ (y), θ (z)] = [θ 2 (x), θ (y), [θ (z), θ (t), θ (u)]] + ε(z, t + u)[θ 2 (x), [θ (y), θ (t), θ (u)], θ (z)] + ε(y + z, t + u)[[θ 2 (x), θ (t), θ (u)], θ (y), θ (z)] = [θ (x), θ (y), θ [θ (z), θ (t), θ (u)]] + ε(z, t + u)[θ (x), θ [θ (y), θ (t), θ (u)], θ (z)] + ε(y + z, t + u)[θ [θ (x), θ (t), θ (u)], θ (y), θ (z)] = {x, y, {z, t, u}} + ε(z, t + u){x, {y, t, u}, z} + ε(y + z, t + u){{x, t, u}, y, z}. This ends the proof.

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Remark 5.7 Observe that the bracket .(c) is nothing but the initial one whenever the map .θ is surjective.

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Definition 5.19 Given a ternary Leibniz color L, an even linear map .P : L → L is said to be a Reynolds operator on L if one has ( [P (x), P (y), P (z)] = P [P (x), P (y), z]+[P (x), y, P (z)]+[x, P (y), P (z)] ) − [P (x), P (y), P (z)] , (5.36)

.

for any .x, y, z ∈ H(L). Theorem 5.2 Let .P : L → L be a Reynolds operator on a ternary Leibniz color algebra L. Then, L is also a ternary Leibniz color for the product {x, y, z} = [P (x), P (y), z] + [P (x), y, P (z)] + [x, P (y), P (z)]

.

− [P (x), P (y), P (z)],

(5.37)

for any .x, y, z ∈ H(L). Proof We have successively, for all .x, y, z, t, u ∈ H(L), {{x, y, z}, t, u} = [P {x, y, z}, P (t), u] + [P {x, y, z}, t, P (u)]

.

+ [{x, y, z}, P (t), P (u)] − [P {x, y, z}, P (t), P (u)] = [[P (x), P (y), P (z)], P (t), u] + [[P (x), P (y), P (z)], t, P (u)] + [{x, y, z}, P (t), P (u)] − [P {x, y, z}, P (t), P (u)] + [[x, P (y), P (z)], P (t), P (u)] − 2[[P (x), P (y), P (z)], P (t), P (u)], {x, y, {z, t, u}} = [P (x), P (y), {z, t, u}] + [P (x), y, P {z, t, u}]

.

+ [x, P (y), P {z, t, u}] − [P (x), P (y), P {z, t, u}] = [P (x), P (y), [P (z), P (t), u]] + [P (x), P (y), [P (z), t, P (u)]] + [P (x), P (y), [z, P (t), P (u)]] − [P (x), P (y), [P (z), P (t), P (u)]] + [P (x), y, [P (z), P (t), P (u)]] + [x, P (y), [P (z), P (t), P (u)]] − [P (x), P (y), [P (z), P (t), P (u)]],

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{x, {y, t, u}, z} = [P (x), P {y, t, u}, z] + [P (x), {y, t, u}, P (z)]

.

+ [x, P {y, t, u}, P (z)] − [P (x), P {y, t, u}, P (z)] = [P (x), [P (y), P (t), P (u)], P (z)] + [P (x), [P (y), P (t), u], P (z)] + [P (x), [P (y), t, P (u)], P (z)] + [P (x), [y, P (t), P (u)], P (z)] − [P (x), [P (y), P (t), P (u)], P (z)] + [x, [P (y), P (t), P (u)], P (z)] − [P (x), [P (y), P (t), P (u)], P (z)], and {{x, t, u}, y, z} = [P {x, t, u}, P (y), z] + [P {x, t, u}, y, P (z)]

.

+ [{x, t, u}, P (y), P (z)] − [P {x, t, u}, P (y), P (z)] = [[P (x), P (t), P (u)], P (y), z] + [[P (x), P (t), P (u)], y, P (z)] + [{x, t, u}, P (y), P (z)] − [P {x, t, u}, P (y), P (z)] + [[x, P (t), P (u)], P (y), P (z)] − 2[[P (x), P (t), P (u)], P (y), P (z)]. It is easy to observe (term by term) that {{x, y, z}, t, u} = {x, y, {[z, t, u}}

.

+ ε(z, t + u){x, {y, t, u]}, z} + ε(y + z, t + u){{x, t, u}], y, z} u n Definition 5.20 An even linear map .R : L → L on a ternary Leibniz color algebra is called a Rota-Baxter operator of weight .λ ∈ K, if for any .x, y, z ∈ H(L), ( [R(x), R(y), R(z)] = R [R(x), R(y), z] + [R(x), y, R(z)] + [x, R(y), R(z)]

.

) + λ[R(x), y, z] + λ[x, R(y), z] + λ[x, y, R(z)] + λ2 [x, y, z] . Theorem 5.3 Given a Rota-Baxter operator .R : L → L on a ternary Leibniz color algebra L, we can make L into another ternary Leibniz color algebra with the bracket {x, y, z} = [R(x), R(y), z] + [R(x), y, R(z)] + [x, R(y), R(z)]

.

5 Ternary Leibniz Color Algebras and Beyond

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+ λ[R(x), y, z]+λ[x, R(y), z]+λ[x, y, R(z)]+λ2 [x, y, z], (5.38) for any .x, y, z ∈ H(L). Proof For all .x, y, z, t, u ∈ H(L), we have {{x, y, z}, t, u} = [R{x, y, z}, R(t), u] + [R{x, y, z}, t, R(u)]

.

+ [{x, y, z}, R(t), R(u)] + λ[R{x, y, z}, t, u] + λ[{x, y, z}, R(t), u] + λ[{x, y, z}, t, R(u)] + λ2 [{x, y, z}, t, u] = [[R(x), R(y), R(z)], R(t), u] + [[R(x), R(y), R(z)], t, R(u)] + [[R(x), R(y), z], R(t), R(u)] + [[R(x), y, R(z)], R(t), R(u)] + [[x, R(y), R(z)], R(t), R(u)] + λ[[R(x), y, z], R(t), R(u)] + λ[[x, y, R(z)], R(t), R(u)] + λ2 [[x, y, z], R(t), R(u)] + λ[[R(x), R(y), R(z)], t, u] + λ[[R(x), R(y), z], R(t), u]. (5.39) + λ[[R(x), y, R(z)], R(t), u] + λ[[x, R(y), R(z)], R(t), u] + λ2 [[R(x), y, z], R(t), u] + λ2 [[x, R(y), z], R(t), u] + λ2 [[x, y, R(z)], R(t), u] + λ3 [[x, y, z], R(t), u] + λ[[R(x), R(y), z], t, R(u)] + λ[[R(x), y, R(z)], t, R(u)] + λ[[x, R(y), R(z)], t, R(u)] + λ2 [[R(x), y, z], t, R(u)] + λ2 [[x, R(y), z], t, R(u)] + λ2 [[x, y, R(z)], t, R(u)] + λ3 [[x, y, z], t, R(u)] + λ2 [[R(x), R(y), z], t, u] + λ2 [[R(x), y, R(z)], t, u] + λ2 [[x, R(y), R(z)], t, u] + λ3 [[R(x), y, z], t, u] + λ3 [[x, R(y), z], t, u] + λ3 [[x, y, R(z)], t, u] + λ4 [[x, y, z], t, u].

(5.40)

{x, y, {z, t, u}} = [R(x), R(y), {z, t, u}] + [R(x), y, R{z, t, u}]

.

+ [x, R(y), R{z, t, u}] + λ[R(x), y, {z, t, u}] + λ[x, R(y), {z, t, u}] + λ[x, y, R{z, t, u}] + λ2 [x, y, {z, t, u}] = [R(x), R(y), [R(z), R(t), u]] + [R(x), R(y), [R(z), t, R(u)]]

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+ [R(x), R(y), [z, R(t), R(u)]] + λ[R(x), R(y), [R(z), t, u]] + λ[R(x), R(y), [z, R(t), u]] + λ[R(x), R(y), [z, t, R(u)]] + λ2 [R(x), R(y), [z, t, u]] + [R(x), y, [R(z), R(t), R(u)]] + [x, R(y), [R(z), R(t), R(u)]] + λ[R(x), y, [R(z), R(t), u]] + λ[R(x), y, [R(z), t, R(u)]] + λ[R(x), y, [z, R(t), R(u)]] + λ2 [R(x), y, [R(z), t, u]] + λ2 [R(x), y, [z, R(t), u]] + λ2 [R(x), y, [z, t, R(u)]] + λ3 [R(x), y, [z, t, u]] + λ[x, R(y), [R(z), R(t), u]] + λ[x, R(y), [R(z), t, R(u)]] + λ[x, R(y), [z, R(t), R(u)]] + λ2 [x, R(y), [R(z), t, u]] + λ2 [x, R(y), [z, R(t), u]] + λ2 [x, R(y), [z, t, R(u)]] + λ3 [x, R(y), [z, t, u]] + λ[x, y, [R(z), R(t), R(u)]] + λ2 [x, y, [R(z), R(t), u]] + λ2 [x, y, [R(z), t, R(u)]] + λ2 [x, y, [z, R(t), R(u)]] + λ3 [x, y, [R(z), t, u]] + λ3 [x, y, [z, R(t), u]] + λ3 [x, y, [z, t, R(u)]] + λ4 [x, y, [z, t, u]]

(5.41)

{x, {y, t, u}, z} = [R(x), R{y, t, u}, z] + [R(x), {y, t, u}, R(z)]

.

+ [x, R{y, t, u}, R(z)] + λ[R(x), {y, t, u}, z] + λ[x, R{y, t, u}, z] + λ[x, {y, t, u}, R(z)] + λ2 [x, {y, t, u}, z] = [R(x), [R(y), R(t), R(u)], z] + [R(x), [R(y), R(t), u], R(z)] + [R(x), [R(y), t, R(u)], R(z)] + [R(x), [y, R(t), R(u)], R(z)] + λ[R(x), [R(y), t, u], R(z)] + λ[R(x), [y, R(t), u], R(z)] + λ[R(x), [y, t, R(u)], R(z)] + λ2 [R(x), [y, t, u], R(z)] + [x, [R(y), R(t), R(u)], R(z)] + λ[R(x), [R(y), R(t), u], z] + λ[R(x), [R(y), t, R(u)], z] + λ[R(x), [y, R(t), R(u)], z] + λ2 [R(x), [R(y), t, u], z] + λ2 [R(x), [y, R(t), u], z] + λ2 [R(x), [y, t, R(u)], z] + λ3 [R(x), [y, t, u], z]

5 Ternary Leibniz Color Algebras and Beyond

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+ λ[x, [R(y), R(t), R(u)], z] + λ[x, [R(y), R(t), u], R(z)] + λ[x, [R(y), t, R(u)], R(z)] + λ[x, [y, R(t), R(u)], R(z)] + λ2 [x, [R(y), t, u], R(z)] + λ2 [x, [y, R(t), u], R(z)] + λ2 [x, [y, t, R(u)], R(z)] + λ3 [x, [y, t, u], R(z)] + λ2 [x, [R(y), R(t), u], z] + λ2 [x, [R(y), t, R(u)], z] + λ2 [x, [y, R(t), R(u)], z] + λ3 [x, [R(y), t, u], z] + λ3 [x, [y, R(t), u], z] + λ3 [x, [y, t, R(u)], z] + λ4 [x, [y, t, u], z]

(5.42)

{{x, t, u}, y, , z} = [R{x, t, u}, R(y), z] + [R{x, t, u}, y, R(z)]

.

+ [{x, t, u}, R(y), R(z)] + λ[R{x, t, u}, y, z] + λ[{x, t, u}, R(y), z] + λ[{x, t, u}, y, R(z)] + λ2 [{x, t, u}, y, z] = [[R(x), R(t), R(u)], R(y), z] + [[R(x), R(t), R(u)], y, R(z)] + [[R(x), R(t), u], R(y), R(z)] + [[R(x), t, R(u)], R(y), R(z)] + [[x, R(t), R(u)], R(y), R(z)] + λ[[R(x), t, u], R(y), R(z)] + λ[[x, R(t), u], R(y), R(z)] + λ[[x, t, R(u)], R(y), R(z)] + λ2 [[x, t, u], R(y), R(z)] + λ[[R(x)], R(t), R(u)], y, z] + λ[[R(x), R(t), u], R(y), z] + λ[[R(x), t, R(u)], R(y), z] + λ[[x, R(t), R(u)], R(y), z] + λ2 [[R(x), t, u], R(u), z] + λ2 [[x, R(t), u], R(y), z] + λ2 [[x, t, R(u)], R(y), z] + λ3 [[x, t, u], R(y), z] + λ[[R(x), R(t), u], y, R(z)] + λ[[R(x), t, R(u)], y, R(z)] + λ[[x, R(t), R(u)], y, R(z)] + λ2 [[R(x), t, u], y, R(z)] + λ2 [[x, R(t), u], y, R(z)] + λ2 [[x, t, R(u)], y, R(z)] + λ3 [[x, t, u], y, R(z)] + λ2 [[R(x), R(t), u], y, z] + λ2 [[R(x), t, R(u)], y, z] + λ2 [[x, R(t), R(u)], y, z] + λ3 [[R(x), t, u], y, z] + λ3 [[x, R(t), u], y, z] + λ3 [[x, t, R(u)], y, z] + λ4 [[x, t, u], y, z]

(5.43)

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Observing the number of times the linear map R appears (or not) in each bracket for given variable(s), the degree of .λ for each term in which .λ occur and the using the identity (5.34), we conclude that the given bracket is well a ternary Leibniz bracket. u n The below theorem asserts that the tensor product of any commutative associative color algebra with any ternary Leibniz color algebra is also a ternary color algebra. Theorem 5.4 Let A be a commutative associative color algebra and L a ternary Leibniz color algebra. Then, for any .a, b, c ∈ H(A), .x, y, z ∈ H(L), the bracket {a ⊗ x, b ⊗ y, c ⊗ z} = ε(x, b + c)ε(y, c)abc ⊗ [x, y, z],

.

(5.44)

make .A ⊗ L into a ternary Leibniz color algebra. Proof The proof of this theorem use essentially, definitions of associative color algebra and ternary Leibniz color algebra, and the .ε-commutativity of associative product. In fact, we have, on the one hand, for any .a, b, c, d, e ∈ H(A), .x, y, z, t, u ∈ H(L), {{a ⊗ x, b ⊗ y, c ⊗ z}, d ⊗ t, e ⊗ u}

.

= ε(x, b + c)ε(y, c){abc ⊗ [x, y, z], d ⊗ t, e ⊗ u} = ε(x, b + c)ε(y, c)ε(x + y + z, d + e)ε(t, e)abcde ⊗ [[x, y, z], t, u] ( = ε(x, b + c)ε(y, c)ε(x + y + z, d + e)ε(t, e)abcde ⊗ [x, y, [z, t, u]] ) + ε(z, t + u)[x, [y, t, u], z] + ε(y + z, t + u)[[x, t, u], y, z] = ε(x, b + c)ε(y, c)ε(x + y + z, d + e)ε(t, e)abcde ⊗ [x, y, [z, t, u]] + ε(x, b + c)ε(y, c)ε(x + y + z, d + e)ε(t, e)ε(z, t + u)abcde ⊗ [x, [y, t, u], z] + ε(x, b + c)ε(y, c)ε(x + y + z, d + e)ε(t, e)ε(y + z, t + u) abcde ⊗ [[x, t, u], y, z].

(5.45)

On the other hand, we have successively, {{a ⊗ x, b ⊗ y, {c ⊗ z, d ⊗ t, e ⊗ u}}

.

= ε(z, d + e)ε(t, e){a ⊗ x, b ⊗ y, cde ⊗ [z, t, u]} = ε(z, d + e)ε(t, e)ε(x, b + c + d + e)ε(y, c + d + e)abcde ⊗ [x, y, [z, t, u]]. (5.46)

ε(c + z, d + t + e + u){a ⊗ x, {b ⊗ y, c ⊗ z, d ⊗ t}, e ⊗ u}

.

= ε(c + z, d + t + e + u)ε(y, d + e)ε(t, e)[a ⊗ x, bde ⊗ [y, t, u], c ⊗ z]

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= ε(c + z, d + t + e + u)ε(y, d + e)ε(t, e)ε(x, b + d + e + c) ε(y + t + u, c)abdec ⊗ [x, [y, t, u], z] = ε(c + z, d + t + e + u)ε(y, d + e)ε(t, e)ε(x, b + d + e + c)ε(y + t + u, c) ε(d + e, c)abcde ⊗ [x, [y, t, u], z]

(5.47)

ε(b + y + c + z, d + t + e + u){{a ⊗ x, d ⊗ t, e ⊗ u}, b ⊗ y, c ⊗ z, }

.

= ε(b + y + c + z, d + t + e + u)ε(x, d + e)ε(t, e)ε(x + t + u, b + c) ε(y, c)adebc ⊗ [[x, y, z], t, u] = ε(b + y + c + z, d + t + e + u)ε(x, d + e)ε(t, e)ε(x + t + u, b + c)ε(y, c) ε(d + e, b + c)abcde ⊗ [[x, y, z], t, u]. By comparing (5.45) to the sum of (5.46)–(5.48), we get the conclusion.

5.4.1.2

(5.48) u n

Bimodules

Now, we introduce bimodule and representation of ternary Leibniz color algebras and establish some properties. To simplify the typography in what follows, we omit the subscript and note most of the multiplications by the same maps. Definition 5.21 A color bimodule M over a ternary Leibniz color algebra (A, ·, [−, −, −], ε) is the given of three even trilinear applications

.

[−, −, −] : M ⊗ L ⊗ L → M,

.

[−, −, −] : L ⊗ M ⊗ L → M,

.

[−, −, −] : L ⊗ L ⊗ M → M

.

satisfying the following sets of axioms [[m, x, y], z, t]] =[m, x, [y, z, t]] + ε(y, z + t)[m, [x, z, t], y]

.

+ ε(x + y, z + t)[[m, z, t], x, y], . [[x, m, y], z, t]] = [x, m, [y, z, t]] + ε(y, z + t)[x, [m, z, t], y] + ε(m + y, z + t)[[x, z, t], m, y],

(5.49)

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[[x, y, m], z, t]] = [x, y, [m, z, t]] + ε(m, z + t)[x, [y, z, t], m] + ε(y + m, z + t)[[x, z, t], y, m], ‘ [[x, y, z], m, t]] = [x, y, [z, m, t]] + ε(z, m + t)[x, [y, m, t], z] + ε(y + z, m + t)[[x, m, t], y, z], [[x, y, z], t, m]] = [x, y, [z, t, m]] + ε(z, t + m)[x, [y, t, m], z] + ε(y + z, t + m)[[x, t, m], y, z],

(5.50)

for all .x, y, z, t ∈ H(L), m ∈ H(M). Theorem 5.5 Let M be a color bimodule over a ternary Leibniz color algebra L. Then .L ⊕ M is a ternary Leibniz color algebra with respect to the multiplication {x1 + m1 , x2 + m2 , x3 + m3 } = [x1 , x2 , x3 ] + [m1 , x2 , x3 ] + [x1 , m2 , x3 ]

.

+ [x1 , x2 , m3 ],

(5.51)

for any .xi + mi ∈ H(L ⊕ M), i = 1, 2, 3. Proof For any .xi + mi ∈ H(L ⊕ M), i = 1, 2, ..., 5, one has {{x1 + m1 , x2 + m2 , x3 + m3 }, x4 + m4 , x5 + m5 }

.

= {[x1 , x2 , x3 ] + [m1 , x2 , x3 ] + [x1 , m2 , x3 ] + [x1 , x2 , m3 ], x4 + m4 , x5 + m5 } = [[x1 , x2 , x3 ], x4 , x5 ] + [[m1 , x2 , x3 ] + [x1 , m2 , x3 ] + [x1 , x2 , m3 ], x4 , x5 ] + [[x1 , x2 , x3 ], m4 , x5 ] + [[x1 , x2 , x3 ], x4 , m5 ] = [[x1 , x2 , x3 ], x4 , x5 ] + [[m1 , x2 , x3 ], x4 , x5 ] + [[x1 , m2 , x3 ], x4 , x5 ] + [[x1 , x2 , m3 ], x4 , x5 ] + [[x1 , x2 , x3 ], m4 , x5 ] + [[x1 , x2 , x3 ], x4 , m5 ]. By axioms (5.34) and (5.49) and (5.50), we get {{x1 + m1 , x2 + m2 , x3 + m3 }, x4 + m4 , x5 + m5 } = [x1 , x2 , [x3 , x4 , x5 ]]

.

+ ε(x3 , x4 + x5 )[x1 , [x2 , x4 , x5 ], x3 ] + ε(x2 + x3 , x4 + x5 )[[x1 , x4 , x5 ], x2 , x3 ] + [m1 , x2 , [x3 , x4 , x5 ]] + ε(x3 , x4 + x5 )[m1 , [x2 , x4 , x5 ], x3 ] + ε(x2 + x3 , x4 + x5 )[[m1 , x4 , x5 ], x2 , x3 ] + [x1 , m2 , [x3 , x4 , x5 ]] + ε(x3 , x4 + x5 )[x1 , [m2 , x4 , x5 ], x3 ] + ε(m2 + x3 , x4 + x5 )[[x1 , x4 , x5 ], m2 , x3 ] + [x1 , x2 , [m3 , x4 , x5 ]] + ε(m3 , x4 + x5 )[x1 , [x2 , x4 , x5 ], m3 ] + ε(x2 + m3 , x4 + x5 )[[x1 , x4 , x5 ], x2 , m3 ] + [x1 , x2 , [x3 , m4 , x5 ]] + ε(x3 , m4 + x5 )[x1 , [x2 , m4 , x5 ], x3 ] + ε(x2 + x3 , m4 + x5 )[[x1 , m4 , x5 ], x2 , x3 ]

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+ [x1 , x2 , [x3 , x4 , m5 ]] + ε(x3 , x4 + m5 )[x1 , [x2 , x4 , m5 ], x3 ] + ε(x2 + x3 , x4 + m5 )[[x1 , x4 , m5 ], x2 , x3 ]. In the above equality, we have a set of six sums; taking the sum of each term of same rank in this set we get, by remembering that .degree(xi ) = degree(mi ), i = 1, ..., 5, {{x1 + m1 , x2 + m2 , x3 + m3 }, x4 + m4 , x5 + m5 }

.

= {x1 + m1 , x2 + m2 , {x3 + m3 , x4 + m4 , x5 + m5 }} + ε(x1 , x4 + x5 ){{x1 + m1 , {x2 + m2 , x3 + m3 }, x4 + m4 }, x5 + m5 } + ε(x1 + x2 , x4 + x5 ){x1 + m1 , x2 + m2 , {x3 + m3 , x4 + m4 , x5 + m5 }}. This completes the proof.

u n

The proof of the following proposition is easy. Proposition 5.15 Let M and N be two color bimodule over the ternary Leibniz color L. Then, .M⊕N is also a color bimodule over L with respect to componentwise operation. In the below proposition, we construct bimodules on ternary Leibniz color algebras from bimodules over Leibniz color algebras. Proposition 5.16 Let M be a bimodule over a Leibniz color algebra L. Let us define [x, y, m] := [x, [y, m]], [x, m, y] := [x, [m, y]], [m, x, y] := [m, [x, y]], (5.52)

.

for all .x, y ∈ H(L) and .m ∈ H(M). Then M is a bimodule over the ternary Leibniz color algebra associates to the Leibniz color algebra L (as in Theorem 5.1). Proof It is proved by a straightforward computation. So, for instance, we only prove axiom (5.50). {{x, y, z}, t, m} = {[x, [y, z]], t, m}

.

= [[x, [y, z]], [t, m]] = [x, [[y, z], [t, m]]] + ε(y + z, t + m)[[x, [t, m]]], [y, z]] = [x, [y, [z, [t, m]]]] + ε(z, t + m)[x, [[y, [t, m]], z]] + ε(y + z, t + m)[[x, [t, m]]], [y, z]] = {x, y, [z, [t, m]]} + ε(z, t + m)[x, [{y, t, u}, z]] + ε(y + z, t + m)[{x, t, m}, [y, z]]

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= {x, y, {z, t, m}} + ε(z, t + m){x, {y, t, u}, z} + ε(y + z, t + m){{x, t, m}, y, z}. u n

This ends the proof.

Theorem 5.6 Let M and N be two color bimodule over the ternary Leibniz color L. Then, .M ⊗ N is also a color bimodule over L with respect to the structure maps: .

{x, y, m ⊗ n} = [x, y, m] ⊗ n + ε(x + y, m)m ⊗ [x, y, n], .

(5.53)

{x, m ⊗ n, y} = ε(n, y)[x, m, y] ⊗ n + ε(x, m)m ⊗ [x, n, y], .

(5.54)

{m ⊗ n, x, y} = ε(n, x + y)[m, x, y] ⊗ n + m ⊗ m ⊗ [n, x, y],

(5.55)

for .x, y ∈ H(L), m ∈ H(M), n ∈ H(N ). Proof We have to prove axioms (5.49) and (5.50) for the tensor product, but we only show axiom (5.49), the others being proved in the same way. Thus, for any .x, y, z, t ∈ H(L), m ∈ H(M) and .n ∈ H(N ), we have: {[x, y, z], t, m ⊗ n} = [[x, y, z], t, m] ⊗ n + ε(x + y + z + t, m)m⊗

.

[[x, y, z], t, n] = [x, y, [z, t, m]] ⊗ n + ε(z, t + m)[x, [y, t, m], z] ⊗ n + ε(y + z, t + m)[[x, t, m], y, z] ⊗ n + ε(x + y + z + t, m)m ⊗ [x, y, [z, t, n]] + ε(x + y + z + t, m)ε(z, t + n)m ⊗ [x, [y, t, n], z] + ε(x + y + z + t, m)ε(y + z, t + n)m ⊗ [[x, t, n], y, z] = [x, y, [z, t, m]] ⊗ n + ε(z, t + m + n)ε(n, z)[x, [y, t, m], z] ⊗ n + ε(y + z, t + m + n)ε(n, y + z)[[x, t, m], y, z] ⊗ n + ε(x + y + z + t, m)m ⊗ [x, y, [z, t, n]] + ε(x + y + t, m)ε(z, t + m + n)m ⊗ [x, [y, t, n], z] + ε(y + z, t + m + n)ε(x + t, m)[[x, t, m], y, z] ⊗ n = {x, y, {z, t, m ⊗ n}} + ε(z, t + m + n){x, {y, t, m ⊗ n}, z} + ε(y + z, t + m + n){{x, t, m ⊗ n}, y, z}. This finishes the proof.

u n

We end this part by the study of representation of ternary Leibniz color algebras. Definition 5.22 Let L be a ternary Leibniz color algebra and M a color vector space. A representation of L on M is the given of three even linear maps .λ : L ×

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L → End(M), (x, y) |→ λx,y , .μ : L ⊗ L → End(M), (x, y) |→ μx,y and ρ : L ⊗ L → End(M), (x, y) |→ ρx,y such that:

.

λ[x,y,z],t (m) = λx,y λz,t (m) + ε(z, t + m)μx,z λy,t (m) + ε(y + z, t + m)ρy,z λx,t (m), . (5.56)

.

μ[x,y,z],t (m) = λx,y μz,t (m) + ε(z, m + t)λx,z μy,t (m) + ε(y + z, m + t)ρy,z μx,t (m), . (5.57) λz,t λx,y (m) = λx,y λz,t (m) + ε(m, z + t)λx,[y,z,t] (m) + ε(y + m, z + t)λ[x,z,t],y (m), . (5.58) ρz,t μx,y (m) = μx,[y,z,t] (m) + ε(y, z + t)μx,y ρz,t (m) + ε(y + m, z + t)μ[x,z,t],y (m), . (5.59) ρz,t ρx,y (m) = ρx,[y,z,t] (m) + ε(y, z + t)ρ[x,z,t],y (m) + ε(x + y, z + t)ρx,y ρz,t (m), (5.60) for any .x, y, z, t ∈ H(L), m ∈ H(M). It is well known that every bimodule M gives rise to a representation .(λ, μ, ρ) of L on M via .λx,y (m) = [x, y, m], .μx,y (m) = [x, m, y] and .ρx,y (m) = [m, x, y]. Conversely, every representation .(λ, μ, ρ) of L on M define an L bimodule structure on M via .[x, y, m] := λx,y (m), .[x, m, y] := μx,y (m) and .[m, x, y] := ρx,y (m). Therefore, we have the following proposition. Proposition 5.17 If M and O M are two representation spaces of a ternary Leibniz color algebra L, then .M N and .M ⊗ N are also representation spaces of L by mean of the following structure maps: (1) For .M ⊕ N, N Ax,y (m + n) := λM x,y (m) + λx,y (n).

(5.61)

N ox,y (m + n) := μM x,y (m) + μx,y (n).

(5.62)

M N wx,y (m + n) := ρx,y (m) + ρx,y (n)

(5.63)

.

for any .x, y ∈ H(L), m + n ∈ H(M ⊕ N). (2) For .M ⊗ N, N Ax,y (m ⊗ n) := λM x,y (m) ⊗ n + ε(x + y, m)m ⊗ λx,y (n).

(5.64)

N ox,y (m ⊗ n) := ε(n, y)μM x,y (m) ⊗ n + ε(x, m)m ⊗ μx,y (n).

(5.65)

M N wx,y (m + n) := ε(n, x + y)ρx,y (m) ⊗ n + m ⊗ ρx,y (n)

(5.66)

.

for any .x, y ∈ H(L), m ∈ H(M), n ∈ H(N ).

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5.4.2 Ternary Leibniz-Poisson Color Algebras We introduce ternary Leibniz-Poisson color algebras, their bimodules and give some procedures of constructions. Definition 5.23 A ternary Leibniz-Poisson color algebra is a quadruple (A, ·, [−, −, −], ε) in which

.

(1) .(A, ·, ε) is an associative color algebra, (2) .(A, [−, −, −], ε) is a ternary Leibniz color algebra, (3) and the following right ternary Leibniz rule [x · y, z, t] = x · [y, z, t] + ε(y, z + t)[x, z, t] · y

(5.67)

.

holds for any .x, y, z, t ∈ H(A). If in addition, the product .· is .ε-commutative, then the ternary Leibniz-Poisson color algebra is said to be commutative. Moreover, if the trilinear map .[−, −, −] is .ε-skew-symmetric for any pair of variables, then .(A, ·, [−, −, −], ε) is called a ternary Poisson color algebra, see [39] for trivial grading. Example 5.14 If .(A, ·, [−, −, −], ε) is a ternary Leibniz-Poisson color algebra, then .(A, ·op , [−, −, −], ε) is also a ternary Leibniz-Poisson color algebra, where op : A ⊗ A → A, x ⊗ y |→ ε(x, y)y · x. .·

5.4.2.1

Constructions

The following results affirm that one can get ternary Leibniz-Poisson color algebras from Leibniz-Poisson color algebras. Theorem 5.7 Let .(A, ·, [−, −], ε) be a Leibniz-Poisson color algebra. Then (A, ·, [−, [−, −]], ε) is a ternary Leibniz-Poisson color algebra.

.

Proof It is clear that .(A, ·, ε) is an associative color algebra and .(A, [−, [−, −]]) is a ternary Leibniz color algebra, thanks to Theorem 5.1. Now, we only need to prove the right ternary Leibniz rule (5.67). For any .x, y, z, t ∈ H(A), we have [x · y, z, t] = [x · y, [z, t]]

.

= x · [y, [z, t]] + ε(y, z + t)[x, [z, t]] · y

(by

(5.33))

= x · [y, z, t] + ε(y, z + t)[x, z, t] · y. This ends the proof.

u n

Corollary 5.5 Let .(A, ·, ε) be a non-commutative associative color algebra. Let us define the even bilinear and trilinear maps .[−, −] : A⊗2 → A and .[−, −, −] : A⊗3 → A respectively by

5 Ternary Leibniz Color Algebras and Beyond

[x, y] := x · y − ε(x, y)y · x

.

227

and

[x, y, z] := [x, [y, z]]

Then .(A, ·, [−, −, −], ε) is a ternary Leibniz-Poisson color algebra. The following proposition asserts that the direct sum of two ternary LeibnizPoisson color algebras is also a ternary Leibniz-Poisson color algebra with componentwise operation. Proposition 5.18 Let .(A, ·, [−, −, −], ε) and .(A, , ·, , [−, −, −], , ε) be two ternary Leibniz-Poisson color algebras. Then .A ⊕ A, is a ternary Leibniz-Poisson color algebra with respect to the operations: (x ⊕ x , ) ∗ (y ⊕ y , ) := (x · y) ⊕ (x , ·, y , )

.

{x ⊕ x , , y ⊕ y , , z ⊕ z, } := [x, y, z] ⊕ [x , , y , , z, ], , for any .x, y, z ∈ H(A) and .x , , y , , z, ∈ H(A, ). Proof We have already proved the ternary Leibniz color algebra structure in Proposition 5.12. The associative product and the right ternary Leibniz rule are u n showed in a similar way. The below theorem states that the tensor product of ternary Leibniz-Poisson color algebra by itself leads to Leibniz-Poisson color algebra. Theorem 5.8 Let .(A, ·, [−, −, −], ε) be a ternary Leibniz-Poisson color algebra. Then .A ⊗ A is endowed with a Leibniz-Poisson color algebra structure for the structure maps defined by (x ⊗ y) ∗ (x , ⊗ y , ) := ε(y, x , )(x · x , ) ⊗ (y · y , )

.

{x ⊗ y, x , ⊗ y , } := x ⊗ [y, x , , y , ] + ε(y, x , + y , )[x, x , , y , ] ⊗ y. Proof It is an easy fact to prove that the product “.∗” is associative. The Leibniz color algebra structure of the bracket .{−, −} is pointed out in Proposition 5.13. Thus, we only need to prove the right Leibniz rule. Then, according to the definition of the bracket .{−, −}, we have for any .x, x , , x ,, , y, y , , y ,, ∈ H(A), {(x ⊗ y) ∗ (x , ⊗ y , ), x ,, ⊗ y ,, } = ε(y, x , ){(x · x , ) ⊗ (y · y , ), x ,, ⊗ y ,, }

.

= ε(y, x , )xx , ⊗ [yy , , x ,, , y ,, ] + ε(y, x , )ε(y + y , , x ,, + y ,, )[xx , , x ,, , y ,, ] ⊗ yy , By relation (5.67), {(x ⊗ y) ∗ (x , ⊗ y , ), x ,, ⊗ y ,, } = ε(y, x , )xx , ⊗ y[y , , x ,, , y ,, ]

.

+ ε(y, x , )ε(y , , x ,, + y ,, )xx , ⊗ [y, x ,, , y ,, ]y , + ε(y, x , )ε(y + y , , x ,, + y ,, )x[x , , x ,, , y ,, ] ⊗ yy ,

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+ ε(y, x , )ε(y + y , , x ,, + y ,, )ε(x , , x ,, + y ,, )[x, x ,, , y ,, ]x , ⊗ yy , By rearranging, {(x ⊗ y) ∗ (x , ⊗ y , ), x ,, ⊗ y ,, } = ε(y, x , )xx , ⊗ y[y , , x ,, , y ,, ]

.

+ ε(y, x , + x ,, + y ,, )ε(y , , x ,, + y ,, )x[x , , x ,, , y ,, ] ⊗ yy , + ε(x , + y , , x ,, + y ,, )ε(y + x ,, + y ,, , x , )xx , ⊗ [y, x ,, , y ,, ]y , + ε(x , + y , , x ,, + y ,, )ε(y, x ,, + y ,, )ε(y, x , )[x, x ,, , y ,, ]x , ⊗ yy , = (x ⊗ y)(x , ⊗ [y , , x ,, , y ,, ] + ε(y , , x ,, + y ,, )[x , , x ,, , y ,, ] + ε(x , + y , , x ,, + y ,, )(x ⊗ [y, x ,, , y ,, ] + ε(y, x ,, + y ,, )[x, x ,, , y ,, ] ⊗ y)(x , ⊗ y , ) = (x ⊗ y) ∗ {x , ⊗ y , , x ,, ⊗ y ,, } + ε(x , + y , , x ,, + y ,, ){x ⊗ y, x ,, ⊗ y ,, } ∗ (x , ⊗ y , ). u n

This achieves the proof.

Corollary 5.6 Let .(A, ·, [−, −]) be a Leibniz-Poisson color algebra. Then .A ⊗ A is also a Leibniz-Poisson algebra with respect to the operations (x ⊗ y) · (x , ⊗ y , ) := ε(y, x , )(x · x , ) ⊗ (y · y , )

.

[x ⊗ y, x , ⊗ y , ] := x ⊗ [y, [x , , y , ]] + ε(y, x , + y , )[x, [x , , y , ]] ⊗ y. Proof It follows from Theorems 5.8 and 5.7.

u n

It is proved in [20, Lemma 2.8] that every Leibniz-Poisson algebra gives rise to ternary Leibniz algebra. In the below theorem, we extend this result to the case Leibniz-Poisson color algebra. Theorem 5.9 Let .(A, ·, [−, −], ε) be a Leibniz-Poisson color algebra, then (A, ·, {−, −, −}, ε) is a ternary Leibniz-Poisson color algebra, with

.

{x, y, z} := [x, y · z]

.

for any .x, y, z ∈ H(A). Proof We only need to prove (5.34) and right ternary Leibniz rule (5.67). Then, for any .x, y, z, t, u ∈ H(A), one has:

5 Ternary Leibniz Color Algebras and Beyond

229

{{x, y, z}, t, u} − {x, y, {z, t, u}} − ε(z, t + u){x, {y, t, u}, z}

.

− ε(y + z, t + u){{x, t, u}, y, z} = [[x, y · z], t · u]] − [x, y · [z, t · u]] − ε(z, t + u)[x, [y, t · u] · z] − ε(y + z, t + u)[[x, t · u], y · z] = [x, y · z], t · u] − [x, y · [z, t · u] − ε(z, t + u)[y, t · u] · z] − ε(y + z, t + u)[[x, t · u], y · z] = [[x, y · z], t · u] − [x, [y · z, t · u] − ε(y + z, t + u)[[x, t · u], y · z]. The last line vanishes thanks to the right Leibniz identity (5.7). Next, {x · y, z, t} − x · {y, z, t} − ε(y, z + t){x, z, t} · y = [x · y, z · t] − x · [y, z · t]

.

+ ε(y, z + t)[x, z · t] · y, u n

which also vanishes by (5.7).

5.4.2.2

Bimodules

In what follows we introduce and give constructions of bimodules over ternary Leibniz-Poisson color algebras. Definition 5.24 A bimodule over a ternary Leibniz-Poisson color algebra (A, ·, [−, −, −], ε) is a bimodule M over the associative color algebra .(A, ·, ε) and a bimodule over the ternary Leibniz color algebra L such that

.

.

[m · x, y, z] = m · [x, y, m] + ε(x, y + z)[m, y, z] · x, .

(5.68)

[x · m, y, z] = x · [m, y, z] + ε(m, y + z)[x, y, z] · m, .

(5.69)

[x · y, m, z] = x · [y, m, z] + ε(y, m + z)[x, m, z] · y, .

(5.70)

[x · y, z, m] = x · [y, z, m] + ε(y, z + m)[x, z, m] · y,

(5.71)

for any .x, y, z, t ∈ H(A) and .m ∈ H(M). In order to have bimodules over ternary Leibniz-Poisson color algebras via morphism, we need the below definition. Definition 5.25 Let .(L, ·, [−, −, −], ε) and .(L, , ·, , [−, −, −], , ε) be two ternary Leibniz-Poisson color algebras. Let .α : L → L, be an even linear map such that, for any .x, y, z ∈ H(L), α(x · y) = α(x) · α(y)

.

and

α([x, y, z]) = [α(x), α(y), α(z)], .

Then .α is called a morphism of ternary Leibniz-Poisson color algebras.

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Then we have the following result. Theorem 5.10 Let .(L, ·, [−, −, −], ε) and .(L, , ·, , [−, −, −], , ε) be two ternary Leibniz-Poisson color algebras and .α : L → L, be a morphism of ternary LeibnizPoisson color algebras. Let us define x ∗ m =α(x) ·, m,

.

m ∗ x = m ·, α(x),

{x, m, y} =[α(x), m, α(y)],

and

{m, x, y} = [m, α(x), α(y)], , . (5.72)

{x, y, m} = [α(x), α(y), m], ,

(5.73)

for any .x, y ∈ H(L) and .m ∈ H(L, ). Then, with these five maps, .L, is a bimodule over L. Proof For all .x, y, z ∈ H(L), we have {[x, y, z], t, m} = [α([x, y, z]), α(t), m], = [[α(x), α(y), α(z)], , α(t), m],

.

= [α(x), α(y), [α(z), α(t), m], ], + ε(z, t + m)[α(x), [α(y), α(t), m], , α(z)], + ε(y + z, t + m)[[α(x), α(t), m], , α(y), α(z)], = {x, y, {z, t, m}} + ε(z, t + m){x, {y, t, m}, z} + ε(y + z, t + m){{x, t, m}, y, z}. The rest of the relations are proved similarly.

u n

Remark 5.8 Any ternary Leibniz-Poisson color algebra is a bimodule over itself. Corollary 5.7 Let .(L, ·, [−, −, −], ε) be a ternary Leibniz-Poisson color algebra and .α : L → L be an endomorphism of L. Then (5.72) and (5.73) define another bimodule structure of L over itself.

5.4.3 Color Lie Triple Systems We start by recalling the definition of Lie triple systems. Definition 5.26 A (right) Lie triple system consists of a vector space L endowed with a trilinear product .[−, −, −] : L × L × L → L satisfying the identities (a) .[x, y, z] = −[x, z, y], (b) .[x, y, z] + [y, z, x] + [z, x, y] = 0, (c) .[[x, y, z], t, u] = [x, y, [z, t, u]] + [x, [y, t, u], z] + [[x, t, u], y, z], for any .x, y, z, t, u ∈ L. Definition 5.27 A color Lie triple system is a ternary color algebra .(L, [−, −, −], ε) satisfying: (i) The ternary color identity (5.34)

5 Ternary Leibniz Color Algebras and Beyond

231

(ii) The .right ε-skew-symmetry [x, y, z] = −ε(y, z)[x, z, y],

.

(5.74)

(iii) and the .ε-Jacobi identity ε(z, x)[x, y, z] + ε(x, y)[y, z, x] + ε(y, z)[z, x, y] = 0,

.

(5.75)

for each .x, y, z ∈ H(L). Theorem 5.11 Let .(A, [−, −], ε) be a Lie color algebra. Define the even trilinear map .[−, −, −] : A ⊗ A ⊗ A → A by [x, y, z] := [x, [y, z]]

.

for any .x, y, z ∈ H(A). Then .(A, [−, −, −], ε) is a color Lie triple system. Proof The right skew-symmetry (5.74) follows from the .ε-skew-symmetry of the Lie color algebra. The ternary .ε-Jacobi identity (5.75) follow from .ε-Jacobi identity of the Lie color algebra.Finally, Theorem 5.1 completes the proof. u n It is well known that an associative color algebra turns to a Lie color algebra. This leads to the following consequence: Corollary 5.8 Let .(A, ·, ε) be an associative color algebra. Then A is a color Lie triple system with respect to the bracket [x, y, z] := x·(y·z)−ε(y, z)x·(z·y)−ε(x, y+z)(y·z)·x+ε(x, y+z)ε(y, z)(z·y)·x.

.

Next we introduce color Jordan triple systems and study their connection with color Lie triple systems. Definition 5.28 A color Jordan triple system is a ternary color algebra (J, [−, −, −], ε) satisfying the outer-.ε-symmetry

.

[x, y, z] = ε(x, y)ε(x, z)ε(y, z)[z, y, x]

.

(5.76)

and the color Jordan triple identity [[x, y, z], t, u] = [x, y, [z, t, u]] − ε(z, t + u)ε(t, u)[x, [y, t, u], z]

.

+ ε(y + z, t + u)[[x, t, u], y, z],

(5.77)

for any .x, y, z ∈ H(J ). Example 5.15 Let .(A, ·, ε) be an associative color algebra. Then .(A, [−, −, −], ε) is a color Jordan triple system with respect to the triple product

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[x, y, z] := x · y · z + ε(x, y)ε(x, z)ε(y, z)z · y · x.

.

Example 5.16 Let .(A, ·, ε) be an associative color algebra and .θ : A → A be an even linear map on A satisfying .θ 2 = I dA and .θ (x · y) = ε(x, y)θ (y) · θ (x) for any .x, y ∈ H(A). Then .(A, [−, −, −], ε) is a color Jordan triple system with the triple product [x, y, z] := x · θ (y) · z + ε(x, y)ε(x, z)ε(y, z)z · θ (y) · x.

.

We have the following result. Theorem 5.12 Let .(J, [−, −, −], ε) be a color Jordan triple system. Define the triple product {x, y, z} := [x, y, z] − ε(y, z)[x, z, y]

.

for any .x, y, z ∈ H(J ). Then .L(J ) = (J, {−, −, −}, ε) is a color Lie triple system. Proof The right .ε-skew-symmetry is immediate. The ternary .ε-Jacobi identity follows from (5.76). It remains to check the identity (5.34) for .{−, −, −}. For any .x, y, z ∈ H(J ), we have {{x, y, z}, t, u} = [{x, y, z}, t, u] − ε(t, u)[{x, y, z}, u, t]

.

= [([x, y, z] − ε(y, z)[x, z, y]), t, u] − ε(t, u)[([x, y, z] − ε(y, z)[x, z, y]), u, t] = [[x, y, z], t, u] − ε(y, z)[[x, z, y], t, u] − ε(t, u)[[x, y, z], u, t] + ε(t, u)ε(y, z)[[x, z, y], u, t]. Similarly, {x, y, {z, t, u}} = [x, y, [z, t, u]] − ε(t, u)[x, y, [z, u, t]]

.

−ε(y, z + t + u)[x, [z, t, u], y] +ε(y, z + t + u)ε(t, u), {x, {y, t, u}, z} = [x, [y, t, u], z] − ε(t, u)[x, [y, u, t], z] −ε(y + t + u, z)[x, z, [y, t, u]] +ε(y + t + u, z)ε(t, u)[x, z, [y, u, t]], {{x, t, u}, t, u} = [[x, t, u], y, z] − ε(t, u)[[x, u, t], y, z] −ε(y, z)[[x, t, u], z, y] +ε(y, z)ε(t, u)[[x, u, t], z, y]. Then, using axiom (5.77), relation (5.34) holds for the bracket .{−, −, −}.

u n

5 Ternary Leibniz Color Algebras and Beyond

233

5.4.4 Comstrans Color Algebras Definition 5.29 A Comstrans color algebra is a vector space A equipped with two ternaries maps [−, −, −], < −, −, − >: A × A × A → A, respectively called the commutator and the translator, and satisfying: (a) < x, y, x >= [x, y, x], (b) [x, y, z] = −[x, z, y], (c) < x, y, z > + < y, z, x > + < z, x, y >= 0. Definition 5.30 A Comstrans color algebra is a G-graded vector space A with a bicharacter ε : G × G → K∗ , two trilinear operations A × A × A → A, the commutator (x, y, z) |→ [x, y, z] and the translator (x, y, z) |→< x, y, z > satisfying the following identities for all x, y, z ∈ H(T ), < x, y, x >= [x, y, x],

(5.78)

[x, y, z] = −ε(y, z)[x, z, y],

(5.79)

.

.

ε(z, x) < x, y, z > +ε(x, y) < y, z, x > +ε(y, z) < z, x, y >= 0.

.

(5.80)

Theorem 5.13 Let (A, ·, ε) be a non-commutative associative color. Then, A carries a structure of Comstrans color algebra with the multiplications [x, y, z] = x · y · z − ε(y, z)x · z · y, .

(5.81)

< x, y, z > = x · y · z − ε(x + y, z)z · x · y,

(5.82)

.

for any x, y, z ∈ H(A). Proof For all x, y, z ∈ H(A), we have (i) Firstly, [x, y, x] = xyx − ε(y, x)xxy = xyx − ε(x + y, x)xxy =< x, y, x > .

.

(ii) Secondly, [x, y, z] = xyz − ε(y, z)xzy = −ε(y, z)(xzy − ε(z, y)xyz)

.

= −ε(y, z)[x, z, y]. (iii) Finally, ε(z, x) < x, y, z > +ε(x, y) < y, z, x > +ε(y, z) < z, x, y >

.

= ε(z, x)xyz − ε(z, x)ε(x + y, z)zxy + ε(x, y)yzx

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− ε(y + z, x)xyz + ε(y, z)zxy − ε(y, z)ε(z + x, y)yzx. It is easy to see, by associativity, that the right hand side vanishes.

u n

Proposition 5.19 Any Lie color algebra (L, [−, −], ε) has an underlying Comstrans color algebra defined by [x, y, z] =< x, y, z >= [x, [y, z]]

.

(5.83)

for any x, y, z ∈ H(L). Proof The first two axioms are immediate. The third comes from the ε-Jacobi identity of the Lie color algebra L. u n Acknowledgments The first author would like to thank Professor Alain Togbé of Purdue University for his material support.

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Chapter 6

(Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary (Hom-)bialgebras Mahouton Norbert Hounkonnou

and Gbevewou Damien Houndedji

6.1 Introduction An n-ary algebra is a linear space endowed with an internal composition law involving n elements, .μ : V ⊗n → V . These algebras, .n ≥ 3, knew a versatile development since the discovery of the Nambu mechanics in 1973 (see [50]) and the work by S. Okubo [51] on Yang-Baxter equation. The n-ary products were also defined by cubic matrices and a generalization of the notion of determinant, called hyperdeterminant, first introduced by Cayley in 1840, then rediscovered and generalized by Sokolov in 1972 [54], and, still later, by Kapranov, Gelfand and Zelevinskii in 1994 [30]. The ternary algebraic structures are particularly of a great potential application in various domains of physics and mathematics, and data processing [9–11, 25, 36, 37, 48, 52]. Their subclass, known as Bagger-Lambert algebras [7], is involved in string theory and M-branes. A good compilation of their applications can be found in the work by Kerner (see [3, 32–35]) on ternary and non-associative structures and their applications in physics. This author investigated the use of .Z3 -graded structures

M. N. Hounkonnou (O) International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin Centre International de Recherches et d’Etudes Avancées en Sciences Mathématiques & Informatiques et Applications (CIREASMIA), Cotonou, Benin e-mail: [email protected] G. D. Houndedji International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_6

237

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instead of .Z2 -graded in physics, leading to interesting results in the construction of gauge theories. The n-ary algebras of associative type were studied by Lister, Loos, Myung and Carlsson (see [12, 13, 41, 43, 49]). Relatively to their structure, these algebras encompass two main classes: totally associative n-ary algebras and partially associative n-ary algebras, which also generate other interesting variants of algebras. Definition 1 A totally associative ternary algebra is a .K-vector space .T endowed with a trilinear operation .μ satisfying, for all .x1 , x2 , x3 , x4 , x5 ∈ A : μ(μ(x1 ⊗ x2 ⊗ x3 ) ⊗ x4 ⊗ x5 ) .

= μ(x1 ⊗ μ(x2 ⊗ x3 ⊗ x4 ) ⊗ x5 ) = μ(x1 ⊗ x2 ⊗ μ(x3 ⊗ x4 ⊗ x5 )).

(6.1)

Example 1 Let .T be a 2-dimensional space vector with a basis .{e1 , e2 }. The trilinear product .μ on .T defined by .

μ(e1 ⊗ e1 ⊗ e1 ) = e1 μ(e1 ⊗ e1 ⊗ e2 ) = e2 μ(e1 ⊗ e2 ⊗ e1 ) = e2 μ(e2 ⊗ e1 ⊗ e1 ) = e2

μ(e2 ⊗ e2 ⊗ e1 ) = e1 + e2 μ(e2 ⊗ e2 ⊗ e2 ) = e1 + 2e2 μ(e1 ⊗ e2 ⊗ e2 ) = e1 + e2 μ(e2 ⊗ e1 ⊗ e2 ) = e1 + e2

defines a totally associative ternary algebra. Definition 2 A weak totally associative ternary algebra is a .K-vector space .W equipped with a trilinear operation .μ satisfying, for all .x1 , x2 , x3 , x4 , x5 ∈ W : μ(μ(x1 ⊗ x2 ⊗ x3 ) ⊗ x4 ⊗ x5 ) = μ(x1 ⊗ x2 ⊗ μ(x3 ⊗ x4 ⊗ x5 )).

.

(6.2)

Remark 1 Naturally, any totally associative ternary algebra is a weak totally associative ternary algebra. Definition 3 A partially associative ternary algebra is a .K-vector space .P with a trilinear operation .μ satisfying, for all .x1 , x2 , x3 , x4 , x5 ∈ P : μ(μ(x1 ⊗ x2 ⊗ x3 ) ⊗ x4 ⊗ x5 ) + μ(x1 ⊗ μ(x2 ⊗ x3 ⊗ x4 ) ⊗ x5 )+ .

μ(x1 ⊗ x2 ⊗ μ(x3 ⊗ x4 ⊗ x5 )) = 0.

(6.3)

Note that the ternary algebras given by subspaces of an associative algebra, closed under the ternary product .(x, y, z) |→ xyz, are linked to the ternary operation defined by Hestenes [27] on a linear space of rectangular matrices .A, B, C ∈ Mm,n , with complex entries by .AB ∗ C, where .B ∗ is the conjugate transpose matrix of .B. This operation, strictly speaking, does not define a ternary algebra product on .Mm,n as it is linear on the first and the third arguments, but conjugate-linear on the second argument. It satisfies identities, sometimes referred to as identities of total associativity of second kind, which only slightly differ from the identities of totally associative algebras.

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

239

The totally associative ternary algebras are also sometimes called associative triple systems. The cohomology of totally associative n-ary algebras was studied by Carlsson through the embedding [13]. In [4], the 1-parameter formal deformation theory was extended to ternary algebras of associative type, while in [5] discussions were made on their cohomologies in the context of deformations. See also [24, 28], and [53] (and references therein). The extension of the notion of associativity to n-ary product is not trivial. The most natural procedure might be based on the notion of totally associativity. Unfortunately, this notion is not auto-dual in the operadic point of view. So, it is necessary to introduce a most general notion of associativity under the concept of partial associativity. In this paper, we focus on the case .n = 3. Of course, the structure of classes of associative n-ary algebras, (for .n ≥ 3), is more complicated than that of associative algebras. Hence, their exhaustive investigation in order to derive their relevant relationships is of some importance in algebra. Besides, the concepts of associative coalgebras and bialgebras are fundamental in the theory of associative algebras. In [1] and [2], Aguiar developed the basic theory of infinitesimal bialgebras and infinitesimal hopf bialgebras. An infinitesimal bialgebra is at the same time an algebra and a coalgebra, in such a way that the comultiplication is a derivation. Aguiar established many properties of ordinary Hopf algebras which possess infinitesimal version. He introduced bicrossproducts, quasitriangular infinitesimal bialgebras, the corresponding infinitesimal YangBaxter equation and a notion of Drinfeld’s double for infinitesimal Hopf algebras. He also showed that non degenerate antisymmetric solutions of associative YangBaxter equations are in one-to-one correspondence with non degenerate cyclic 2-cocycles. Furthermore, Baï established a clear analogy between the antisymmetric infinitesimal bialgebras and the dendriform D-bialgebras [8]. Motivated by all these studies, we define, in this work, the concepts of totally and partially coassociative ternary coalgebras and totally and partially associative ternary infinitesimal bialgebras, and investigate their main properties as well as their relationships with associative ternary infinitesimal bialgebras. The hom-algebraic structures were originated from quasi-deformations of Lie algebras of vector fields. The latter gave rise to quasi-Lie algebras, defined as generalized Lie structures in which the skew-symmetry and Jacobi conditions are twisted. The first examples were concerned with q-deformations of the Witt and Virasoro algebras (see [14–19, 29, 31, 42]). Motivated by those works and examples that use the general quasi-deformation construction provided by Larsson and coworkers [26, 38, 39], and by the possibility of studying, within the same framework, such well-known generalizations of Lie algebras as the color and super Lie algebras, the quasi-Lie algebras, subclasses of quasi-hom-Lie algebras and hom-Lie algebras were introduced in [26, 38–40]. It is worth noticing here that, in the subclass of hom-Lie algebras, the skew-symmetry is untwisted, whereas the Jacobi identity is twisted by a single linear map and contains three terms as in Lie algebras, reducing to ordinary Lie algebras when the twisting linear map is the identity map.

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The hom-associative algebras are generalizations of the associative algebras, where the associativity law is twisted by a linear map. In [44], it was shown in particular that the commutator product, defined using the multiplication in a hom-associative algebra, naturally leads to a hom-Lie algebra. The hom-Lieadmissible algebras and more general G-hom- associative algebras with subclasses of hom-Vinberg and pre-hom-Lie algebras, generalizing twisted Lie-admissible algebras, G-associative algebras, Vinberg and pre-Lie algebras were also introduced. For these classes of algebras, the operation of taking commutator leads to hom-Lie algebras as well. The enveloping algebras of hom-Lie algebras were discussed in [55]. The fundamentals of the formal deformation theory and associated cohomology structures for hom-Lie algebras were recently considered in [46]. Simultaneously, Yau developed elements of homology for hom-Lie algebras in [56, 57]. In [45] and [47], Ataguema et al. elaborated the theory of hom-coalgebras and related structures. They introduced hom-coalgebra structure, leading to the notions of hom-bialgebras and hom-Hopf algebras, proved some fundamental properties and provided examples. They also defined the concepts of hom-Lie admissible homcoalgebra generalizing the admissible coalgebra introduced in [23], and performed their classification based on subgroups of the underlying symmetric group. The present work also aims at defining and discussing properties of partially and totally hom-coassociative ternary coalgebras. The infinitesimal bialgebraic structures of hom-associative ternary algebras and their relation with corresponding hom-associative ternary algebras are investigated. The dual structures of such categories of algebras are also studied. The paper is organized as follows. In Sect. 6.2, we construct partially and totally coassociative ternary coalgebras and discuss their main properties. Section 6.3 is devoted to the construction of trimodules and matched pairs of totally and partially associative ternary algebras. Then, in Sect. 6.4, we define the partially and totally associative ternary infinitesimal bialgebras and investigate their relation with associative ternary algebras. Besides, in Sect. 6.5, we construct partially and totally hom-coassociative ternary hom-coalgebras and discuss their main properties. Section 6.6 is devoted to the construction of trimodules and matched pairs of totally and partially hom-associative ternary algebras. Then, in Sect. 6.7, we define the partially and totally associative ternary infinitesimal bialgebras and investigate their relation with associative ternary algebras. Section 6.8 is devoted to concluding remarks.

6.2 Coassociative Ternary Coalgebras In this section, we introduce and develop the concepts of totally coassociative ternary coalgebra, weak totally coassociative ternary coalgebra and partially coassociative ternary coalgebra.

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

241

6.2.1 Definitions Let us start with the following definitions. Definition 4 A totally associative ternary algebra is a .F-vector space .T equipped with a trilinear operation .μ satisfying μ ◦ (μ ⊗ id ⊗ id) = μ ◦ (id ⊗μ ⊗ id) = μ ◦ (id ⊗ id ⊗μ).

.

(6.4)

Definition 5 A weak totally associative ternary algebra is a .F-vector space .W endowed with a trilinear operation .μ satisfying μ ◦ (μ ⊗ id ⊗ id) = μ ◦ (id ⊗ id ⊗μ).

.

(6.5)

Definition 6 A partially associative ternary algebra is a .F-vector space .P endowed with a trilinear operation .μ satisfying μ ◦ (μ ⊗ id ⊗ id + id ⊗μ ⊗ id + id ⊗ id ⊗μ) = 0.

.

(6.6)

Let .(A, μ) be an associative ternary algebra and .A∗ be its dual space. Then, we get the dual mapping .μ∗ : A∗ → A∗ ⊗ A∗ ⊗ A∗ of .μ, for every .x, y, z ∈ A and ∗ .ξ, η, γ ∈ A , = , .

.

= ,

(6.7) (6.8)

where . is the natural nondegenerate symmetric bilinear form on the vector space A ⊕ A∗ defined by . = ξ(x), ξ ∈ A∗ , x ∈ A. Further, by the definition of associative ternary algebras, we have ∗ ∗ ∗ ∗ .I m(μ ) ⊆ A ⊗ A ⊗ A , and

.

(μ∗ ⊗ id ⊗ id) ◦ μ∗ = (id ⊗μ∗ ⊗ id) ◦ μ∗ = (id ⊗ id ⊗μ∗ ) ◦ μ∗

(6.9)

(μ∗ ⊗ id ⊗ id + id ⊗μ∗ ⊗ id + id ⊗ id ⊗μ∗ ) ◦ μ∗ = 0,

(6.10)

.

and

.

for the totally associative ternary algebras and the partially associative ternary algebras, respectively. That is, for all .x1 , x2 , x3 , x4 , x5 ∈ A and .ξ ∈ A∗ , we have ∗

.

= = = =

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and .

= ,

respectively. Provided the above, the following definitions are in order. Definition 7 A totally coassociative ternary coalgebra .(T, A) is a vector space .T with a linear mapping .A : T → T ⊗ T ⊗ T satisfying (A ⊗ id ⊗ id) ◦ A = (id ⊗A ⊗ id) ◦ A = (id ⊗ id ⊗A) ◦ A.

(6.11)

.

Equation (6.11) is also called standard ternary coassociativity condition in [20]. Definition 8 A weak totally coassociative ternary coalgebra .(W, A) is a vector space .W with a linear mapping .A : W → W ⊗ W ⊗ W satisfying (A ⊗ id ⊗ id) ◦ A = (id ⊗ id ⊗A) ◦ A.

(6.12)

.

Definition 9 A partially coassociative ternary coalgebra .(P, A) is a vector space .P with a linear mapping .A : P → P ⊗ P ⊗ P satisfying (A ⊗ id ⊗ id + id ⊗A ⊗ id + id ⊗ id ⊗A) ◦ A = 0.

(6.13)

.

6.2.2 Main Results Let us now examine the partially coassociative ternary coalgebras in terms of structure constants. For that, let .(A, A) be a partially coassociative ternary coalgebra with a basis .e1 , . . . .., en . Assume that A(el ) =

E

.

l l crst er ⊗ es ⊗ et , crst ∈ F, 1 ≤ l ≤ n.

(6.14)

1≤r,s,t≤n

Then, we obtain (A ⊗ id ⊗ id + id ⊗A ⊗ id + id ⊗ id ⊗A) ◦ A(el ) ⎛ .

= (A ⊗ id ⊗ id + id ⊗A ⊗ id + id ⊗ id ⊗A) ⎝

E

1≤r,s,t≤n

⎞ l crst er ⊗ es ⊗ et ⎠

(6.15)

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

243

leading to

.

n E l r l r l r (crst cij k + cirt cj ks + cij r ckst ) = 0, 1 ≤ i, j, k, s, t, l ≤ n.

(6.16)

r=1

By a similar discussion, for a totally coassociative ternary coalgebra, we get n E .

l r crst cij k =

r=1

n E

l cirt cjr ks =

r=1

n E

l r cij r ckst , 1 ≤ i, j, k, s, t, l ≤ n,

(6.17)

r=1

while for a weak totally coassociative ternary coalgebra, n E .

l r crst cij k

=

r=1

n E

l r cij r ckst , 1 ≤ i, j, k, s, t, l ≤ n.

(6.18)

r=1

Therefore, we infer the following statement. Theorem 1 Let .A be an n-dimensional vector space with a basis .e1 , . . . .., en , and .A : A → A ⊗ A ⊗ A be defined as (6.14). Then, 1. .(A, A) is a partially coassociative ternary coalgebra if and only if the constants l .c ij k , .1 ≤ i, j, k ≤ n, satisfy the identity (6.16); 2. .(A, A) is a totally coassociative ternary coalgebra if and only if the constants l .c ij k , .1 ≤ i, j, k ≤ n, satisfy the identity (6.17); 3. .(A, A) is a weak totally coassociative ternary coalgebra if and only if the l , .1 ≤ i, j, k ≤ n, satisfy the identity (6.18). constants .cij k Now, let .(A, μ) be a partially associative ternary algebra with a basis e1 , e2 , . . . .., en , and the mutiplication .μ of .A in this basis be defined as follows:

.

μ(er , es , et ) =

n E

.

l l crst el , crst ∈ F, 1 ≤ r, s, t ≤ n.

(6.19)

l=1

Using the condition (6.6) we have: .

μ(μ(er , es , et ), ei , ej ) + μ(er , μ(es , et , ei ), ej ) + μ(er , es , μ(et , ei , ej )) =

n n E E k=1 l=1

En

l k crst clij ek +

n n E E k=1 l=1

l k csti crlj ek +

n n E E

l k ctij crsl ek = 0

k=1 l=1

l ck + cl ck + cl ck ) = 0, i. e. , .{cl yielding . l=1 (crst lij sti rlj tij rsl i1 i2 i3 ,1≤i1 ,i2 ,i3 ≤n } satisfies the identity (6.16). Similarly, for a totally associative ternary algebra and a weak totally associative ternary algebra, we derive, respectively:

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E l ck = En cl ck , i. e. , .{cl = nl=1 csti i1 i2 i3 ,1≤i1 ,i2 ,i3 ≤n } satisfies En rljl k l=1 Etijn rsl l k l the identity (6.17), and . l=1 crst clij = l=1 ctij crsl , i. e. , .{ci1 ,i2 ,i3 ,1≤i1 ,i2 ,i3 ≤n } satisfies the identity (6.18). Let .A∗ be the dual space of partially associative ternary algebra .(A, μ), and ∗ ∗ ∗ .e , . . . .., en be the dual basis of .e1 , . . . .., en , = δij , 1 ≤ i, j ≤ n. Assume 1 1 ∗ ∗ ∗ ∗ ∗ that .μ : A .→ .A ⊗ A ⊗ A is the dual mapping of .μ defined by (6.7). Then, for every .1 ≤ l ≤ n, we have En

.

k l l=1 crst clij

μ∗ (el∗ ) =

E

.

l crst er∗ ⊗ es∗ ⊗ et∗ .

(6.20)

1≤r,s,t≤n

Following the identities (6.14) and (6.16), (.A∗ , μ∗ ) is a partially coassociative ternary coalgebra. Conversely, if .(A, A) is a partially coassociative ternary coalgebra with a basis ∗ is the dual space of .A with the dual basis .e1 , . . . .., en satisfying (6.14), .A ∗ ∗ .e , . . . ., en . The dual mapping 1 ∗ ∗ ∗ ∗ ∗ ∗ .A : .A .→ .A ⊗ A ⊗ A of .A satisfies, for all .ξ, η, γ ∈ A , x ∈ A, = .

.

Then, .A∗ (er∗ , es∗ , et∗ ) = identity (6.16).

En

∗ l l l=1 crst el , crst

(6.21)

∈ F, 1 ≤ r, s, t, l ≤ n and .A∗ satisfies

Remark 2 The above constructions and discussions on the totally associative ternary algebra and weak totally associative ternary algebra also remain valid in this case. Therefore, the following results are true. Theorem 2 Let .A be a vector space over a field .F, and .A : A → A ⊗ A ⊗ A. Then, 1. .(A, A) is a partially coassociative ternary coalgebra if and only if .(A∗ , A∗ ) is a partially associative ternary algebra. 2. .(A, A) is a totally coassociative ternary coalgebra if and only if .(A∗ , A∗ ) is a totally associative ternary algebra. 3. .(A, A) is a weak totally coassociative ternary coalgebra if and only if .(A∗ , A∗ ) is a weak totally associative ternary algebra. We can also give an equivalence description of (2) as below. Theorem 3 Let .A be a vector space over a field .F, and .μ : A ⊗ A ⊗ A → A be a trilinear mapping. Then, 1. .(A, μ) is a partially associative ternary algebra if and only if .(A∗ , μ∗ ) is a partially coassociative ternary coalgebra. 2. .(A, μ) is a totally associative ternary algebra if and only if .(A∗ , μ∗ ) is a totally coassociative ternary coalgebra.

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

245

3. .(A, μ) is a weak totally associative ternary algebra if and only if .(A∗ , μ∗ ) is a weak totally coassociative ternary coalgebra. Example 2 Let .(T∗ , μ∗ ) be the dual of totally associative ternary algebra .(T, μ) in Example (10). The product .μ∗ on .T∗ is given by μ∗ (e1∗ ) = e1∗ ⊗ e1∗ ⊗ e1∗ + e1∗ ⊗ e2∗ ⊗ e2∗ + e2∗ ⊗ e2∗ ⊗ e1∗

.

∗

μ

(e2∗ )

+ e2∗ ⊗ e2∗ ⊗ e2∗ + e2∗ ⊗ e1∗ ⊗ e2∗ = e1∗ ⊗ e1∗ ⊗ e2∗ + e1∗ ⊗ e2∗ ⊗ e2∗ + e2∗ ⊗ e1∗ ⊗ e1∗ + e2∗ ⊗ e2∗ ⊗ e1∗ + e2∗ ⊗ e1∗ ⊗ e2∗ + e1∗ ⊗ e2∗ ⊗ e1∗ + 2e2∗ ⊗ e2∗ ⊗ e2∗ .

(T∗ , μ∗ ) is a totally coassociative ternary coalgebra.

.

Example 3 Let .(P∗ , μ∗ ) be the dual of partially associative ternary algebra .(P, μ) in Example (9). The product .μ∗ on .P∗ is given by μ∗ (e2∗ ) = e1∗ ⊗ e1∗ ⊗ e1∗ ; μ(e1∗ ) = 0.

.

(P∗ , μ∗ ) is a partially coassociative ternary coalgebra.

.

Let us recall that a coassociative ternary coalgebra is a partially coassociative ternary coalgebra, or a totally coassociative ternary coalgebra, or a weak totally coassociative ternary coalgebra. Definition 10 Let (.A1 , A1 ) and (.A2 , A2 ) be two coassociative ternary coalgebras. If there is a linear isomorphism .ϕ : A1 → A2 satisfying (ϕ ⊗ ϕ ⊗ ϕ)(A1 (e)) = A2 (ϕ(e)), for all e ∈ A1 ,

.

(6.22)

then .(A1 , A1 ) is isomorphic to .(A2 , A2 ), and .ϕ is called a coassociative ternary coalgebra isomorphism, i.e., (ϕ ⊗ ϕ ⊗ ϕ)

E

.

i

(ai ⊗ bi ⊗ ci ) =

E

ϕ(ai ) ⊗ ϕ(bi ) ⊗ ϕ(ci ).

(6.23)

i

Theorem 4 Let (.A1 , A1 ) and (.A2 , A2 ) be two coassociative ternary coalgebras. Then, .ϕ : A1 → A2 is a coassociative ternary coalgebra isomorphism from (.A1 , A1 ) to (.A2 , A2 ) if and only if the dual mapping .ϕ ∗ : A∗2 → A∗1 is an associative ternary algebra isomorphism from (.A∗2 , A∗2 ) to (.A∗1 , A∗1 ), where for every .ξ ∈ A∗2 , .v ∈ A1 , . = . Proof Let (.A1 , A1 ) and (.A2 , A2 ) be two coassociative ternary coalgebras. It follows that (.A∗1 , A∗1 ) and (.A∗2 , A∗2 ) are two associative ternary algebras. Let .ϕ : A1 → A2 be a coassociative ternary coalgebra isomorphism from (.A1 , A1 ) to (.A2 , A2 ). Hence, the dual mapping .ϕ ∗ : A∗2 → A∗1 is a linear isomorphism and for

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all .ξ, η, γ ∈ A∗2 , .x ∈ A1 : =

.

= = = . Then, .ϕ ∗ A∗2 (ξ, η, γ ) = A∗1 (ϕ ∗ (ξ ), ϕ ∗ (η), ϕ ∗ (γ )), that is, .ϕ ∗ is an associative ternary algebra isomorphism. u n

6.3 Trimodules and Matched Pairs of Associative Ternary Algebras The concept of trimodule is a particular case of the concept of module over an algebra over an operad defined in [21]. For the more general context of n-ary algebras, see [22].

6.3.1 Trimodules and Matched Pairs of Totally Associative Ternary Algebras Definition 11 A trimodule structure over totally associative ternary algebra (A, μ) on a vector space V is defined by the following three linear multiplication mappings: .

Lμ : A ⊗ A ⊗ V → V , Rμ : V ⊗ A ⊗ A → V , Mμ : A ⊗ V ⊗ A → V

satisfying the following compatibility conditions: ∀a, b, c, d, x, y, z ∈ A, v ∈ V , Lμ (a, b)(Lμ (c, d)(v)) = Lμ (μ(a, b, c), d)(v)

.

Lμ (a, b)(Lμ (c, d)(v)) = Lμ (a, μ(b, c, d))(v), .

(6.24)

Rμ (c, d)(Rμ (a, b)(v)) = Rμ (a, μ(b, c, d))(v) Rμ (c, d)(Rμ (a, b)(v)) = Rμ (μ(a, b, c), d)(v), . Mμ (a, z)(Mμ (b, y)(Mμ (c, x)(v)) = Mμ (μ(a, b, c), μ(x, y, z))(v), .

(6.25) (6.26)

Mμ (a, d)(Lμ(b, c)(v)) = Lμ (a, b)(Mμ (c, d)(v)) Mμ (a, d)(Lμ(b, c)(v)) = Mμ (μ(a, b, c), d)(v), .

(6.27)

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

247

Mμ (a, d)(Rμ (b, c)(v)) = Rμ (c, d)(Mμ (a, b)(v)) Mμ (a, d)(Rμ (b, c)(v)) = Mμ (a, μ(b, c, d))(v), .

(6.28)

Rμ (c, d)(Lμ (a, b)(v)) = Lμ (a, b)(Rμ (c, d)(v)) Rμ (c, d)(Lμ (a, b)(v)) = Mμ (a, d)(Mμ (b, c)(v)).

(6.29)

Proposition 1 (Lμ , Mμ , Rμ , V ) is a trimodule of an associative totally ternary algebra (A, μ) if and only if the direct sum (A ⊕ V , τ ) of the underlying vector spaces of A and V is turned into an associative totally ternary algebra τ given by τ [(x + a), (y + b), (z + c)] = μ(x, y, z) + Lμ (x, y)(c)

.

+Mμ (x, z)(b) + Rμ (y, z)(a), for all x, y, z ∈ A, a, b, c ∈ V . We denote it by A xLμ ,Mμ ,Rμ V . Proof Let v1 , v2 , v3 , v4 , v5 ∈ V and x1 , x2 , x3 , x4 , x5 ∈ A. Set .

τ [τ [(x1 + v1 ), (x2 + v2 ), (x3 + v3 )], (x4 + v4 ), (x5 + v5 )] = τ [(x1 + v1 ), τ [(x2 + v2 ), (x3 + v3 ), (x4 + v4 )], (x5 + v5 )] = τ [(x1 + v1 ), (x2 + v2 ), τ [(x3 + v3 ), (x4 + v4 ), (x5 + v5 )]].

After computation, we obtain Eqs. (6.24)–(6.29) while Eq. (6.26) is satisfied with the specification of the action of Mμ . Then (Lμ , Mμ , Rμ , V ) is a trimodule of the associative totally ternary algebra (A, μ) if and only if the direct sum (A ⊕ V , τ ) is a totally ternary algebra. u n Remark 3 In the case where Eq. (6.26) is not satisfied, we refer to the name quasi trimodule structure instead of simply trimodule structure. Theorem 5 Let (A, μA ) and (B, μB ) be two associative totally ternary algebras. Suppose that there are linear maps LμA : A ⊗ A ⊗ B → B, RμA : B ⊗ A ⊗ A → B, MμA : A ⊗ B ⊗ A → B and LμB : B ⊗ B ⊗ A → A, RμB : A ⊗ B ⊗ B → A, MμB : B ⊗ A ⊗ B → A such that (LμA , MμA , RμA , B) is a quasi trimodule of the associative totally ternary algebra (A, μA ) and (LμB , MμB , RμB , A) is a quasi trimodule of the associative totally ternary algebra (B, μB ) satisfying the following conditions: ∀x, y, z ∈ A, ∀a, b, c ∈ B, .μA (LμB (a, b)(x), y, z)

= LμB [a, RμA (x, y)(b)](z) = LμB (a, b)(μA (x, y, z)), .

(6.30)

μA (MμB (a, b)(x), y, z) = LμB [a, MμA (x, y)(b)](z) = MμB [a, RμA (y, z)(b)](x), .

(6.31)

μA (RμB (a, b)(x), y, z) = μA [x, LμB (a, b)(y), z] = RμB [a, RμA (y, z)(b)](x), .

(6.32)

LμB [LμA (x, y)(a), b](z) = μA [x, RμB (a, b)(y), z] = μA [x, y, LμB (a, b)(z)], .

(6.33)

LμB [MμA (x, y)(a), b](z) = μA [x, MμB (a, b)(y), z] = RμB [a, MμA (y, z)(b)](x), .

(6.34)

LμB [RμA (x, y)(a), b](z) = LμB [a, LμA (x, y)(b)](z) = MμB [a, MμA (y, z)(b)](x), .

(6.35)

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MμB [LμA (x, y)(a), b](z) = RμB [MμA (y, z)(a), b](x) = μA [x, y, MμB (a, b)(z)], .

(6.36)

MμB [MμA (x, y)(a), b](z) = RμB [RμA (y, z)(a), b](x) = RμB [a, LμA (y, z)(b)](x), .

(6.37)

MμB [RμA (x, y)(a), b](z) = MμB (a, b)(μA (x, y, z)) = MμB [a, LμA (y, z)(b)](x), .

(6.38)

RμB (a, b)(μA (x, y, z)) = RμB [LμA (y, z)(a), b](x) = μA [x, y, RμB (a, b)(z)], .

(6.39)

μB [LμA (x, y)(a), b, c] = LμA [x, RμB (a, b)(y)](c) = LμA (x, y)(μB (a, b, c)), .

(6.40)

μB [MμA (x, y)(a), b, c] = LμA [x, MμB (a, b)(y)](c) = MμA [x, RμB (b, c)(y)](a), .

(6.41)

μB [RμA (x, y)(a), b, c] = μB [a, LμA (x, y)(b), c] = RμA [x, RμB (b, c)(y)](a), .

(6.42)

LμA [LμB (a, b)(x), y](c) = μB [a, RμA (x, y)(b), c] = μB [a, b, LμA (x, y)(c)], .

(6.43)

LμA [MμB (a, b)(x), y](c) = μB [a, MμA (x, y)(b), c] = RμA [x, MμB (b, c)(y)](a), .

(6.44)

LμA [RμB (a, b)(x), y](c) = LμA [x, LμB (a, b)(y)](c) = MμA [x, MμB (b, c)(y)](a), .

(6.45)

MμA [LμB (a, b)(x), y](c) = RμA [MμB (b, c)(x), y](a) = μB [a, b, MμA (x, y)(c)], .

(6.46)

MμA [MμB (a, b)(x), y](c) = RμA [RμB (b, c)(x), y](a) = RμA [x, LμB (b, c)(y)](a), .

(6.47)

MμA [RμB (a, b)(x), y](c) = MμA (x, y)(μB (a, b, c)) = MμA [x, LμB (b, c)(y)](a), .

(6.48)

RμA (x, y)(μB (a, b, c)) = RμA [LμB (b, c)(x), y](a) = μB [a, b, RμA (x, y)(c)].

(6.49)

Then, there is an associative totally ternary algebra structure on the direct sum A ⊕ B of the underlying vector spaces of A and B given by the product τ defined by ∀x, y, z ∈ A, and ∀a, b, c ∈ B, τ [(x + a), (y + b), (z + c)] = [μA (x, y, z) + LμB (a, b)(z) + MμB (a, c)(y) + RμB (b, c)(x)] + [μB (a, b, c) + LμA (x, y)(c) + MμA (x, z)(b) + RμA (y, z)(a)].

.

Lμ ,Mμ ,Rμ

Let A > = = , i = 1, 2.

.

Then, .ϕ ∗ A∗2 (ξ, η, γ ) = A∗1 (ϕ ∗ (ξ ), ϕ ∗ (η), ϕ ∗ (γ )) and .ρ ∗ ◦ αi'∗ = αi∗ ◦ ρ ∗ , i = 1, 2, proving that .ϕ ∗ is a hom-associative ternary algebra isomorphism. u n

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6.6 Trimodules and Matched Pairs of Hom-associative Ternary Algebras In this section we introduce the concept of trimodules over hom-associative ternary algebra and give their matched pairs. Recall that a hom-module is a pair .(V , β) in which V is a vector space and .β : V → V is a linear map. We give the definition of bihom-module as follows. Definition 31 A bihom-module is a triple .(V , β1 , β2 ) in which V is a vector space and .β1 , β2 : V → V are linear maps.

6.6.1 Trimodules and Matched Pairs of Totally Hom-associative Ternary Algebras Definition 32 A trimodule structure over totally hom-associative ternary algebra (A, μ, α1 , α2 ) on a bihom-module (V , β1 , β2 ) is defined by the following three linear multiplication mappings: .

Lμ : A ⊗ A ⊗ V → V , Rμ : V ⊗ A ⊗ A → V , Mμ : A ⊗ V ⊗ A → V

satisfying the following compatibility conditions: ∀a, b, c, d, x, y, z ∈ A, v ∈ V , Lμ (α1 (a), α2 (b))(Lμ (c, d)(v)) = Lμ (μ(a, b, c), α1 (d))β2 (v)

.

= Lμ (α1 (a), μ(b, c, d))β2 (v), .

(6.126)

Rμ (α1 (c), α2 (d))(Rμ (a, b)(v)) = Rμ (α2 (a), μ(b, c, d))β1 (v) = Rμ (μ(a, b, c), α2 (d))β1 (v), .

(6.127)

Mμ (α1 (a), α2 (d))(Lμ(b, c)(v)) = Lμ (α1 (a), α2 (b))(Mμ (c, d)(v)) = Mμ (μ(a, b, c), α2 (d))β1 (v), .

(6.128)

Mμ (α1 (a), α2 (d))(Rμ (b, c)(v)) = Rμ (α1 (c), α2 (d))(Mμ (a, b)(v)) = Mμ (α1 (a), μ(b, c, d))β2 (v), .

(6.129)

Rμ (α1 (c), α2 (d))(Lμ (a, b)(v)) = Lμ (α1 (a), α2 (b))(Rμ (c, d)(v)) = Mμ (α1 (a), α2 (d))(Mμ (b, c)(v)),

(6.130)

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

273

Mμ (α1 (a), α2 (z))(Mμ (α1 (b), α2 (y))(Mμ (α1 (c), α2 (x))β1 (v)) .

= Mμ (μ(α1 (a), α1 (b), α1 (c)), μ(α2 (x), α2 (y), α2 (z)))β1 (v),

.

Mμ (α1 (a), α2 (z))(Mμ (α1 (b), α2 (y))(Mμ (α1 (c), α2 (x))β2 (v)) = Mμ (μ(α1 (a), α1 (b), α1 (c)), μ(α2 (x), α2 (y), α2 (z)))β2 (v),

(6.131)

(6.132)

β1 (Lμ (a, b)v) = Lμ (α1 (a), α2 (b))β1 (v),

.

β2 (Lμ (a, b)v) = Lμ (α1 (a), α2 (b))β2 (v).

(6.133)

β1 (Mμ (a, b)v) = Mμ (α1 (a), α2 (b))β1 (v), β2 (Mμ (a, b)v) = Mμ (α1 (a), α2 (b))β2 (v).

(6.134)

β1 (Rμ (a, b)v) = Rμ (α1 (a), α2 (b))β1 (v), β2 (Rμ (a, b)v) = Rμ (α1 (a), α2 (b))β2 (v).

(6.135)

Remark 7 In the case where only Eqs. (6.126)–(6.130) are satisfied, we refer to the name of quasi trimodule structure instead of simply trimodule structure. Proposition 6 (Lμ , Mμ , Rμ , β1 , β2 , V ) is a quasi trimodule of totally homassociative ternary algebra (A, μ, α1 , α2 ) if and only if the direct sum (A⊕V , τ, α1 +β1 , α2 +β2 ) of the underlying vector spaces of A and V is turned into a totally hom-associative ternary algebra τ given, for all x, y, z ∈ A, a, b, c ∈ V , by τ [(x + a), (y + b), (z + c)] = μ(x, y, z) + Lμ (x, y)(c)

.

+Mμ (x, z)(b) + Rμ (y, z)(a). We denote it by A xLμ ,Mμ ,Rμ ,β1 ,β2 V . Proof Let v1 , v2 , v3 , v4 , v5 ∈ V and x1 , x2 , x3 , x4 , x5 ∈ A. Set .

τ [τ [(x1 + v1 ), (x2 + v2 ), (x3 + v3 )], (α1 (x4 ) + β1 (v4 )), (α2 (x5 ) + β2 (v5 ))] = τ [(α1 (x1 ) + β1 (v1 )), τ [(x2 + v2 ), (x3 + v3 ), (x4 + v4 )], (α2 (x5 ) + α2 (v5 ))] = τ [(α1 (x1 ) + β1 (v1 )), (α2 (x2 ) + β2 (v2 )), τ [(x3 + v3 ), (x4 + v4 ), (x5 + v5 )]].

After computation, we obtain Eqs. (6.126)–(6.130). Then (Lμ , Mμ , Rμ , β1 , β2 , V ) is a quasi trimodule of totally hom-associative ternary algebra (A, μ, α1 , α2 ) if and only if (A ⊕ V , τ, α1 + β1 , α2 + β2 ) is a totally hom-associative ternary algebra. n u Example 11 Let (A, μ, α1 , α2 ) be a totally multiplicative hom-associative ternary algebra. Then (Lμ , 0, 0, α1 , α2 , A), (0, 0, Rμ , α1 , α2 , A) and (Lμ , Mμ , Rμ , α1 , α2 , A) are quasi trimodules of totally hom-associative ternary algebra (A, μ, α1 , α2 ).

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Theorem 19 Let (A, μA , α1 , α2 ) and (B, μB , β1 , β2 ) be two totally homassociative ternary algebras. Suppose that there are linear maps LμA : A ⊗ A ⊗ B → B, RμA : B ⊗ A ⊗ A → B, MμA : A ⊗ B ⊗ A → B and LμB : B ⊗ B ⊗ A → A, RμB : A ⊗ B ⊗ B → A, MμB : B ⊗ A ⊗ B → A such that (LμA , MμA , RμA , β1 , β2 , B) is a quasi trimodule of the totally hom-associative ternary algebra (A, μA , α1 , α2 ) and (LμB , MμB , RμB , α1 , α2 , A) is a quasi trimodule of the totally hom-associative ternary algebra (B, μB , β1 , β2 ), satisfying the following conditions: μA (LμB (a, b)(x), α1 (y), α2 (z)) = LμB [β1 (a), RμA (x, y)(b)]α2 (z) .

.

= LμB (β1 (a), β2 (b))(μA (x, y, z)), (6.136)

μA (MμB (a, b)(x), α1 (y), α2 (z)) = LμB [β1 (a), MμA (x, y)(b)]α2 (z) = MμB [β1 (a), RμA (y, z)(b)]α2 (x), (6.137)

μA (RμB (a, b)(x), α1 (y), α2 (z)) = μA [α1 (x), LμB (a, b)(y), α2 (z)] .

.

= RμB [β2 (a), RμA (y, z)(b)]α1 (x), LμB [LμA (x, y)(a), β1 (b)]α2 (z) = μA [α1 (x), RμB (a, b)(y), α2 (z)]

= μA [α1 (x), α2 (y), LμB (a, b)(z)],

(6.138)

(6.139)

LμB [MμA (x, y)(a), β1 (b)]α2 (z) = μA [α1 (x), MμB (a, b)(y), α2 (z)] .

.

= RμB [β2 (a), MμA (y, z)(b)]α1 (x), (6.140) LμB [RμA (x, y)(a), β1 (b)]α2 (z) = LμB [β1 (a), LμA (x, y)(b)]α2 (z) = MμB [β1 (a), MμA (y, z)(b)]α2 (x), (6.141)

MμB [LμA (x, y)(a), β2 (b)]α1 (z) = RμB [MμA (y, z)(a), β2 (b)]α1 (x) .

.

= μA [α1 (x), α2 (y), MμB (a, b)(z)], (6.142)

6 (Hom-)(co)associative Ternary (Co)algebras and Infinitesimal Ternary. . .

275

MμB [MμA (x, y)(a), β2 (b)]α1 (z) = RμB [RμA (y, z)(a), β2 (b)]α1 (x) = RμB [β2 (a), LμA (y, z)(b)]α1 (x), (6.143)

MμB [RμA (x, y)(a), β2 (b)]α1 (z) = MμB (β1 (a), β2 (b))(μA (x, y, z) .

.

= MμB [β1 (a), LμA (y, z)(b)]α2 (x), (6.144) RμB (β1 (a), β2 (b))(μA (x, y, z)) = RμB [LμA (y, z)(a), β2 (b)]α1 (x) = μA [α1 (x), α2 (y), RμB (a, b)(z)], (6.145)

μB [LμA (x, y)(a), β1 (b), β2 (c)] = LμA [α1 (x), RμB (a, b)(y)]β2 ((c) .

.

= LμA (α1 (x), α2 (y))(μB (a, b, c)),

μB [MμA (x, y)(a), β1 (b), β2 (c)] = LμA [α1 (x), MμB (a, b)(y)]β2 (c) = MμA [α1 (x), RμB (b, c)(y)]β2 (a), μB [RμA (x, y)(a), β1 (b), β2 (c)] = μB [β1 (a), LμA (x, y)(b), β2 (c)] .

.

= RμA [α2 (x), RμB (b, c)(y)]β1 (a),

LμA [LμB (a, b)(x), α1 (y)]β2 (c) = μB [β1 (a), RμA (x, y)(b), β2 (c)] = μB [β1 (a), β2 (b), LμA (x, y)(c)],

(6.146)

(6.147)

(6.148)

(6.149)

LμA [MμB (a, b)(x), α1 (y)]β2 (c) = μB [β1 (a), MμA (x, y)(b), β2 (c)] .

.

= RμA [α2 (x), MμB (b, c)(y)]β1 (a), (6.150) LμA [RμB (a, b)(x), α1 (y)]β2 (c) = LμA [α1 (x), LμB (a, b)(y)]β2 (c) = MμA [α1 (x), MμB (b, c)(y)]β2 (a), (6.151)

MμA [LμB (a, b)(x), α2 (y)]β1 (c) = RμA [MμB (b, c)(x), α2 (y)]β1 (a) .

.

= μB [β1 (a), β2 (b), MμA (x, y)(c)], (6.152)

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MμA [MμB (a, b)(x), α2 (y)]β1 (c) = RμA [RμB (b, c)(x), α2 (y)]β1 (a) = RμA [α2 (x), LμB (b, c)(y)]β1 (a), (6.153)

MμA [RμB (a, b)(x), α2 (y)]β1 (c) = MμA (α1 (x), α2 (y))(μB (a, b, c)) .

.

= MμA [α1 (x), LμB (b, c)(y)]β2 (a), (6.154) RμA (α1 (x), α2 (y))(μB (a, b, c)) = RμA [LμB (b, c)(x), α2 (y)]β1 (a) = μB [β1 (a), β2 (b), RμA (x, y)(c)], (6.155)

for all x, y, z ∈ A, a, b, c ∈ B. Then, there is a totally hom-associative ternary algebra structure on the direct sum A ⊕ B of the underlying vector spaces of A and B given by the product τ defined, for all x, y, z ∈ A, a, b, c ∈ B, by .

τ [(x + a), (y + b), (z + c)] = [μA (x, y, z) + LμB (a, b)(z) + MμB (a, c)(y) + RμB (b, c)(x)]+ [μB (a, b, c) + LμA (x, y)(c) + MμA (x, z)(b) + RμA (y, z)(a)].

Proof Let x1 , x2 , x3 , x4 , x5 ∈ A and y1 , y2 , y3 , y4 , y5 ∈ B. By definition, we have .

τ [(x + a), (y + b), (z + c)] = [μA (x, y, z) + LμB (a, b)(z) + MμB (a, c)(y) + RμB (b, c)(x)]+ [μB (a, b, c) + LμA (x, y)(c) + MμA (x, z)(b) + RμA (y, z)(a)]

for all x, y, z ∈ A, a, b, c ∈ B. Setting the conditions: .

τ [τ [(x1 + y1 ), (x2 + y2 ), (x3 + y3 )], (α1 (x4 ) + β2 (y4 )), (α2 (x5 ) + β2 (y5 ))] = τ [(α1 (x1 ) + β1 (y1 )), τ [(x2 + y2 ), (x3 + y3 ), (x4 + y4 )], (α2 (x5 ) + β2 (y5 ))] = τ [(α1 (x1 ) + β1 (y1 )), (α2 (x2 ) + β2 (y2 )), τ [(x3 + y3 ), (x4 + y4 ), (x5 + y5 )]],

we obtain by direct computation Eqs. (6.136)–(6.155). Then, there is a totally homassociative ternary algebra structure on the direct sum A ⊕ B of the underlying vector spaces of A and B if and only if Eqs. (6.136)–(6.155) are satisfied. u n Lμ ,Mμ ,Rμ ,α1 ,α2

Let A > l. Then for m = j + 1 we have

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Aj +1

.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

ν k−1 Ar+1 , j = r + (k − 1)(l − r) k−1 r+2 j = r + 1 + (k − 1)(l − r) ν A , = m ≥ l, for some k > 1. ... ⎪ ⎪ k−1 Al−1 , ⎪ j = l − r − 2 + (k − 1)(l − r) ν ⎪ ⎪ ⎩ ν k−1 Al = ν k Ar , j = l − r − 1 + (k − 1)(l − r), u n n E

Example 2 Let AB = BF (A) where F (t) =

δi t i , for some integer n > 3,

i=0

δi ∈ R, i = 0, . . . , n. Suppose that A3 = νA, where ν /= 0. We claim that { k−1 ν A, m = 2k − 1 m for some k ∈ N. .A = ν k−1 A2 , m = 2k, In fact, it holds true for m = 1, 2, 3. Suppose that it is true for some positive integer m = j . Then for m = j + 1 we have { ν k−1 A2 , if j = 2k − 1 j +1 for some k ∈ N. .A = if j = 2k, ν k−1 A3 = ν k A, Then we have n

[ n+1 2 ]

AB = δ0 B + BA

E

.

δ2k−1 ν

k−1

+ BA

2

k=1

[2] E

δ2k ν k−1 ,

k=1

where [m] is the greatest integer not greater than m. Example 3 Let AB = BF (A) where F (t) =

n E

δi t i , for some integer n > 8, δi ∈

i=0

R, i = 0, . . . , n. Suppose that A8 = νA5 , where ν /= 0. By applying Corollary 1 we have ⎧ k−1 5 ⎨ ν A , n = 3k + 2 n .A = n ≥ 8, for some k ∈ N \ {1}. ν k−1 A6 , n = 3k + 3 ⎩ k−1 7 ν A , n = 3k + 4, So the relation AB = BF (A) can be written as follows AB = B

7 E

.

[ n−2 3 ]

δi Ai + BA5

i=0

E i=2

[ n−4 3 ]

+BA7

E i=2

δ3i+4 ν i−1

[ n−3 3 ]

δ3i+2 ν i−1 + BA6

E i=2

δ3i+3 ν i−1

9 Algebraic Properties of Representations

=B

4 E

385

[ n−2 3 ]

δi Ai + BA5

i=0

E

[ n−3 3 ]

δ3i+2 ν i−1 + BA6

E

δ3i+3 ν i−1

i=1

i=1 [ n−4 3 ]

+BA7

E

δ3i+4 ν i−1 .

i=1

That is, AB = BH (A) where H (t) =

7 E

αi t i and

i=0

⎧ δi , i = 0, 1, 2, 3, 4 ⎪ ⎪ ⎪ n−2 ⎪ [ ] ⎪ 3 ⎪ E ⎪ ⎪ ⎪ δ3k+2 ν k−1 , i=5 ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎨ n−3 [ 3 ] E .αi = ⎪ δ3k+3 ν k−1 , i=6 ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ [ n−4 ] ⎪ 3 ⎪ ⎪ ⎪E ⎪ δ3k+4 ν k−1 , i = 7. ⎪ ⎩ k=1

9.3.2 Additivity Property of Elements in the Algebra Containing Representations 9.3.2.1

Binomial Expansion

Lemma 1 Let A1 , A2 be elements of an associative algebra over the field C (the set of complex numbers). Suppose that A1 and A2 anti-commute, that is, A1 A2 = −A2 A1 then the following are equivalent: 1. 2. 3. 4. 5.

(A1 + A2 )3 = A31 + A32 ; A2 A1 A1 = A2 A1 A2 ; A1 A2 A2 = A1 A2 A1 ; A2 A2 A1 = −A2 A1 A1 ; A1 A2 A2 = −A1 A1 A2 .

Proof (1) −→ (2) Suppose that (A1 + A2 )3 = A31 + A32 holds. By using the anticommutativity of elements A1 and A2 , we have (A1 + A2 )3 = (A1 + A2 )(A1 + A2 )2

.

= (A1 + A2 )(A1 A1 + A1 A2 + A2 A1 + A2 A2 ) = A31 + A1 A2 A2 + A2 A1 A1 + A32 .

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D. Djinja et al.

Then, (A1 + A2 )3 = A31 + A32 if and only if A1 A2 A2 + A2 A1 A1 = 0.

.

(9.4)

By using the anti-commutativity of A1 , A2 and associativity, Equality (9.4) can be rewritten as follows 0 = A1 A2 A2 + A2 A1 A1 = −A2 A1 A2 + A2 A1 A1 .

.

Thus, A2 A1 A2 = A2 A1 A1 . (2) −→ (3) Suppose that A2 A1 A2 = A2 A1 A1 holds. By using the anticommutativity of elements A1 , A2 and associativity, we have 0 = −A2 A1 A2 + A2 A1 A1 = A1 A2 A2 + A2 A1 A1 = A1 A2 A2 − A1 A2 A1 .

.

That is A1 A2 A2 = A1 A2 A1 . (3) −→ (4) Suppose that A1 A2 A2 = A1 A2 A1 holds. By using the anticommutativity of A1 , A2 and associativity, we have 0 = A1 A2 A2 − A1 A2 A1

.

= A1 A2 A2 + A2 A1 A1 = −A2 A1 A2 + A2 A1 A1 = A2 A2 A1 + A2 A1 A1 . That is A2 A2 A1 = −A2 A1 A1 . (4) −→ (5) Suppose that A2 A2 A1 = −A2 A1 A1 holds. By using the anticommutativity of A1 , A2 and associativity, we have 0 = A2 A2 A1 + A2 A1 A1 = −A2 A1 A2 − A1 A2 A1 = A1 A2 A2 + A1 A1 A2 .

.

That is A1 A2 A2 = −A1 A1 A2 . (5) −→ (1) Suppose that A1 A2 A2 = −A1 A1 A2 holds. By using the anticommutativity of A1 , A2 and associativity we get (A1 + A2 )3 = (A1 + A2 )(A1 A1 + A1 A2 + A2 A1 + A2 A2 )

.

= A31 + A1 A2 A2 + A2 A1 A1 + A32 = A31 + A1 A2 A2 − A1 A2 A1 + A32 = A31 + A1 A2 A2 + A1 A1 A2 + A32 = A31 + A32 . u n Lemma 2 Let A1 , A2 be elements of an associative algebra over the field C. If A1 An2 + A2 An1 = 0,

.

n = 1, 2

holds, then A1 An2 + A2 An1 = 0,

.

n = 3, 4, 5, . . .

(9.5)

9 Algebraic Properties of Representations

387

Proof We proceed by induction. We notice that from (9.5) we get A2 A21 = −A22 A1 . Then, by using this, anti-commutativity, associativity, distributivity properties and the hypothesis we get for n = 3 A1 A32 + A2 A31 = (A1 A2 )A22 + (A2 A1 A1 )A1

.

= −A2 A1 A22 − A2 A2 A1 A1 = −A2 (A1 A22 + A2 A21 ) = 0. Suppose that A1 Ak2 + A2 Ak1 = 0,

.

for some integer n = k ≥ 3.

Then, using A2 A1 A1 = −A2 A2 A1 , anti-commutativity, associativity and distributivity properties, we get then for n = k + 1: A1 Ak+1 + A2 Ak+1 = A1 A2 Ak2 + A2 A1 A1 Ak−1 = −A2 A1 Ak2 + A2 A1 A1 Ak−1 2 1 1 1

.

= −A2 A1 Ak2 − A2 A2 A1 Ak−1 = −A2 (A1 Ak2 + A2 Ak1 ) = 0. 1 u n Lemma 3 Let A1 , A2 be elements of an associative algebra over the field C. n E Let F : R → R be a polynomial defined as follows F (t) = δi t i , δi ∈ R, i=1

i = 1, . . . , n. If (9.5) holds, that is, A1 Ak2 + A2 Ak1 = 0,

.

k = 1, 2

then F (A1 + A2 ) = F (A1 ) + F (A2 )

.

(9.6)

also holds. Proof We first prove for monomial M(t) = t n , n = 1, 2, 3, . . .. Thus, we need to prove that elements A1 and A2 satisfy M(A1 ) + M(A2 ) = M(A1 + A2 ).

.

We proceed by induction. The case n = 1 follows from the fact that M(t) = t is a linear map. Suppose that for some integer n = k ≥ 1 M(A1 + A2 ) = (A1 + A2 )k = Ak1 + Ak2 = M(A1 ) + M(A2 ).

.

Then, by using Lemma 2 we have for n = k + 1:

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D. Djinja et al.

M(A1 + A2 ) = (A1 + A2 )k+1 = (A1 + A2 )(A1 + A2 )k = (A1 + A2 )(Ak1 + Ak2 )

.

= Ak+1 + A1 Ak2 + A2 Ak1 + Ak+1 1 2 = Ak+1 + Ak+1 = M(A1 ) + M(A2 ). 1 2 By applying this we get F (A1 + A2 ) =

n E

.

δi (A1 + A2 )i =

i=1

n E

δi (Ai1 + Ai2 ) = F (A1 ) + F (A2 ).

i=1

u n Corollary 2 Let A1 , A2 be elements of an associative algebra over the field C. n E Let F : R → R be a polynomial defined as follows F (t) = δi t i , δi ∈ R, i=1

i = 1, . . . , n. If A1 A2 = −A2 A1.

(9.7)

A1 A2 = A1 A1

(9.8)

.

then (9.6) holds, that is, F (A1 + A2 ) = F (A1 ) + F (A2 ). Furthermore, in this case A31 = 0. Proof Suppose that (9.7) and (9.8) hold true. Then, by multiplying on the left in relation (9.8) by A2 we get A2 A1 A2 = A2 A1 A1

.

By using anti-commutativity and Lemma 1, Relation (9.9) is equivalent to A1 A2 A2 = −A1 A1 A2 .

.

Using anti-commutativity, we get A1 A2 A2 + A1 A1 A2 = 0 ⇔ A1 A2 A2 − A1 A2 A1 = 0

.

⇔ A1 A2 A2 + A2 A1 A1 = 0 ⇔ A1 A22 + A2 A21 = 0. Hence relation (9.5) holds. Applying the Lemma 3 we get F (A1 + A2 ) = F (A1 ) + F (A2 ).

.

In order to prove that A31 = 0, we right-multiply both sides of the equality A1 A2 = A1 A1 by A1 and apply Relations (9.7) and (9.8). We get

(9.9)

9 Algebraic Properties of Representations

389

A1 A2 A1 = A1 A1 A1 ⇔ −A2 A1 A1 = A1 A1 A2 ⇔ −A2 (A21 ) = −A1 A2 A1

.

⇔ −A2 (A21 ) = A2 (A21 ) ⇔ A2 (A21 ) = 0. Then, we have A1 A2 A1 = A1 A1 A1 ⇔ −A2 (A21 ) = A31 =⇒ A31 = 0.

.

u n Corollary 3 Let A1 , A2 be elements of an associative algebra over the field C. n E Let F : R → R be a polynomial defined as follows F (t) = δi t i , δi ∈ R, i=1

i = 1, . . . , n. If A1 A2 = −A2 A1

.

A2 A1 = A2 A2

(9.10)

hold then (9.6) also holds. Furthermore, in this case A32 = 0. Proof Suppose that A1 A2 = −A2 A1 and A2 A1 = A2 A2 hold true. Then, by multiplying on the left in Relation 9.10 by A1 we get A1 A2 A1 = A1 A2 A2

.

(9.11)

By working on the left hand side of (9.11) we get A1 A2 A1 = A1 A2 )A1 = −(A2 A1 )A1 = −A2 A21 .

.

Substituting this into (9.11) we get; A1 A22 + A2 A21 = 0.

.

Hence Relation (9.5) holds. By applying Lemma 3 we get F (A1 + A2 ) = F (A1 ) + F (A2 ).

.

In order to prove that A32 = 0, we left-multiply on both sides of the equality A2 A1 = A2 A2 by A2 and, apply anti-commutativity and Relation (9.10). We get A2 A2 A1 = A2 A2 A2 ⇔ −A2 A1 A2 = A2 A1 A2 ⇔ A1 (A22 ) = −A1 A2 A2

.

⇔ −A1 (A22 ) = A1 (A22 ) ⇔ A1 (A22 ) = 0. Thus, we have

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D. Djinja et al.

A2 A2 A1 = A2 A2 A2 ⇔ −A1 (A22 ) = A32 =⇒ A32 = 0.

.

u n Proposition 4 Let A1 : E → E, A2 : E → E be linear operators such that the composition of any pair of these is well defined. If A1 A2 = −A2 A1 and A1 A2 = A1 A1 hold true, then 1. Im(A21 ) ⊆ ker(A2 ); 2. Im(A22 ) ⊆ ker(A1 ), where Im(A) is the range of operator A and ker(A) is its kernel. Proof By multiplying A1 A2 = A1 A1 on the right by A1 and applying these relations we get A1 A2 A1 = A1 A1 A1 ⇔ −A2 A1 A1 = A1 A1 A2 ⇔ −A2 (A21 ) = −A1 A2 A1

.

⇔ −A2 (A21 ) = A2 (A21 ) ⇔ A2 (A21 ) = 0. Then Im(A21 ) ⊆ ker(A2 ). By multiplying A1 A2 = A1 A1 on the right by A2 and, applying anticommutativity and relation (9.8) we get A1 A2 A2 = A1 A1 A2 ⇔ A1 (A2 A2 ) = −A1 A2 A1 ⇔ A1 (A22 ) = A2 (A21 ).

.

Since A2 (A21 ) = 0 we get A1 (A22 ) = 0, then Im(A22 ) ⊆ ker(A1 ). 9.3.2.2

u n

Construction of Representations

Let E be a normed linear space, .Ai : E → E, .i = 1, 2, .B : E → E linear operators and .F : R → R a real-valued polynomial. In general, it is not true for all operators .A1 , A2 , B and polynomials F that Ai B = BF (Ai ),

.

i = 1, 2

(9.12)

implies that (A1 + A2 )B = BF (A1 + A2 ).

.

(9.13)

Nevertheless, there are operators .A1 , .A2 and B which satisfy (9.12) and (9.13) for a monomial .F (t) = t n , .t ∈ R, .n > 1. The following Proposition gives sufficient conditions for those operators .A1 , A2 that satisfy (9.12) and (9.13). Proposition 5 Let .A1 : E → E, .A2 : E → E, .B : E → E be linear operators such that the composition of any pair of these is well defined and so is the

9 Algebraic Properties of Representations

391

composition of B with any powers of .Ai , .i = 1, 2. Suppose that for some polynomial n E .F (t) = δi t i , .δi ∈ R, .i = 1, . . . , n i=1

Ai B = BF (Ai ), i = 1, 2

.

holds. If .n ≥ 2 and B(Ak1 A2 + Ak2 A1 ) = 0,

.

k = 1, . . . , n − 1,

(9.14)

then (A1 + A2 )B = BF (A1 + A2 ).

.

For .n = 1, this is true independent of condition (9.14). Proof We first prove for monomial .M(t) = t n , .n = 2, 3, . . .. That is, we need to prove that elements .A1 and .A2 satisfy B(M(A1 ) + M(A2 )) = BM(A1 + A2 ).

.

We proceed by induction. For .n = 2, using the fact that .B(A1 A2 + A2 A1 ) = 0, we have BM(A1 + A2 ) = B(A1 + A2 )2 = B(A21 + A1 A2 + A2 A1 + A22 ) =

.

= BA21 + B(A1 A2 + A2 A1 ) + BA22 = BA21 + BA22 = B(M(A1 ) + M(A2 )). For .n = 3 we have BM(A1 + A2 ) = B(A1 + A2 )3 = BA31 + BA21 A2 + BA1 A2 A1 + BA1 A22 +

.

+BA2 A21 + BA2 A1 A2 + BA22 A1 + BA32 . u n Multiplying on the equality .B(A1 A2 + A2 A1 ) = 0 on the right by .A1 and .A2 , we get the following: BA1 A2 A1 + BA2 A21 = 0, and BA1 A22 + BA2 A1 A2 = 0.

.

Using this, the equality .B(A21 A2 + A22 A1 ) = 0 and commutativity of elements addition, we get BM(A1 + A2 ) = B(A1 + A2 )3 = BA31 + BA21 A2 + BA1 A2 A1 + BA1 A22 +

.

+ BA2 A21 + BA2 A1 A2 + BA22 A1 + BA32 = BA31 + BA32 = B(M(A1 ) + M(A2 )).

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D. Djinja et al. j

j

Suppose that .B(A1 + A2 )j = B(A1 + A2 ) for all integers j such that .j = 2, . . . , n − 1. Then, BM(A1 + A2 ) = B(A1 + A2 )n = B(A1 + A2 )n−1 (A1 + A2 )

.

= B(An−1 + An−1 1 2 )(A1 + A2 ) n−1 n = BAn1 + BAn−1 1 A2 + BA2 A1 + BA2

= BAn1 + BAn2 = B(M(A1 ) + M(A2 )). By applying this and the fact that .BM(A1 + A2 ) = BM(A1 ) + BM(A2 ) for monomial .M(t) = t, we get BF (A1 + A2 ) =

n E

.

δi B(A1 + A2 )i =

i=1

n E

δi B(Ai1 + Ai2 ) = BF (A1 ) + BF (A2 ).

i=1

u n Corollary 4 Let .A1 : E → E, .A2 : E → E, .B : E → E be linear operators such that the composition of any pair of these is well defined and so is the composition of B with any powers of .Ai , .i = 1, 2. Suppose that for some polynomial .F (t) = n E δi t i , .δi ∈ R, .i = 1, . . . , n i=1

Ai B = BF (Ai ), i = 1, 2.

.

If .F (A1 + A2 ) = F (A1 ) + F (A2 ), in particular if A1 Ak2 + A2 Ak1 = 0,

.

for .k = 1, 2, then (A1 + A2 )B = BF (A1 + A2 ).

.

Proof Using the hypothesis and applying Lemma 3 we get (A1 + A1 )B = BF (A1 ) + BF (A2 ) = B(F (A1 ) + F (A2 )) = BF (A1 + A2 ).

.

u n Corollary 5 Let .A1 : E → E, .A2 : E → E, .B : E → E be linear operators such that the composition of any pair of these is well defined and so is the composition of B with any powers of .Ai , .i = 1, 2. Suppose that for some polynomial .F (t) = n E δi t i , .δi ∈ R, .i = 1, 2, . . . , n, i=1

Ai B = BF (Ai ), i = 1, 2.

.

9 Algebraic Properties of Representations

393

If A1 A2 = −A2 A1

.

A1 A2 = A1 A1 then .(A1 + A2 )B = BF (A1 + A2 ). Proof By applying Corollary 2 we have .F (A1 + A2 ) = F (A1 ) + F (A2 ). Then, by applying Corollary 4 we complete the proof. u n Corollary 6 Let .A1 : E → E, .A2 : E → E, .B : E → E be linear operators such that the composition of any pair of these is well defined and so is the composition of B with any powers of .Ai , .i = 1, 2. Suppose that for some polynomial .F (t) = n E δi t i , .δi ∈ R, .i = 1, . . . , n, .t ∈ R i=1

Ai B = BF (Ai ), i = 1, 2.

.

If .A2 A1 = A2 A2 and .A1 A2 = −A2 A1 hold then .(A1 + A2 )B = BF (A1 + A2 ). Proof Applying Corollary 3 we have .F (A1 + A2 ) = F (A1 ) + F (A2 ). Then we apply Corollary 4 to complete the proof. n u We now present some examples when such situation occurs. Example 4 Consider .n × n dimensional matrices ⎛

⎛ ⎞ 0 α1 0 0 ... ⎜0 0 ... 0 α2 ⎟ ⎜ ⎟ .. .. ⎟ , A = ⎜ .. .. .. ⎜. . . ⎟ 2 . . ⎟ ⎜ ⎝0 0 0 ⎠ 0 0 αn−1 00 0 0 0 00 0 ⎛ ⎞ 0 0 . . . 0 β1 ⎜ 0 0 . . . 0 β2 ⎟ ⎜ ⎟ ⎜ ⎟ B = ⎜ ... ... ... ... ... ⎟ , ⎜ ⎟ ⎝ 0 0 0 0 βn−1 ⎠ 00 0 0 0

00 ⎜0 0 ⎜ ⎜. . .A1 = ⎜ . . ⎜. . ⎝0 0

... ... .. .

where .αi , γi , βi , .i = 1, . . . , n − 1 are constants. We have A1 A2 = −A2 A1 = 0

.

A1 A2 = A1 A1 = 0 An1 = An2 = 0,

n = 2, 3, . . .

⎞ 0 γ1 0 γ2 ⎟ ⎟ .. .. ⎟ , . . ⎟ ⎟ 0 γn−1 ⎠ 0 0

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D. Djinja et al.

Ai B = BAni = 0,

i = 1, 2,

n = 1, 2, 3, . . .

(A1 + A2 )B = B(An1 + An2 ) = B(A1 + A2 )n = 0,

n = 1, 2, 3, . . .

where 0 is the null matrix. Remark 1 The conclusion in Example 4 is not true in general for nilpotent matrices that do not satisfy the commutation relation. For instance, the following matrices ( A1 = A2 =

.

) 01 , 00

( B=

00 10

)

are nilpotent of order two however ⎧( 0 ⎪ ⎪ ( ) ⎨ 0 10 .Ai B = /≡ BAni = ( ⎪ 0 00 ⎪ ⎩ 0

) 0 , i = 1, 2, if n = 1 1 ) 0 , i = 1, 2, if n ≥ 2. 0

and ( (A1 + A2 )B =

.

20 00

) /≡ B(A1 + A2 )n =

⎧( ) ⎪ 00 ⎪ ⎪ , if n = 1 ⎪ ⎪ ⎨ 02 ( ) ⎪ ⎪ ⎪ 00 ⎪ ⎪ , if n ∈ N \ {1} ⎩ 00

Example 5 Let .A1 , A2 , B be .n × n matrices. If .A1 , A2 are nilpotent of order two such that .A1 + A2 is also nilpotent of order two and .Ai B = BF (Ai ), .i = 1, 2, for n E δi t i , .δi ∈ R, .i = 1, . . . , n, .t ∈ R, then some polynomial .F (t) = i=1

(A1 + A2 )B = BF (A1 + A2 ).

.

In fact, from the identities .A21 = A22 = 0, .(A1 + A2 )2 = 0 we get .A1 A2 + A2 A1 = 0 and .A1 A22 + A2 A21 = 0. Hence, from Corollary 4 we have .(A1 + A2 )B = BF (A1 + A2 ). In particular, the following matrices ( A1 =

.

α −α α −α

(

) ,

A2 =

γ −γ γ −γ

)

( ,

B=

β0 −β1 β0 −β1

) ,

for real constants .α, β0 , β1 , γ satisfy A21 = A22 = 0, (A1 + A2 )2 = 0, Ai B = BA2i = 0, i = 1, 2,

.

9 Algebraic Properties of Representations

395

(A1 + A2 )B = B(A1 + A2 )2 . Example 6 Consider .Ai : Lp [α, β] → Lp [α, β], .i=1, 2, .B : Lp [α, β] → Lp [α, β], 1 ≤ p < ∞ defined as follows

.

fβ (A1 x)(t) =

a1 (t)c1 (s)x(s)ds,

.

α

fβ (A2 x)(t) =

a2 (t)c2 (s)x(s)ds, α

fβ (Bx)(t) =

b(t)e(s)x(s)ds, α

where .α, β ∈ R, .α < β, .a1 , a2 , b ∈ Lp [α, β] and .c1 , c2 , e ∈ Lq [α, β], where 1 < q < ∞ such that . p1 + q1 = 1. If the following pairs .(c1 , a2 ), .(c2 , a1 ), .(a1 , c1 ), are orthogonal, that is,

.

fβ

fβ c1 (s)a2 (s)ds =

.

α

fβ c2 (s)a1 (s)ds =

c1 (s)a1 (s)ds = 0

α

α

fβ

fβ

and if fβ c1 (s)b(s)ds =

.

α

c2 (s)b(s)ds = α

a2 (s)c2 (s)ds = 0 α

then for each monomial .F (t) = δt n , .δ ∈ R \ {0}, .t ∈ R, where .n > 1 operators .A1 , .A2 B satisfy the relation .Ai B = BF (Ai ), .i = 1, 2. Moreover, we have (A1 + A2 )B = δB(A1 + A2 )n .

.

In fact, in this case we have A1 A2 = −A2 A1 = 0,

.

Ai B = δBAni = 0, (A1 + A2 )B =

δB(An1

where 0 is the zero operator.

+ An2 )

A1 A2 = A1 A1 = 0

i = 1, 2,

n = 2, 3, . . .

= δB(A1 + A2 )n = 0,

n = 2, 3, . . .

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9.4 Representations of Polynomial Covariant Commutation Relations on lp Spaces We start with a finite-dimensional vector space. The following representations can be found in [35]. Consider .F (t) = δ0 + δ1 t + δ2 t 2 + . . . + δn t n , A, B .n × n real matrices given by ⎛ ⎜ ⎜ .A = ⎜ ⎝

a1 0 .. .

0 a2 .. .

... ... .. . 0 0 ...

⎞

0 0⎟ ⎟ ⎟, 0⎠ an

⎛

⎞ ... 0 α ... 0 0⎟ ⎟ ⎟ .. . 0 0⎟ ⎟. . . .. .. ⎟ . . .⎠ 0 0 ... 1 0

00 ⎜1 0 ⎜ ⎜ B=⎜ ⎜0 1 ⎜. . ⎝ .. ..

We have ⎛

0 ⎜ a2 ⎜ ⎜ .AB = ⎜ 0 ⎜ ⎜ . ⎝ ..

0 ... 0 ... . a3 . . .. . . . . 0 0 ...

⎞ 0 αa1 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ ⎟, .. .. ⎟ . . ⎠

⎛

F (a1 ) 0 ⎜ 0 F (a2 ) ⎜ ⎜ F (A) = ⎜ 0 ⎜ 0 ⎜ . .. . ⎝ . .

an 0

0

0

⎞ ... 0 0 ... 0 0 ⎟ ⎟ ⎟ .. .0 0 ⎟ ⎟ .. ⎟ . . .. . . . ⎠ . . . 0 F (an )

and ⎛

0 0 ⎜ F (a1 ) 0 ⎜ ⎜ .BF (A) = ⎜ ⎜ 0 F (a2 ) ⎜ . .. ⎝ .. . 0

0

... ... .. . ..

0 0 0 .. .

. . . . F (an−1 )

⎞ αF (an ) 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟. ⎟ .. ⎠ . 0

Therefore, .AB = BF (A) implies that F (a1 ) = a2 ,

.

F (a2 ) = a3 , . . . , F (an−1 ) = an ,

F (an ) = a1 .

When we take .B t (the transpose matrix of B) instead of B we get the following conditions: F (a1 ) = an ,

.

F (a2 ) = a1

F (a3 ) = a2

...

F (an ) = an−1 .

Example 7 Consider .A : l2 → l2 , .B : l2 → l2 defined as follows:

9 Algebraic Properties of Representations

397

⎛

⎛

0 ... a2 . . . . 0 .. .. . . . .

a1 ⎜0 ⎜ .A = ⎜ ⎜0 ⎝ .. .

⎞ 0... 0... ⎟ ⎟ ⎟, 0... ⎟ ⎠ an . . .

00 ⎜1 0 ⎜ ⎜ ⎜0 1 ⎜ B =⎜. . ⎜ .. .. ⎜ ⎜0 0 ⎝ .. .. . .

⎞ ... 0 0 ... ... 0 0 ...⎟ ⎟ ⎟ .. . 0 0 ...⎟ ⎟ . . . .. .. .. ⎟ . . . . ⎟ ⎟ ... 1 0 ...⎟ ⎠ .. . . . 1 .

(9.15)

The above operators can be written as Ax = (a1 x1 , . . . , an xn , 0, 0, . . .),

.

Bx = (0, x1 , x2 , x3 , . . .),

for .x = (x1 , x2 , . . .) ∈ l2 . Thus, .AB = BF (A) implies that F (a1 ) = a2 ,

F (a2 ) = a3 , . . . , F (an−1 ) = an , . . .

.

Now, for certain constants .ai , i ∈ N, we have Ax = (a1 x1 , . . . , an xn , an+1 xn+1 , . . .) ∈ lp ,

.

Bx = (0, x1 , x2 , x3 , . . .) ∈ lp ,

for .x = (x1 , x2 , . . .) ∈ lp , .1 ≤ p ≤ ∞. Thus, .AB = BF (A) if and only if F (a1 ) = a2 ,

F (a2 ) = a3 , . . . , F (an−1 ) = an , F (an ) = 0, F (0) = 0.

.

Remark 2 The operators in (9.15) are also representations of relation (9.1) in .lp , 1 ≤ p ≤ ∞.

.

Remark 3 Suppose that the sequence .{ai } is bounded, that is, there exists a positive M such that |ai | ≤ M, i = 1, 2, 3, . . .

.

then, for .1 ≤ p < ∞ and for .x ∈ lp , we have ||Ax||p =

∞ E

.

|ai xi |p ≤ M p

i=1

∞ E

|xi |p < ∞

i=1

since .x ∈ lp . If .p = ∞, we have ||Ax|| = sup |(Ax)i | = sup |ai xi | ≤ sup |ai | · sup |xi | = M · ||x|| < ∞.

.

i∈N

i∈N

i∈N

i∈N

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We can generalize the previous case in the following theorem. For an infinite sequence of real numbers .{bi } i ∈ N, we set Jb = {i ∈ N : bi /= 0}.

.

(9.16)

Theorem 1 Let .A : lp → lp , .Br : lp → lp , .1 ≤ p ≤ ∞ defined as follows Ax = (a1 x1 , a2 x2 , a3 x3 , . . .),

.

Br x = (0, 0, . . . , 0, br+1 x1 , br+2 x2 , . . .), ' '' ' r

where .r = 0, 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . Let .F : R → R be defined as follows .F (t) = δ0 + δ1 t + . . . + δn t n , .δj ∈ R, j is a non negative integer. If .b = (b1 , b2 , . . .) is given, then .ABr = Br F (A) if and only if for all positive integers i F (ai ) = ar+i ,

for r + i ∈ Jb and ai , ar+i are free for r + i /∈ Jb ,

.

where .Jb is defined in (9.16). Proof We first verify that the operators A and .Br act from .lp to .lp . For .1 ≤ p < ∞ we have p .||Ax|| lp

=

∞ E

|ai xi | ≤ M p

i=1

p

∞ E

|xi |p = M p ||x||p < ∞,

i=1

where .M = sup |ai | < ∞ because .a ∈ l∞ . For .p = ∞ we have i∈N

||Ax||l∞ = sup |ai xi | ≤ M||x||l∞ < ∞.

.

i∈N

Analogously, one can prove that the operator .Bk acts from .lp to .lp . Now, consider a polynomial .F (t) = δ0 + δ1 t + . . . + δn t n , where .δi ∈ R, .i = 0, . . . , n. We compute F (A)x = (F (a1 )x1 , F (a2 )x2 , . . .),

.

A(Br x) = (0, 0, . . . , 0, ar+1 br+1 x1 , ar+2 br+2 x2 , . . .), ' '' ' r

Br (F (A)x) = (0, 0, . . . , 0, br+1 F (a1 )x1 , br+2 F (a2 )x2 , . . .), ' '' ' r

where .r = 0, 1, 2, . . . is fixed. The operators .F (A), AB, .BF (A) are well defined and they act from .lp to .lp , .1 ≤ p ≤ ∞. Thus, .ABr x = Br F (A)x for all .x ∈ lp if and only if for all .i ∈ N ar+i br+i = br+i F (ai ).

.

9 Algebraic Properties of Representations

399

If .b = (b1 , b2 , . . .) is given and if .br+i /= 0 for a fixed .i ∈ N, that is, .r + i ∈ Jb , then ar+i br+i = br+i F (ai ) ⇔ ar+i = F (ai ).

.

Otherwise if .br+i = 0 for a fixed .i ∈ N, that is, .r + i /∈ Jb , then the equation ar+i br+i = br+i F (ai ) is satisfied for any value of .ai or .ar+i . u n

.

Theorem 2 Let .A : lp → lp , .Bl : lp → lp , .1 ≤ p ≤ ∞ defined as follows Ax = (a1 x1 , a2 x2 , a3 x3 , . . .),

.

Bl x = (bl+1 xl+1 , bl+2 xl+2 , bl+3 xl+3 , . . .),

where .l = 0, 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . Let F : R → R be defined as follows .F (t) = δ0 + δ1 t + . . . + δn t n , .δj ∈ R, j is a non negative integer. If .b = (b1 , b2 , . . .) is given then .ABl = Bl F (A) if and only if for all positive integers i

.

F (al+i ) = ai ,

.

for l + i ∈ Jb and ai , al+i are free for l + i /∈ Jb ,

where .Jb is defined in (9.16). Proof Consider a polynomial .F (t) = δ0 + δ1 t + . . . + δn t n , where .δi ∈ R, .i = 0, 1, . . . , n. We compute F (A)x = (F (a1 )x1 , F (a2 )x2 , . . .),

.

A(Bl x) = A(bl+1 xl+1 , bl+2 xl+2 , bl+3 xl+3 , . . .) = = (a1 bl+1 xl+1 , a2 bl+2 xl+2 , a3 bl+3 xl+3 , . . .) Bl (F (A)x) = Bl (F (a1 )x1 , F (a2 )x2 , . . .) = (bl+1 F (al+1 )xl+1 , bl+2 F (al+2 )xl+2 , . . .), Thus, .ABl x = Bl F (A)x for all .x ∈ lp if and only if for all .i ∈ N ai bl+i = bl+i F (al+i ).

.

If .b = (b1 , b2 , . . .) is given and if .bl+i /= 0 for a fixed .i ∈ N, that is, .l + i ∈ Jb then ai bl+i = bl+i F (al+i ) ⇔ ai = F (al+i ).

.

If .bl+i = 0 for a fixed .i ∈ N, that is, .l + i /∈ Jb , then equation .ai bl+i = bl+i F (al+i ) is satisfied for any value of .ai or .al+i . u n

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D. Djinja et al.

Example 8 Consider .A : lp → lp , .B : lp → lp , .1 ≤ p ≤ ∞ defined as follows Ax = (a1 x1 , a2 x2 , a3 x3 , . . .),

.

Bx = (x2 , x3 , x4 , . . .)

where .{ai } is a bounded real sequence. For a polynomial .F : R → R defined by n E .F (t) = δk t k , .δk ∈ R, .k = 0, . . . , n we have k=0

AB = BF (A)

.

if and only if F (a2 ) = a1 , F (a3 ) = a2 , F (a4 ) = a3 , . . . , F (an+1 ) = an , . . .

.

Theorem 3 Let .Al : lp → lp , .Br : lp → lp , .1 ≤ p ≤ ∞ defined as follows Al x = (al+1 xl+1 , al+2 xl+2 , . . .),

.

Br x = (br+1 xr+1 , br+2 xr+2 , . . .),

where .r = 0, 1, 2, 3, . . ., .l = 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . If .b = (b1 , b2 , . . .) is given, then for some .n ∈ N and some .δ ∈ R \ {0}, n .Al Br = δBr A if and only if one of the following holds: l 1. if .n = 1 then (a) if .br+i /= 0 for a fixed .i ∈ N then al+r+i = al+i

.

br+l+i δbr+i

(b) if .br+i = 0 and .br+l+i /= 0 for a fixed .i ∈ N, then .al+i = 0 (c) if .r + i /∈ Jb and .r + l + i /∈ Jb then .al+i and .a1+r+i can be freely chosen. 2. if .n /= 1 then (a) if .br+l+i /= 0 for a fixed .i ∈ N then .al+i = 0. Otherwise, if .br+l+i = 0 then .al+i can be freely chosen; (b) if .br+i /= 0 for a fixed .i ∈ N, then .akl+r+i = 0 for some integer k, .1 ≤ k ≤ n. Otherwise, if .br+i = 0 then for all .1 ≤ k ≤ n, .akl+r+i can be freely chosen. Proof We compute A2l x = Al (a1+1 xl+1 , al+2 xl+2 , . . .) = (al+1 a2l+1 x2l+1 , al+2 a2l+2 x2l+2 , . . .).

.

Inductively, we have Anl x = (al+1 a2l+1 · . . . · anl+1 xnl+1 , al+2 a2l+2 · . . . · anl+2 xnl+2 , . . .).

.

(9.17)

9 Algebraic Properties of Representations

401

Also Al (Br x) = (al+1 br+l+1 xr+l+1 , al+2 br+l+2 xr+l+2 , . . .),

.

δBr (Anl x) = δ(br+1 al+r+1 · . . . · anl+r+1 xnl+r+1 , br+2 al+r+2 · . . . · anl+r+2 xnl+r+2 , . . .). Thus, .Al Br x = δBr Anl x for all .x ∈ lp , .1 ≤ p ≤ ∞ if and only if one of the following holds: 1. if .r + l + 1 = r + nl + 1, that is, .n = 1 (since .l /= 0) then for all .i ∈ N we have al+i br+l+i = δal+r+i br+i .

(9.18)

.

r+l+i . If .br+i /= 0, that is, .r + i ∈ Jb , then Eq. (9.18) is equivalent to .al+r+i = al+iδbbr+i If .br+i = 0, that is, .r + i /∈ Jb , then Eq. (9.18) reduces to .al+i br+l+i = 0 which is equivalent to .al+i = 0, if .br+l+i /= 0. Otherwise, if .br+l+i = 0 then Eq. (9.18) is satisfied, hence .al+i and .ar+l+i can be freely chosen. 2. if .r + l + 1 /= r + nl + 1, that is, .n /= 1 (since .l /= 0 ) then for all .i ∈ N we have

0 = al+i br+l+i

.

0 = br+i a1+r+i a2l+r+i · . . . · anl+r+i . If .br+l+i /= 0, that is, .(r + l + i ∈ Jb ), then .al+i br+l+i = 0 if and only if .al+i = 0. Otherwise, equation .0 = al+i br+l+i is satisfied and .al+i can be chosen freely. If n n || || .br+i /= 0, that is, .(r +i ∈ Jb ), then .br+i · akl+r+i = 0 reduces to . akl+r+i = 0, where .

n ||

k=1

k=1

αk = α1 · α2 · . . . · αn . This is equivalent to .akl+r+i = 0 for some integer

k=1

k, .1 ≤ k ≤ n. If .br+i = 0 then for all .1 ≤ k ≤ n, .akl+r+i can be freely chosen.

u n

Example 9 Let .A1 : lp → lp , .B1 : lp → lp , .1 ≤ p ≤ ∞ defined as follows A1 x = (a2 x2 , a3 x3 , . . .),

.

B1 x = (b2 x2 , b3 x3 , . . .)

where .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , 0, b4 , 0, b6 , 0, . . .) ∈ l∞ . Then .A1 B1 x = B1 A21 x if .b2k /= 0 and .a2k+1 = 0 for all positive integers k. Indeed this by Theorem 3 when .r = l = 1 and .n = 2. Theorem 4 Let .Al : lp → lp , .Br : lp → lp , .1 ≤ p ≤ ∞ defined as follows Al x = (0, 0, . . . , 0, al+1 x1 , al+2 x2 , . . .), ' '' '

.

l

Br x = (0, 0, . . . , 0, br+1 x1 , br+2 x2 , . . .), ' '' ' r

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D. Djinja et al.

where .r = 1, 2, 3, . . ., .l = 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . If .b = (b1 , b2 , . . .) is given, then for some .n ∈ N and .δ ∈ R \ {0} we have n .Al Br = δBr A if and only if one of the following holds: l 1. if .n = 1 then, (a) if .br+i /= 0 for a fixed .i ∈ N, then ar+l+i =

.

δbr+l+i al+i . br+i

(b) if .br+i = 0 and .br+l+i /= 0 for a fixed .i ∈ N, then .al+i = 0; (c) if .br+i = 0 and .br+l+i = 0 for a fixed .i ∈ N, then .al+i and .ar+l+i can be freely chosen. 2. if .n /= 1 then (a) if .br+i /= 0 for a fixed .i ∈ N, then .ar+l+i = 0. Otherwise, if .br+i = 0 then .ar+l+i can be freely chosen. (b) if .br+nl+i /= 0 for a fixed .i ∈ N, then .akl+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, .br+nl+i = 0 then for all .1 ≤ k ≤ n, .akl+i can be freely chosen. Proof We compute A2l x = Al (Al x) = (0, 0, . . . , 0, al+1 x1 , al+2 x2 , . . .) ' '' '

.

2l

Inductively, we have Anl x = (0, 0, . . . , 0, ' '' '

n ||

.

nl

alk+1 x1 ,

k=1

n ||

(9.19)

alk+2 x2 , . . .)

k=1

and Al (Br x) = (0, 0, . . . , 0, ar+l+1 br+1 x1 , ar+l+2 br+2 x2 , . . .), ' '' '

.

r+l

Br (Anl x) = (0, 0, . . . , 0, br+nl+1 ' '' ' r+nl

n ||

akl+1 x1 , br+nl+2

k=1

n ||

akl+2 x2 , . . .).

k=1

It follows that, .Al Br x = δBl Anr x for all .x ∈ lp , .1 ≤ p ≤ ∞ if and only if one of the following holds: 1. if .r + l = r + nl, that is, .n = 1 (since .l /= 0) then for all .i ∈ N we have ar+l+i br+i = δbr+l+i al+i .

.

9 Algebraic Properties of Representations

403

We consider the following cases: . If .r + i ∈ Jb , that is, .br+i /= 0 for a fixed .i ∈ N, then ar+l+i =

.

δbr+l+i al+i . br+i

. If .r + i /∈ Jb , that is, .br+i = 0 for a fixed .i ∈ N, then equation ar+l+i br+i = δbr+l+i+ al+i

.

reduces to .br+l+i al+i = 0. If .br+l+i /= 0 then .br+l+i al+i = 0 if and only if al+i = 0. Otherwise if .br+l+i = 0 then .0 = br+l+i+ al+i is satisfied, hence .al+i and .ar+l+i can be freely chosen.

.

2. if .r + l /= r + nl, that is, .n /= 1 (since .l /= 0 ) then for all .i ∈ N we have 0 = ar+l+i br+i

.

0 = δbr+nl+i+

n ||

akl+i .

k=1

We consider the following cases: . if .r + i ∈ Jb , where .Jb is defined in (9.16), then .0 = ar+l+i br+i if and only if .ar+l+i = 0. Otherwise, the equation .0 = ar+l+i br+i is satisfied, hence .ar+l+i can be freely chosen. n || . if .r + nl + i ∈ Jb then .0 = δbr+nl+i akl+i if and only if .akl+i = 0 for some k=1

integer k such that .1 ≤ k ≤ n. Otherwise, the equation .0 = δbr+nl+i satisfied, hence for all .1 ≤ k ≤ n, .akl+i can be freely chosen.

n ||

akl+i is

k=1

u n

Corollary 7 Let .Ar : lp → lp , .Br : lp → lp , .1 ≤ p ≤ ∞ defined as follows Ar x = (0, 0, . . . , 0, ar+1 x1 , ar+2 x2 , . . .), ' '' '

.

r

Br x = (0, 0, . . . , 0, br+1 x1 , br+2 x2 , . . .), ' '' ' r

where .r = 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . If .b = (b1 , b2 , . . .) is given, then for some .n ∈ N and some .δ ∈ R \ {0} we have n .Ar Br = δBr Ar if and only if one of the following holds: 1. if .n = 1 then, (a) if .br+i /= 0 for a fixed .i ∈ N, then

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D. Djinja et al.

a2r+i =

.

δb2r+i ar+i ; br+i

(b) if .br+i = 0 and .b2r+i /= 0 for a fixed .i ∈ N, then .ar+i = 0; (c) if .br+i = 0 and .b2r+i = 0 for a fixed .i ∈ N then .ar+i and .a2r+i can be freely chosen. 2. if .n /= 1 then (a) if .br+i /= 0 for a fixed .i ∈ N, then .a2r+i = 0. Otherwise, if .br+i = 0 for a fixed .i ∈ N, then .a2r+i can be freely chosen; (b) if .b(1+n)r+i /= 0 for a fixed .i ∈ N, then .akr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, .b(1+n)r+i = 0 for a fixed .i ∈ N, then for all .1 ≤ k ≤ n, .akr+i can be freely chosen. Proof This follows by Theorem 4 when .r = l.

u n

Example 10 Let .A1 : lp → lp , .B1 : lp → lp , .1 ≤ p ≤ ∞ defined as follows A1 x = (0, a2 x1 , a3 x2 , . . .),

.

B1 x = (0, b2 x1 , b3 x2 , . . .)

where .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , 0, b4 , 0, b6 , 0, . . .) ∈ l∞ . Then .A1 B1 x = B1 A21 x if .b2k /= 0 and .a2k+1 = 0 for all positive integers k. Indeed this follows by Corollary 7 when .n = 2. Theorem 5 Let .Al : lp → lp , .Br : lp → lp , .1 ≤ p ≤ ∞ defined as follows Al x = (al+1 xl+1 , al+2 xl+2 , . . .),

.

Br x = (0, 0, . . . , 0, br+1 x1 , br+2 x2 , . . .), ' '' ' r

where .r = 0, 1, 2, 3, . . ., .l = 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . If .b = (b1 , b2 , . . .) is given, then for some .n ∈ N, and some .δ ∈ R \ {0} we have n .Al Br = δBr A if and only if one of the following holds: l 1. if .r > l then (a) if .r = nl then, if .br+l+i /= 0 for a fixed .i ∈ N such that .1 ≤ i ≤ l then

ar+l+i =

.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

δbr+i

n ||

0, if 1 ≤ i ≤ l, akl+i

k=1

br+l+i

, if i ≥ l + 1;

if .br+i /= 0 for a fixed .i ∈ N such that .1 ≤ i ≤ l then .ar+i = 0. Otherwise, if .br+i = 0 then .ar+i can be freely chosen; if .br+l+i = 0 and .br+i /= 0 for a fixed .i ∈ N such that .i ≥ l + 1 then .akl+i = 0 for some integer k such that .1 ≤ k ≤ n; if .br+l+i = 0 and .br+i /= 0 for a fixed .i ∈ N then for all .1 ≤ k ≤ n, .akl+i can be freely chosen.

9 Algebraic Properties of Representations

405

(b) if .r /= nl then .br+i /= 0 implies .ar+i = 0 and .akl+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, if .br+i = 0 then for all .1 ≤ k ≤ n, .akl+i = 0 and .ar+i = 0 can be freely chosen. 2. if .l = r then (a) if .n = 1 then, if .bl+i /= 0 for a fixed integer i such that .1 ≤ i ≤ l then .al+i = 0. Otherwise, if .bl+i /= 0 for a fixed integer i such that .1 ≤ i ≤ l then .al+i can be freely chosen; if .b2l+i /= 0 for a fixed positive integer i then { a2l+i =

.

0, if 1 ≤ i ≤ l, if i ≥ l + 1.

δal+i ·bl+i b2l+i ,

if .bl+i /= 0 and .b2l+i = 0 for a fixed positive integer i such that .i ≥ l + 1 then .al+i = 0. if .bl+i = 0 and .b2l+i = 0 for a fixed positive integer i then .al+i and .a2l+i can be freely chosen. (b) if .n /= 1 then, if .bl+i /= 0 for a fixed positive integer i then .al+i = 0. Otherwise, if .bl+i = 0 then .al+i can be freely chosen. 3. if .r < l then (a) if .n = 1 then, if .br+i /= 0 for a fixed positive integer i such that .1 ≤ i ≤ r then .ar+i = 0. Otherwise, if .br+i = 0 then .ar+i can be freely chosen; if δal+i br+i .b2r+i /= 0 for a fixed positive integer i then .a2r+i = b2r+i ; if .br+i /= 0 and .b2r+i = 0 for a fixed positive integer i such that .i ≥ r +1 then .al+i = 0; if .br+i = 0 and .b2r+i = 0 then .al+i and .a2r+i can be freely chosen. (b) if .n /= 1 then, if .br+i /= 0 for a fixed positive integer i then .ar+i = 0 and .akl+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, if .br+i = 0 then for all .1 ≤ k ≤ n, .akl+i and .ar+i can be freely chosen. Proof We compute Al Br x = Al (0, 0, . . . , 0, br+1 x1 , br+2 x2 , . . .) = ' '' '

.

r

⎧ ⎪ (0, 0, . . . , 0, ar+1 br+1 x1 , ar+2 br+2 x2 , . . .) if r > l ⎪ ⎪ ' '' ' ⎪ ⎪ ⎨ r−l = (br+1 ar+1 x1 , br+2 ar+2 x2 , . . .) if r = l ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a b x if r < l. l+1 l+1 l−r+1 , al+2 bl+2 xl−r+2 , . . .) From (9.17) we have Anl x = (al+1 a2l+1 · . . . · anl+1 xnl+1 , al+2 a2l+2 · . . . · anl+2 xnl+2 , . . .).

.

So, we compute

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δBr Anl x = δBr (al+1 xl+1 , al+2 xl+2 , . . .)

.

= δ(0, . . . , 0, al+1 · a2l+1 · . . . · anl+1 · br+1 xnl+1 , . . .). ' '' ' r

Thus, .Al Br x = δBr Anl x for all .x ∈ lp , .1 ≤ p ≤ ∞ if and only if one of the following holds: 1. if .r > l then, if .l + 1 + (r − l) = nl + 1, that is, .r = nl then we have ar+i br+i = 0, i = 1, 2, . . . , l

.

ar+l+i br+l+i = δbr+i

n ||

akl+i , ∀i ∈ N.

(9.20)

k=1

|| where . nk=1 αk = α1 · α2 · . . . · αn . We split this into the following cases: . if .r + i ∈ Jb , that is, .br+i /= 0 for a fixed .i ∈ N such that .1 ≤ i ≤ l then .ar+i br+i = 0 if and only if .ar+i = 0. If .r + i /∈ Jb then equation .ar+i br+i = 0 is fulfilled, hence .ar+i can be freely chosen. . if .i ∈ N such that .1 ≤ i ≤ l we substitute .ar+i br+i = 0 in Eq. (9.20), since .r = nl, so this reduces to .ar+l+i br+l+i = 0. Then, if .r + l + i ∈ Jb , that is, .br+l+i /= 0, then .ar+l+i br+l+i = 0 if and only if .ar+l+i = 0. If .r + l + i /∈ Jb , then equation .ar+l+i br+l+i = 0 is satisfied. . if .r + l + i ∈ Jb for a fixed .i ∈ N such that .i ≥ l + 1 then Eq. (9.20) holds if and only if ar+l+i = δ

.

br+i

||n

k=1 akl+i

br+l+i

.

If .r + l + i /∈ Jb for a fixed .i ∈ N such that .i ≥ l + 1 then Eq. (9.20) reduces to br+i al+i a2l+i · . . . · anl+i = 0 which holds true if and only .akl+i = 0 for some integer k such that .1 ≤ k ≤ n, if .r + i ∈ Jb . Otherwise, Eq. (9.20) is fulfilled, hence for all .1 ≤ k ≤ n, .akl+i and .ar+l+i can be freely chosen.

.

If .r /= nl then for all .i ∈ N we have ar+i br+i = 0 and δbr+i al+i a2l+i · . . . · anl+i = 0.

.

If .r + i ∈ Jb for a fixed .i ∈ N then .ar+i br+i = 0 and .δbr+i al+i a2l+i · . . . · anl+i = 0 if and only if .ar+i = 0 and .akl+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise equations .ar+i br+i = 0 and .δbr+i al+i a2l+i · . . . · anl+i = 0 are fulfilled, hence for all .1 ≤ k ≤ n, .akl+i and .ar+i can be freely chosen. 2. If .r = l then, if .nl + 1 = l + 1, that is, .n = 1 (since .l /= 0) then we have

9 Algebraic Properties of Representations

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al+i bl+i = 0, i = 1, 2, . . . , l

.

a2l+i b2l+i = δal+i bl+i , ∀i ∈ N. Therefore, we consider the following cases: . if .bl+i /= 0 for a fixed .i ∈ N such that .1 ≤ i ≤ n then .al+i bl+i = 0 if and only if .al+i = 0. Otherwise, if .bl+i = 0 for a fixed .i ∈ N such that .1 ≤ i ≤ l then equation .al+i bl+i = 0 is satisfied, hence .al+i can be freely chosen. . if .i ∈ N such that .1 ≤ i ≤ n then from .al+i bl+i = 0 follows that equation .a2l+i b2l+i = δal+i bl+i reduces to .a2l+i b2l+i which is equivalent to .a2l+i = 0 if .b2l+i /= 0; . if .i ∈ N such that .i ≥ l + 1 and .b2r+i /= 0 then equation .a2l+i b2l+i = δal+i bl+i is equivalent to a2l+i =

.

δal+i bl+i . b2l+i

If .b2r+i = 0 then equation .a2l+i b2l+i = δal+i bl+i reduces to equation al+i bl+i = 0 which is equivalent to .al+i = 0 if .bl+i /= 0 for a fixed .i ∈ N such that .i ≥ l + 1. Otherwise, equation .al+i bl+i = 0 is satisfied, hence .al+i and .a2l+i can be freely chosen. .

If .n /= 1 then for all .i ∈ N we have al+i bl+i = 0 and δbl+i al+i a2l+i · . . . · anl+i = 0.

.

This reduces to the equation .al+i bl+i = 0 for all .i ∈ N. Hence, if .bl+i /= 0 then al+i bl+i = 0 if and only if .al+i = 0. Otherwise equation .al+i bl+i = 0 is satisfied, hence .al+i can be freely chosen. 3. If .r < l then, if .l − r + r + 1 = nl + 1, that is, .n = 1 (since .l /= 0) then

.

ar+i br+i = 0, i = 1, 2, . . . , r

.

a2r+i b2r+i = δal+i br+i , ∀ i ∈ N.

(9.21)

We consider the following cases: . if .r + i ∈ Jb for a fixed integer i such that .1 ≤ i ≤ r then .ar+i br+i = 0 if and only if .ar+i = 0. Otherwise, if .r + i /∈ Jb for a fixed integer i such that .1 ≤ i ≤ r then equation .ar+i br+i = 0 is satisfied, hence .ar+i can be freely chosen. . if .2r + i ∈ Jb for a fixed integer i such that .1 ≤ i ≤ r then (9.21) holds if and br+i only if .a2r+i = δabl+i . Otherwise, if .2r + i /∈ Jb for a fixed integer i such 2r+i that .1 ≤ i ≤ r and Eq. (9.21) reduces to .δal+i br+i = 0 which is equivalent to .al+i = 0 if .br+i /= 0. Otherwise, if .br+i = 0 and .b2r+i = 0 for a fixed integer i such that .1 ≤ i ≤ r then equation .δal+i br+i = 0 is satisfied, hence .al+i and .a2r+i can be freely chosen.

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. if .2r + i ∈ Jb for a fixed integer i such that .i ≥ r + 1 then Eq. (9.21) is equivalent to a2r+i = δ

.

al+i br+i . b2r+i

Otherwise, that is, if .2r + i /∈ Jb for a fixed integer i such that .i ≥ r + 1 then (9.21) reduces to δal+i br+i = 0

.

which is equivalent to .al+i = 0 if .r + i ∈ Jb . Otherwise, equation .δal+i br+i = 0 is satisfied, hence .al+i and .a2r+i can be freely chosen. if .n /= 1 then for all .i ∈ N we have ar+i br+i = 0 and δal+i · a2l+i · . . . · anl+i · br+i = 0.

.

If .r + i ∈ Jb then .ar+i br+i = 0 and .δal+i · a2l+i · . . . · anl+i · br+i = 0 if and only if ar+i = 0 and .akl+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, if .r +i /∈ Jb then .br+i = 0 so equations .ar+i br+i = 0 and .δal+i · a2l+i · . . . · anl+i · br+i = 0 are fulfilled. Hence for all .1 ≤ k ≤ n, .akl+i and .ar+i can be freely chosen. u n

.

Example 11 Let .A1 : lp → lp , .B1 : lp → lp , .1 ≤ p ≤ ∞ defined as follows A1 x = (a2 x2 , a3 x3 , . . .),

.

B1 x = (0, b2 x1 , b3 x2 , . . .)

where .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , 0, b4 , 0, b6 , 0, . . .) ∈ l∞ . Then .A1 B1 x = B1 A21 x if .b2k /= 0 and .a2k = 0 for all positive integers k. Indeed this follows by Theorem 5 when .r = l = 1 and .n = 2. Theorem 6 Let .Ar : lp → lp , .Bl : lp → lp , .1 ≤ p ≤ ∞ defined as follows Ar x = (0, 0, . . . , 0, ar+1 x1 , ar+2 x2 , . . .), ' '' '

.

Bl x = (bl+1 xl+1 , bl+2 xl+2 , . . .)

r

where .r = 1, 2, 3, . . ., .l = 0, 1, 2, 3, . . ., .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . If .b = (b1 , b2 , . . . ) is given, then for some .n ∈ N and some .δ ∈ R \ {0} we have n .Ar Bl = δBl Ar if and only if one of the following holds: 1. if .nr > l then (a) if .nr − l = r and .l = 0, that is, .n = 1, then we have for a fixed .i ∈ N, .ar+i = 0 when .δbr+i /= bi . Otherwise, if .δbr+i = bi then .ar+i can be freely chosen. (b) if .nr − l = r and .l /= 0, that is, .n /= 1 then, if .bl+i /= 0 for a fixed .i ∈ N then .ar+i = 0. Moreover, if .bl+i = 0 and .bnr+i /= 0 for a fixed .i ∈ N then .akr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, if .bl+i = 0

9 Algebraic Properties of Representations

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and .bnr+i = 0 for a fixed .i ∈ N then for all .1 ≤ k ≤ n .akr+i can be freely chosen. (c) if .nr − l < r then, if .n = 1 then if .br+i /= 0 for a fixed integer i such that .1 ≤ i ≤ l, then .ar+i = 0. Otherwise, if .br+i = 0 for a fixed integer i such that .1 ≤ i ≤ l, then .ar+i can be freely chosen. If .br+l+i /= 0 for a fixed positive integer i then ar+l+i =

.

bl+i ar+i . δbr+l+i

Otherwise, if .br+l+i = 0 and .bl+i /= 0 for a fixed positive integer i then ar+i = 0; Otherwise, if .br+l+i = 0 and .bl+i = 0 for a fixed positive integer i then .ar+i and .ar+l+i can be freely chosen. (d) if .n /= 1 then, if .bl+i /= 0 for a fixed .i ∈ N then .ar+i = 0. If .bl+i = 0 and .bnr+i /= 0 for a fixed .i ∈ N then .akr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, if .bl+i = 0 and .bnr+i = 0 for a fixed .i ∈ N, then for all .1 ≤ k ≤ n, .akr+i can be freely chosen. (e) if .nr − l > r then, if .bl+i /= 0 for a fixed .i ∈ N, then .ar+i = 0. Otherwise, if .bl+i = 0 and .bnr+i /= 0 for a fixed .i ∈ N, then .akr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, if .bl+i = 0 and .bnr+i = 0 for a fixed .i ∈ N, then for all .1 ≤ k ≤ n, .akr+i can be freely chosen. .

2. if .nr = l then (a) if .n = 1 then, if .br+i /= 0 for a fixed integer i such that .1 ≤ i ≤ r, then .ar+i = 0. Otherwise, if .br+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, then .ar+i can be freely chosen. Moreover, if .b2r+i /= 0 for a fixed positive integer i then { a2r+i =

.

0, if 1 ≤ i ≤ r if i > r;

ar+i br+i δb2r+i ,

Otherwise, if .b2r+i = 0 and .br+i /= 0 for a fixed integer i such that .i ≥ r + 1 then .ar+i = 0. Otherwise, if .b2r+i = 0 and .br+i = 0 for a fixed integer i such that .i ≥ r + 1 then .ar+i and .a2r+i can be freely chosen. (b) if .n /= 1 then, if .bl+i /= 0 for a fixed .i ∈ N, then .ar+i = 0. Otherwise, if .bl+i /= 0 for a fixed .i ∈ N, then for each integer k such that .1 ≤ k ≤ n, .akr+i can be freely chosen. 3. if .nr < l then (a) if .n = 1 then, if .bl+i /= 0 for a fixed integer i such that .1 ≤ i ≤ r, then .al+i = 0. Otherwise, if .bl+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, then .al+i can be freely chosen. Moreover, if .br+l+i /= 0 for a fixed positive integer i then

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ar+l+i =

.

ar+i bl+i . δbr+l+i

Otherwise, if .br+l+i = 0 and .bl+i /= 0 for a fixed positive integer i then ar+i = 0. Otherwise, if .br+l+i = 0 and .bl+i = 0 for a fixed positive integer i then .ar+i and .ar+i can be freely chosen. (b) if .n /= 1 and if for a fixed .i ∈ N such that .bl+i /= 0, then either .ar+i = 0 or .akr+l−nr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise if, .bl+i = 0 for a fixed .i ∈ N then for all integer k such that .1 ≤ k ≤ n, .akr+l−nr+i and .ar+i can be freely chosen. .

Proof We compute Ar Bl x = Ar (bl+1 xl+1 , bl+2 xl+2 , . . .)

.

= (0, 0, . . . , 0, ar+1 bl+1 xl+1 , ar+2 bl+2 xl+2 , . . .). ' '' ' r

We have Anr x = (0, 0, . . . , 0, ' '' '

n ||

.

nr

akr+1 x1 ,

k=1

n ||

akr+2 x2 , . . .),

k=1

|| where . nk=1 αk = α1 ·α2 ·. . .·αn with the usual multiplication in .R. So, we compute δBl Anr x = δB.l (0, 0, . . . , 0, ' '' ' nr

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

n || k=1

akr+1 x1 ,

n ||

akr+2 x2 , . . .) =

k=1

|| δ(0, . . . , 0, nk=1 akr+1 · bl+nr−l+1 x1 , . . .), if nr > l ' '' ' || nr−l || = δ( nk=1 akr+1 · bl+1 x1 , nk=1 akr+2 · bl+2 x2 , . . .), if nr = l || ⎪ ⎪ ⎪ δ(bl+1 nk=1 akr+l−nr+1 xl−nr+1 , ⎪ ⎪ || ⎩ if nr < l. bl+1 nk=1 akr+l−nr+2 xl−nr+2 . . .), Thus, .Al Br x = δBr Anl x for all .x ∈ lp , .1 ≤ p ≤ ∞ if and only if one of the following holds: 1. if .nr > l then, if .nr − l = r and .l + 1 = 1, that is, .n = 1 and .l = 0, then for all .i ∈ N we have .δar+i br+i = ar+i bi . This is equivalent to .ar+i = 0 if .δbr+i /= bi for a fixed .i ∈ N. Otherwise, if .δbr+i = bi for a fixed .i ∈ N, then equation .δar+i br+i = ar+i bi is satisfied, hence .ar+i can be freely chosen. If .nr − l = r and .l /= 0 then for all .i ∈ N we have δ

n ||

.

k=1

akr+i bnr+i = 0 and ar+i bl+i = 0.

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If .bl+i /= 0 for a fixed .i ∈ N then .ar+i bl+i = 0 if and only if .ar+i = 0 and so n || akl+i bnr+i = 0 is satisfied. Otherwise, if .bl+i = 0 and .bnr+i /= 0, equation .δ k=1

for a fixed .i ∈ N, then equation .ar+i bl+i = 0 is satisfied and .δ

n ||

akl+i bnr+i = 0

k=1

is equivalent to .akr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, that is, if .bl+i = 0 and .bnr+i = 0 for a fixed .i ∈ N, then both equations .ar+i bl+i = 0 n || and .δ akr+i bnr+i = 0 are satisfied, hence for all .1 ≤ k ≤ n, .akr+i can be freely k=1

chosen. If .nr − l < r then, if .r − nr + l = l, that is, .n = 1 (since .r /= 0), then δar+i br+i = 0,

.

i = 1, 2, . . . , l

δar+l+i br+l+i = bl+i ar+i ,

∀ i ∈ N.

(9.22)

We consider the following cases: . if .br+i /= 0 for a fixed integer i such that .1 ≤ i ≤ l, then since .δ /= 0, .δar+i br+i = 0 if and only if .ar+i = 0. Otherwise, that is, .br+i = 0 for a fixed integer i such that .1 ≤ i ≤ l, hence equation .δar+i br+i = 0 is satisfied. . If .br+l+i /= 0 for a fixed positive integer i then Eq. (9.22) is equivalent to ar+l+i =

.

bl+i ar+i ; δbr+l+i

. if .br+l+i = 0 for a fixed positive integer i such that .1 ≤ i ≤ l then Eq. (9.22) reduces to .ar+i bl+i = 0. If .br+i /= 0, then the system of equations have solution if and only if .ar+i = 0 and so equation .ar+i bl+i = 0 is satisfied. If .br+i = 0 and .bl+i /= 0 for a fixed integer i such that .1 ≤ i ≤ l then .ar+i bl+i = 0 if and only if .ar+i = 0. Otherwise, the equation .ar+i bl+i = 0 is satisfied. . if .br+l+i = 0 for a fixed integer i such that .i ≥ l + 1, then Eq. (9.22) reduces to .ar+i bl+i = 0. If .bl+i /= 0 for a fixed positive integer .i ≥ l + 1 then .ar+i bl+i = 0 is equivalent to .ar+i = 0. Otherwise, that is, .bl+i = 0 for a fixed positive integer .i ≥ l + 1, equation .ar+i bl+i = 0 is satisfied, hence .ar+i and .ar+l+i can be freely chosen. If .r − nr + l /= l, that is, .n /= 1 then for all .i ∈ N we have δ

n ||

.

akr+i bnr+i = 0 and ar+i bl+i = 0.

k=1

If .bl+i /= 0 then .ar+i bl+i = 0 if and only if .ar+i = 0 and so equation n || .δ akr+i bnr+i = 0 is satisfied. Otherwise, equation .ar+i bl+i = 0 is satisfied and k=1

if .bnr+i /= 0, for a fixed .i ∈ N, then since .δ /= 0, .δ

n || k=1

akr+i bnr+i = 0 if and only if

akr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, that is, .bnr+i = 0 for

.

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some positive integer i then equation .δ

n ||

akr+i bnr+i = 0 is satisfied, hence for all

k=1

1 ≤ k ≤ n, .akr+i can be freely chosen. If .nr − l > r then for all .i ∈ N we have

.

n ||

δ

akr+i bnr+i = 0 and ar+i bl+i = 0.

.

k=1

Analogously, this is equivalent to .ar+i = 0 if .bl+i /= 0. And if .bl+i = 0 and .bnr+i /= 0 for a fixed .i ∈ N then .akr+i = 0 for some integer k such that .1 ≤ k ≤ n. And if n || .bl+i = 0 and .bnr+i = 0 for a fixed .i ∈ N then both equations .δ akr+i bnr+i = 0 k=1

and .ar+i bl+i = 0 are satisfied, hence for all .1 ≤ k ≤ n, .akr+i can be freely chosen. 2. if .nr = l then, if .n = 1, that is, .l = r then ar+i br+i = 0,

.

i = 1, 2, . . . , r.

δa2r+i b2r+i = ar+i br+i ,

∀ i ∈ N.

We consider the following cases: . if .br+i /= 0 for a fixed integer i such that .1 ≤ i ≤ r, then .ar+i br+i = 0 if and only if .ar+i = 0. Otherwise, that is, .br+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, equation .ar+i br+i = 0 is satisfied, hence .ar+i can be freely chosen; . if .b2r+i /= 0 for a fixed positive integer i then equation .δa2r+i b2r+i = ar+i br+i is equivalent to a2r+i =

.

ar+i br+i , δb2r+i

since .δ /= 0; . if .b2r+i = 0 for a fixed positive integer i then equation .δa2r+i b2r+i = ar+i br+i reduces to .ar+i br+i = 0 which is equivalent to .ar+i = 0 if .br+i /= 0. If .br+i = 0 equation .ar+i br+i = 0 is satisfied, hence .ar+i and .a2r+i can be freely chosen. If .n /= 1, that is, .l /= r then for all .i ∈ N we have δ

n ||

.

akr+i bnr+i = 0 and ar+i bl+i = 0.

k=1

Since .δ /= 0 and .l = rn, this reduces to .ar+i bl+i = 0 for all .i ∈ N. Thus, if .bl+i /= 0 for a fixed .i ∈ N then .ar+i bl+i = 0 if and only if .ar+i = 0. Otherwise, equation .ar+i bl+i = 0 is satisfied, hence for all .1 ≤ k ≤ n, .akr+i can be freely chosen. 3. if .nr < l then, if .l − nr + r + 1 = l + 1, that is, .n = 1 then δbl+i al+i = 0,

.

i = 1, 2, . . . , r

δbl+r+i al+r+i = bl+i ar+i ,

∀ i ∈ N.

9 Algebraic Properties of Representations

413

We consider the following cases: . if .bl+i /= 0 for a fixed integer i such that .1 ≤ i ≤ r, then .al+i bl+i = 0 if and only .al+i = 0. Otherwise, that is, .bl+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, equation .al+i bl+i = 0 is satisfied, hence .al+i can be freely chosen. . if .br+l+i /= 0 for a fixed positive integer i then equation .δal+r+i bl+r+i = ar+i bl+i is equivalent to ar+l+i =

.

ar+i bl+i , δbr+l+i

since .δ /= 0. . if .br+l+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, then equation δal+r+i bl+r+i = ar+i bl+i

.

reduces to .ar+i bl+i = 0 which is equivalent to .ar+i = 0 if .bl+i /= 0. Otherwise, that is, .bl+i = 0 for a fixed positive integer i, equation is satisfied, hence .ar+l+i and .ar+i can be freely chosen. If .l − nr + 1 + r /= l + 1, that is, .n /= 1, then for all .i ∈ N we have bl+i

n ||

.

akr+l−nr+i = 0 and ar+i bl+i = 0.

(9.23)

k=1

If .b|| l+i /= 0 for a fixed .i ∈ N then the System of Equations (9.23) is equivalent to . nk=1 akr+l−nr+i = 0 and .ar+i = 0. Which is equivalent to either .ar+i = 0 or .akr+l−nr+i = 0 for some integer k such that .1 ≤ k ≤ n. Otherwise, the System of Equations (9.23) is satisfied, hence for all .1 ≤ k ≤ n, .ar+i and .akr+l−nr+i can be freely chosen. u n Corollary 8 Let .Ar : lp → lp , .Br : lp → lp , .1 ≤ p ≤ ∞ defined as follows Ar x = (0, 0, . . . , 0, ar+1 x1 , ar+2 x2 , . . .), ' '' '

.

Br x = (br+1 xr+1 , br+2 xr+2 , . . .)

r

where .r ∈ N, .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , . . .) ∈ l∞ . If .b = (b1 , b2 , . . .) is given, then for some .n ∈ N, .δ ∈ R \ {0} .Ar Br = δBr Anr if and only if one of the following holds: 1. if .n = 1 then (a) if .br+i /= 0 for a fixed integer i such that .1 ≤ i ≤ r, then .ar+i = 0. If .br+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, then .ar+i can be freely chosen; (b) if .b2r+i /= 0 for a fixed positive integer i then

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{ a2r+i =

.

0, ar+i br+i δb2r+i ,

1≤i≤r i > r;

(c) if .b2r+i = 0 and .br+i /= 0 for a fixed integer i such that .i ≥ r + 1 then .ar+i = 0; (d) if .br+i = 0 and .b2r+i = 0 for a fixed positive integer i, then .ar+i and .a2r+i can be freely chosen; 2. if .n /= 1 then (a) if .br+i /= 0 for a fixed .i ∈ N then .ar+i = 0. If .br+i = 0 for a fixed integer i such that .1 ≤ i ≤ r, then .ar+i can be freely chosen; (b) if .br+i = 0 and .bnr+i /= 0 for a fixed .i ∈ N then .akr+i = 0 for some integer k such that .1 ≤ k ≤ n; (c) if .br+i = 0 and .bnr+i = 0 for a fixed positive integer i, then for each integer k such that .1 ≤ k ≤ n, .akr+i can be freely chosen. Proof By applying Theorem 6 when .r = l we have the cases: .nr − r = r, that is, n = 2, .nr − r > r, that is, .n > 1 and .nr = r, that is, .n = 1. The conclusion follows by Theorem 6 items 1b, 1d and 2a. u n

.

Example 12 Let .A1 : lp → lp , .B1 : lp → lp , .1 ≤ p ≤ ∞ defined as follows A1 x = (0, a2 x1 , a3 x2 , . . .),

.

B1 x = (b2 x2 , b3 x3 , . . .)

where .a = (a1 , a2 , . . .) ∈ l∞ , .b = (b1 , b2 , 0, b4 , 0, b6 , 0, . . .) ∈ l∞ . Then .A1 B1 x = B1 A21 x if .b2k /= 0 and .a2k = 0 for all positive integers k. Indeed this follows by Corollary 8 on the case when .n = 2. Acknowledgments This work was supported by the Swedish International Development Cooperation Agency (Sida) bilateral program with Mozambique and International Science Programme (ISP). Alex and Domingos are grateful to the Mathematics and Applied Mathematics research environment MAM, Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalens University for excellent environment for research in Mathematics.

References 1. Adams, M., Gullemin, V.: Measure Theory and Probability. Birkhäuser (1996) 2. Akhiezer, N. I., Glazman, I. M.: Theory of Linear Operators in Hilbert Spaces. Volume I, Pitman Advanced Publ. (1981) 3. Bratteli, O., Evans, D. E., Jorgensen, P. E. T.: Compactly supported wavelets and representations of the Cuntz relations. Appl. Comput. Harmon. Anal. 8(2), 166–196 (2000) 4. Bratteli, O., Jorgensen, P. E. T.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Amer. Math. Soc. 139(663), x+89 (1999)

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5. Bratteli, O., Jorgensen, P. E. T.: Wavelets through a looking glass. The world of the spectrum. Applied and Numerical Harmonic Analysis. Birkhauser Boston, Inc., Boston, MA, xxii+398 (2002) 6. Carlsen, T. M., Silvestrov, S.: C ∗ -crossed products and shift spaces, Expo. Math. 25, no. 4, 275–307 (2007) 7. Carlsen, T. M., Silvestrov, S.: On the Exel crossed product of topological covering maps. Acta Appl. Math. 108, no. 3, 573–583 (2009) 8. Carlsen, T. M., Silvestrov, S.: On the K-theory of the C ∗ -algebra associated with a one-sided shift space. Proc. Est. Acad. Sci. 59, no. 4, 272–279 (2010) 9. Dutkay, D. E., Jorgensen, P. E. T.: Martingales, endomorphisms, and covariant systems of operators in Hilbert space. J. Operator Theory, 58(2), 269–310 (2007) 10. Dutkay, D. E., Jorgensen, P. E. T., Silvestrov, S.: Decomposition of wavelet representations and Martin boundaries. J. Funct. Anal. 262(3), 1043–1061 (2012). (arXiv:1105.3442 [math.FA], 2011) 11. Dutkay, D. E., Larson, D. R., Silvestrov, S: Irreducible wavelet representations and ergodic automorphisms on solenoids. Oper. Matrices 5(2), 201–219 (2011) (arXiv:0910.0870 [math.FA], 2009) 12. Dutkay, D. E., Silvestrov, S.: Reducibility of the wavelet representation associated to the Cantor set. Proc. Amer. Math. Soc. 139(10), 3657–3664 (2011). (arXiv:1008.4349 [math.FA], 2010). 13. Folland, G.: Real Analysis: Modern techniques and their applications. 2nd ed, John Wiley & Sons Inc. (1999) 14. Hutson, V., Pym, J. S., Cloud, M. J.: Applications of Functional Analysis and Operator Theory. 2nd edition, Elsevier (2005) 15. Jorgensen, P. E. T.: Analysis and probability: wavelets, signals, fractals. Graduate Texts in Mathematics, 234. Springer, New York, xlviii+276 (2006) 16. Jorgensen, P. E. T.: Operators and Representation Theory. Canonical Models for Algebras of Operators Arising in Quantum Mechanics. North-Holland Mathematical Studies 147 (Notas de Matemática 120), Elsevier Science Publishers, viii+337 (1988) 17. Jorgensen, P. E. T., Moore, R. T.: Operator Commutation Relations. Commutation Relations for Operators, Semigroups, and Resolvents with Applications to Mathematical Physics and Representations of Lie Groups. Springer Netherlands, xviii+493 (1984) 18. Dutkay, D. E., Silvestrov, S.: Wavelet Representations and Their Commutant. In: Åström, K., Persson, L-E., Silvestrov, S. D. (eds.): Analysis for science, engineering and beyond. Springer proceedings in Mathematics Vol. 6, Springer, Berlin, Heidelberg, Ch. 9, pp 253–265 (2012) 19. de Jeu, M., Svensson, C., Tomiyama, J.: On the Banach ∗-algebra crossed product associated with a topological dynamical system. J. Funct. Anal. 262(11), 4746–4765 (2012) 20. de Jeu, M., Tomiyama, J.: Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system. Studia Math. 208(1), 47–75 (2012) 21. Kantorovitch, L. V., Akilov, G. P.: Functional Analysis. 2nd ed, Pergramond Press Ltd, England (1982) 22. Kolmogorov, A. N., Fomin, S. V.: Elements of the theory of functions and Functional Analysis. 1st vol, Graylock press. (1957) 23. Mackey, G. W.: Induced Representations of Groups and Quantum Mechanics. W. A. Benjamin, New York, Editore Boringhieri, Torino (1968) 24. Mackey, G. W.: The Theory of Unitary Group Representations. University of Chicago Press (1976) 25. Mackey, G. W.: Unitary Group Representations in Physics, Probability, and Number Theory. Addison-Wesley (1989) 26. Mansour, T., Schork, M.: Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press (2016) 27. Musonda, J.: Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators. PhD thesis, Mälardalen University, (2018)

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45. Svensson, C., Silvestrov, S., de Jeu, M.: Dynamical systems associated with crossed products. Acta Appl. Math. 108(3), 547–559 (2009) (Preprints in Mathematical Sciences, Centre for Mathematical Sciences, Lund University 2007:22, LUTFMA-5088–2007; Leiden Mathematical Institute report 2007-30; arXiv:0707.1881 [math.OA]). 46. Svensson, C., Tomiyama, J.: On the commutant of C(X) in C ∗ -crossed products by Z and their representations. J. Funct. Anal. 256(7), 2367–2386 (2009) 47. Tomiyama, J.: Invitation to C ∗ -algebras and topological dynamics. World Scientific (1987) 48. Tomiyama, J.: The interplay between topological dynamics and theory of C ∗ -algebras. Lecture Notes Series, 2, Seoul National University Research Institute of Mathematics, Global Anal. Research Center, Seoul (1992) 49. Tomiyama, J.: The interplay between topological dynamics and theory of C∗ -algebras. II., S¯urikaisekikenky¯usho K¯oky¯uroku (Kyoto Univ.) 1151, 1–71 (2000) 50. Tumwesigye, A. B.: Dynamical Systems and Commutants in Non-Commutative Algebras. PhD thesis, Mälardalen University, (2018) 51. Vaysleb, E. Ye., Samo˘ılenko, Yu. S.: Representations of operator relations by unbounded operators and multi-dimensional dynamical systems. Ukrain. Math. Zh. 42(8), 1011–1019 (1990) (Russian). English transl.: Ukr. Math. J. 42 899–906 (1990)

Chapter 10

Applications of Quasigroups in Cryptography and Coding Theory Smile Markovski, Verica Bakeva, Vesna Dimitrova, Aleksandra Mileva, Aleksandra Popovska-Mitrovikj, and Hristina Mihajloska Trpcheska

10.1 Introduction Quasigroups can be used to design almost all cryptographic products needed in cryptography in order to obtain secure communications. This short survey will present only some of them to obtain the main ideas on how suitable cryptographic primitives can be constructed. In more (but not complete) details, we explain some constructions of S-boxes, block and stream ciphers, hash functions, message authentication codes, authenticated encryption ciphers, public key cryptography and pseudo-random string generators. It is worth mentioning that almost all the primitives explained here are part of some secret-key cryptography competitions with a long tradition. The quasigroups are also suitable for constructing error detecting and error correcting code and their applications in transmitting images and audio signals. We start in Sect. 10.2 with definitions of quasigroup string transformations that are used to transform a given string .α in an alphabet A into a string .β. Usually, sequences of such transformations are used to obtain suitable security. The quasigroup transformations are applied mainly on two types of quasigroups: “small

S. Markovski · V. Bakeva · V. Dimitrova · A. Popovska-Mitrovikj · H. M. Trpcheska (O) Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] A. Mileva Faculty of Computer Science, Goce Delcev University, Stip, North Macedonia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_10

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quasigroups” that are of order 256 or less, and “huge quasigroups” that are of order higher than 256. (For instance, some applications use quasigroups of order .2512 or higher.) There are several ways how to construct a huge quasigroup effectively [40]; one is given in Table 10.4 of Sect. 10.6, where the hash function Edon-.R is described. This section also presents the matrix presentation of quasigroups of order four and the Boolean functions presentation of quasigroups. The paper is organized in 6 sections. Section 10.3 contains several constructions of crypto primitives. There are 8 subsections there. Section 10.3.1 consider two constructions of S-boxes, where quasigroups of order 4 and 16 are used. Three types of hash functions are given in Sect. 10.3.2: Edon-.R, NaSHA and GAGE. A message authentication code is given in Sect. 10.3.3, and a pseudo random generator in Sect. 10.3.4. The block cipher defined by matrix presentation of quasigroup (BCDMPQ), and the stream cipher Edon80, are given in Sects. 10.3.5 and Sect. 10.3.6 correspondingly. Two authenticated encryption with associated data, so called .π -Cipher and InGAGE, are presented in Sect. 10.3.7, while public-key cryptography is discussed in Sect. 10.3.8. Section 10.4 deals with the error-correcting codes RCBQ. It is an error-correcting code resistant to an intruder attack. RCBQ depends on several parameters (pattern of redundancy, length of the key, chosen quasigroup, ...) and all of them are considered. Several methods are presented for improving the efficiency of decoding. Applications of RCBQ in image and audio signal processing are given too. Some error-detecting codes based on quasigroups are given in Sect. 10.5, and Sect. 10.6 is the conclusion section.

10.2 Quasigroups and Quasigroup Transformations Here we give needed definitions of some types of quasigroups we are using in the sequel. Also, so called quasigroup transformations of strings are defined. We deal in this article only with finite quasigroups, so all definitions are for that class of quasigroups.

10.2.1 Notation of Quasigroups Let .Q = {a1 , a2 , . . . , ad } be a finite set of d elements. A quasigroup .(Q, ∗) is a groupoid satisfying the law: for all .a, b ∈ Q, there exist unique .x, y ∈ Q, so that .a ∗ x = b and .y ∗ a = b. This means that every row and every column in the multiplication table of .(Q, ∗) is a permutation of Q. To every finite quasigroup with d elements .(Q, ∗), given by its Cayley table, an equivalent combinatorial structure .d × d Latin square can be associated, consisting of the matrix formed by the interior of the table (an .n × n

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Latin square is made up of n distinct elements, each of them appears exactly once in each row and exactly once in each column). Given a quasigroup .(Q, ∗), its left and right parastrophe operations .\, / are defined by x∗y =z

.

⇔

x \ z = y,

z/y = x.

Then .(Q, \) and .(Q, /) are quasigroups too. A left quasigroup .(Q, ∗) is a groupoid satisfying the law: for all .a, b ∈ Q, there exist unique .x ∈ Q, so that .a ∗ x = b. A right quasigroup .(Q, ∗) is a groupoid satisfying the law: for all .a, b ∈ Q, there exist unique .x ∈ Q, so that .x ∗ a = b. An n-ary groupoid .(n ≥ 1) is an algebra .(Q, f ) on a nonempty set Q as its universe and with one .n−ary operation .f : Qn → Q. An n-ary groupoid .(Q, f ) is an n-ary quasigroup (of order .|Q|) if any n elements of the .a1 , a2 , . . . , an+1 ∈ Q, satisfying the equality f (a1 , a2 , . . . , an ) = an+1 ,

.

uniquely specifies the remaining one [4]. One most desirable property of a quasigroup in cryptography is its shapelessness [49]. This means that the quasigroup .(Q, ∗) should not be associative, commutative, idempotent, not have (left, right) unit, it should not have proper subquasigroups, and it should not satisfy identities of kind x ∗ (x ∗ . . . (x ∗ (x ∗y))) = y .(((y ∗ x) ∗ x) ∗ . . . ) ∗ x = y, ' ' ' '' ' '' k

for some .k < 2n, where .n = |Q|.

10.2.1.1

k

Matrix Representation of Quasigroups of Order 4

All quasigroup operations on the set .Q = {0, 1, 2, 3} have so called matrix representations in the following form, given in the next theorem [62]: Theorem 1 Each quasigroup .(Q, ∗) of order 4 has a matrix representation of form x ∗ y = mT + AxT + ByT + CAxT ◦ CByT ,

.

(10.1)

where .x = (x1 , x2 ), y = (y1 , y2 ) ∈ Q (.xi , yi denotes bit variables), .m = (m1 , m2 ) is some constant from Q, | A|and | B are | |nonsingular | | |2-dimensional matrices of bits, C 00 01 00 11 is one of the matrices . , , , , and .◦ denotes the component00 00 10 11 wise multiplication of vectors. (Note: The addition and multiplication are in the field .GF (2).)

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The matrix presentation of the parastrophe operation .\ of the quasigroup operation .∗ g is the following: x \ z = B −1 mT + B −1 (I + C)AxT + B −1 (CmT ◦ CAxT )+ .

+B −1 zT + B −1 (CAxT ◦ CzT ),

(10.2)

where I denotes the identity | | matrix. 00 If the matrix C is . , then we say that the quasigroup is linear. 00

10.2.1.2

Boolean Functions Presentations of Quasigroups

The elements of a finite quasigroups .(Q, ∗) of order .2d can be represented binary as bit strings of d bits. Now, the binary operation .∗ can be interpreted as a vector valued Boolean operation .∗vv : {0, 1}2d → {0, 1}d defined as: x ∗ y = z ⇔ ∗vv (x1 , . . . , xd , y1 , . . . , yd ) = (z1 , . . . , zd ),

.

where .x1 . . . xd , y1 . . . yd , z1 . . . zd are binary representations of .x, y, z. Each .zi depends of the bits .x1 , x2 , . . . , xd , y1 , y2 , . . . , yd and is uniquely determined by them. So, each .zi can be seen as a 2d-ary Boolean function 2d → {0, 1} strictly depends .zi = fi (x1 , . . . , xd , y1 , . . . , yd ), where .fi : {0, 1} on, and is uniquely determined by, .∗. A k-ary Boolean function .f (x1 , . . . , xk ) can be represented in a unique way by its algebraic normal form (ANF) as a sum of products in the field .GF (2): ANF (f ) =

E

.

αI xI ,

I ⊆{1,2,...,k}

where .αI ∈ {0, 1} and .xI is the product of all variables .xi such that .i ∈ I. The ANFs of the functions .fi give us information about the complexity of the quasigroup .(Q, ∗) via the degrees of the Boolean functions .fi . A quasigroup .(Q, ∗) of order .2n is called Multivariate Quadratic Quasigroup (MQQ) of type .Quadn−k Link if exactly .(n − k) of the polynomials .fi are of degree 2 (i.e., they are quadratic) and k of them are of degree 1 (i.e., they are linear), where .0 ≤ k ≤ n.

10.2.2 Quasigroup Transformations Quasigroup String Transformations were introduced in [33] and were investigated in several other papers [35–38].

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Consider an alphabet (i.e., a finite set) Q, and denote by Q+ = {a1 a2 . . . . . . an | ai ∈ Q}

.

the set of all nonempty words (i.e. finite strings) formed by the elements of Q. (If there is no misunderstanding, we identify .a1 a2 . . . an and .(a1 , a2 , . . . , an ).) Let .∗ be a quasigroup operation on the set Q, i.e. consider a quasigroup .(Q, ∗). For each .a ∈ Q we define two functions .ea,∗ , da,∗ : Q+ −→ Q+ as follows. Let .ai ∈ Q, α = a1 a2 . . . an . Then ea,∗ (α) = b1 b2 . . . bn ⇐⇒ b1 = a ∗ a1 , b2 = b1 ∗ a2 , . . . , bn = bn−1 ∗ an ,

.

da,∗ (α) = c1 c2 . . . cn ⇐⇒ c1 = a ∗ a1 , c2 = a1 ∗ a2 , . . . , cn = an−1 ∗ an .

.

The functions .ea,∗ , da,∗ are called e- and d-transformation of .Q+ based on the operation .∗ with leader a. Example 1 Take .Q = {0, 1, 2, 3} and let the quasigroup .(Q, ∗) and its parastrophes .(Q, \) and .(Q, /) be given by the multiplication schemes in Fig. 10.1. Consider the string .α = 1 0 2 1 0 0 0 0 0 0 0 0 0 1 1 2 1 0 2 2 0 1 0 1 0 3 0 0 and choose the leader 0. We present two consecutive applications of the e-transformation on Table 10.1. After that we apply two times the transformation .d0,\ on the last obtained string 2 .β = e0,∗ (α) (see Table 10.2): Notice that we have obtained . α = d0,\ 2 (β) = d0,\ 2 (e0,∗ 2 (α)) = (d0,\ 2 ◦ e0,∗ 2 )(α). .O In fact, the following property is true: Fig. 10.1 A quasigroup and its parastrophes .(Q, \) and .(Q, /) .(Q, ∗)

Table 10.1 Consecutive e-transformations .Leader

0 0

021000000000112102201010300=α .1 3 2 2 1 3 0 2 1 3 0 2 1 0 1 1 2 1 1 1 3 3 0 1 3 1 3 0 = e0,∗ (α) 2 .1 2 3 2 2 0 2 3 3 1 3 2 2 1 0 1 1 2 2 2 0 3 0 1 2 2 0 2 = e0,∗ (α) .1

Table 10.2 Consecutive d-transformations 232202331322101122203012202=β 2 3 2 2 0 2 3 3 1 3 2 2 1 0 1 1 2 2 2 0 3 0 1 2 2 0 2 = d0,\ (β) 2 .1 0 2 1 0 0 0 0 0 0 0 0 0 1 1 2 1 0 2 2 0 1 0 1 0 3 0 0 = d0,\ (β)

.Leader

.1

0 0

.1

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Theorem 2 Let .(Q, ∗, \, /) be a finite quasigroup. Then for each string .α ∈ Q+ and for each leader .l ∈ Q we have that .el,∗ and .dl,\ are mutually inverse permutations of .Q+ , i.e., .dl,\ (el,∗ (α)) = α = el,∗ (dl,\ (α)). By Theorem 2 we conclude that the transformations .ea,∗ and .da,\ can be used for defining suitable functions for encryption and decryption. We can define in the similar way several pairs of quasigroup string transformations suitable for ' , d' cryptography. Thus, let .a, a1 , . . . , an ∈ Q and define the functions .ea,∗ a,∗ : + + Q −→ Q as follows: ' .ea,∗ (α) = b1 b2 . . . bn ⇐⇒ bn = an ∗ a, bn−1 = an−1 ∗ bn , . . . , b1 = a1 ∗ b2 , ' .da,∗ (α) = c1 c2 . . . cn ⇐⇒ cn = an ∗ a, cn−1 = an−1 ∗ an , . . . , c1 = a1 ∗ a2 . ' , d' Then, Theorem 2 holds for the functions .ea,∗ a,\ too. Also, for encryp' , d ' ), (e , d ), tion/decryption purposes, the pairs of functions .(ea,∗ , da,/ ), (ea,∗ a,∗ a,\ a,/ ' ' .(ea,∗ , d ), can be defined in an obvious way. a,\ There are also some other types of quasigroup transformations. In order to exploit more completely all isotopes of a quasigroup, the so called parastrophic quasigroup transformation is given in [11, 29]. Quasigroup string transformations are also defined on ternary quasigroup [53]. By considering growing the periods of the strings .el,∗ t (α) .(t = 1, 2, 3, . . . ) of a periodic string .α, the class of all quasigroups of order n can be classified in two disjoint classes, the class of exponential and the class of linear quasigroups. A quasigroup is said to be exponential if the period of the string .el,∗ t (α) is down bounded by an exponential function .const · 2at , where const and a are positive constant. We note that for some quasigroups the constant a is big enough, so they can be used to produce suitable crypto primitives. Also, in [10] authors classify the set of all quasigroups of given finite order n into 2 disjoint classes, the class of so called fractal quasigroups, and the class of so called non-fractal quasigroups. A graphical presentation of these classes are defined there too. The class of fractal quasigroup is not recommended to be used for producing cryptographic primitives, but it is convenient for defining error-detecting codes (see Sect. 10.5). The method for obtaining graphical presentation of .e−transformations is the following. Let Q be a quasigroup of given order. If we take a periodical sequence .s = a1 . . . am a1 . . . am . . . a1 . . . am of elements of Q of length t, then by consecutive application of e-transformation k times, we obtain a .k × t matrix with elements from Q. If we treat the elements of Q as a pixels with the corresponding color, then we have images that present that e-transformation for Q. Example 2 For the quasigroups of order 4 with lexicographic numbers 5 and 106 given in Fig. 10.2 and the periodical starting sequence .s = 12341234 . . . 1234 with length .t = 100, leader .l = 4, and .k = 100 times of e-transformations, the corresponding images for the e-transformations obtained by this module are shown on Fig. 10.3.

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Fig. 10.2 Quasigroups with lexicographic numbers 5 and 106

Fig. 10.3 A fractal and a non-fractal images of quasigroups

The left image on this figure is an example of fractal quasigroup and the right image is an example of non-fractal quasigroup.

10.3 Application of Quasigroups in Cryptography 10.3.1 S-boxes Defined by Quasigroups The main point of security in symmetric cryptography in almost all modern block ciphers are the substitution boxes (S-boxes). S-boxes have to confuse the input data into the cipher. Since S-boxes contain a small amount of data, the construction of an S-box should be made very carefully in order the needed cryptographic properties to be satisfied. It is especially important when an ultra-lightweight block cipher is designed, like PRESENT [6]. PRESENT S-boxes are derived as a result of an exhaustive search of all .16! bijective 4-bit S-boxes. Then 16 different classes are obtained and all S-boxes in that classes are optimal with respect to linear and differential properties. Instead of an exhaustive search of all .16! bijections of 16 elements as it was done for the design of PRESENT, quasigroups of order 4 can be applied for construction of cryptographically strong S-boxes, called Q-S-boxes [48]. S-boxes are defined as lookup tables that are interpreted as vector valued Boolean q functions or Boolean maps .f : Fn2 → F2 , where .F2 is a Galois field with two elements. For S-boxes, so-called linear and differential potential can be computed, and according to them, the corresponding resistance against linear and differential attacks can be measured. We already mentioned that quasigroups of order .2n have vector valued representation. For example, the given quasigroup of order 4

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∗ 0 .1 2 3

0 0 1 2 3

1 1 0 3 2

2 3 2 0 1

3 2 3 1 0

has representation with the following pair of Boolean functions f (x0 , x1 , y0 , y1 ) = (x0 + y0 , x1 + y0 + y1 + x0 y0 ).

.

The algebraic degree of this quasigroup is 2, since the Boolean function f2 (x0 , x1 , y0 , y1 ) = x1 + y0 + y1 + x0 y0

.

has degree 2. Generally, the quasigroups of order 4 can have algebraic degree 1 (144 of them, so called linear) and 2 (432 of them, so called nonlinear), [14]. Only nonlinear quasigroups can be used for the construction of suitable S-boxes, i.e., QS-boxes. Note that quasigroups of order 4 are .4 × 2-bit S-boxes. Cryptographically strong .4×4-bit S-boxes can be generated by using quasigroups of order 4. One criterion for a good S-box is to have the highest possible algebraic degree (degree 3 for all output bits), and optimal values for linear and differential potential [30]. To obtain .4 × 4-bit S-boxes, e-transformations will be used to raise the algebraic degree of the final bijections produced. As it is shown in Fig. 10.4, one non-linear quasigroup of order 4 and at least 4 e-transformations will be used to reach the desired degree of 3 for all the bits in the final output block. This methodology can easily construct Q-S-boxes that satisfy the condition of algebraic degree. If the other conditions are satisfied (linear and differential potential), the found Q-S-box can be put in the set of optimal Q-S-boxes. Apparently, by increasing the number of leaders and rounds, the number of optimal Q-S-boxes also increases. Fig. 10.4 Four e-trans that bijectively transforms 4 bits into 4 bits by a quasigroup of order 4

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Table 10.3 Examples of optimal Q-S-boxes given in its hexadecimal notation x S(x)

0 D

1 9

2 F

3 C

4 B

5 5

6 7

7 6

8 3

9 8

A E

B 2

C 0

D 1

E 4

F A

x S(x)

0 5

1 E

2 6

3 D

4 7

5 4

6 2

7 A

8 8

9 C

A 0

B 9

C 1

D B

E F

F 3

Some representative of optimal Q-S-boxes are given in Table 10.3. Similar construction of S-boxes is used in the design of an authenticated encryption scheme in Sect. 10.3.7 (InGAGE AEAD cipher). Mileva et al. [50] presented a new construction of N -bit S-boxes, which is a generalization of the constructions with e quasigroup string transformations. They mixed two different layers alternately—the layer of bijectional quasigroup string transformations, and the layer of modular addition with N -bit constants (see the algorithm bellow). First, the starting quasigroup should be chosen, and it should be of order q, such that .N = w · log2 (q), or in any other case, .q = N and .w = 1. Second, a vector of n bijectional quasigroup string transformations .T = (qst1 , qst2 , . . . , qstn ), which has a corresponding vector of inverse quasigroup string transformations .T −1 = (qst1−1 , qst2−1 , . . . , qstn−1 ), should be chosen, together with the vectors of n leaders .L = (l1 , l2 , . . . ln ) and n N -bit constants −1 , is also .C = (c0 , c1 , c2 , . . . cn ). The algorithm for finding the inverse S-box, .S given in the paper [50]. A lot of experiments are made for producing 8-bit S-boxes from quasigroups of order 4 and 16. An algorithm for a new construction of N -bit S-box Input: Q - quasigroup of order q, vector of n bijectional quasigroup string transformations .T = (qst1 , qst2 , . . . , qstn ), vector of leaders .L = (l1 , l2 , . . . ln ) and vector of N -bit constants .C = (c0 , c1 , c2 , . . . cn ) Output: S For all possible input blocks .x1 , x2 , . . . , xw in lexicographic ordering N .(p1 , p2 , . . . , pw ) = (x1 , x2 , . . . , xw ) + c0 (mod 2 ) For .i = 1 to n If i is odd N .(t1 , t2 , . . . , tw ) = (qsti )li (p1 , p2 , . . . , pw ) + ci (mod 2 ) else .(pw , . . . , p2 , p1 ) = (qsti )li (tw , . . . , t2 , t1 ) N .(p1 , p2 , . . . , pw ) = (p1 , p2 , . . . , pw ) + ci (mod 2 ) Use all output blocks from the last round to generate S

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10.3.2 Hash Functions Hash functions on a set A are mappings .h : A+ → An that take a variablesize input messages and map them into fixed-size output, known as hash result, message digest, hash-code etc. They are used in checking data integrity, digital signature schemes, commitment schemes, password based identification systems, digital timestamping schemes, pseudo-random string generation, key derivation, one-time passwords etc. Here we will explain the construction of the candidates of the NIST SHA-3 competition [52], Edon-.R and NaSHA. Edon-.R, is one of those whose design is based on huge quasigroup transformations. Edon-.R Edon-.R [26] is wide-pipe iterative hash function with standard message digest straightening. It was the fastest First round candidate of NIST SHA-3 competition. The chaining value .Hi and the message input .Mi for the ith round are composed of two q-bits blocks, .q = 256, 512, i.e. .Hi = (Hi1 , Hi2 ) and .Mi = (Mi1 , Mi2 ), and the new chaining value .Hi+1 is produced as follows 1 2 1 2 1 2 .Hi+1 = (H i+1 , Hi+1 ) = R(Hi , Hi , Mi , Mi ), .R is little bit modified reverse string transformation, in a sense that two parts from the message are taken reversed when are used like leaders, and the order of leaders is ¯ 2 , H 1 , H 2 , M¯ 1 . The compression function .R uses two huge quasigroups of order .M i i i i 256 and .2512 . Algorithmic description of the quasigroup of order .2256 is given in .2 Table 10.4. There .Xi , .Yi and .Zi are 32-bit variables, so .X = (X0 , X1 , . . . , X7 ), .Y = (Y0 , Y1 , . . . , Y7 ) and .Z = (Z0 , Z1 , . . . , Z7 ) are 256-bits variables. (Note that the operation is .X ∗ Y = Z.) Operation “+” denotes addition modulo .232 , operation r .⊕ is the logical operation of bitwise exclusive or and the operation .ROT L (Xi ) is the operation of bit rotation of the 32-bit .Xi , to the left for r positions. NaSHA NaSHA [41] is another First round candidate to the NIST SHA-3 competition based on quasigroups . It is also wide-pipe iterative hash function with standard message digest straightening. NaSHA-.(m, k, r) has three parameters .m, k, r, where m denotes message length, k is the number of elementary quasigroup string r transformations of type A and RA, and r is from the order .22 of used quasigroups. Elementary quasigroup additive and reverse additive string transformations .A, RA : Q+ → Q+ with leader l are defined in [41] as follows: A(x1 x2 . . . xt ) = (z1 z2 , . . . zt ) ⇐⇒ zj = (zj −1 + xj ) ∗ xj , 1 ≤ j ≤ t, z0 = l,

.

RA(x1 x2 . . . xt ) = (z1 z2 . . . zt ) ⇐⇒ zj = xj ∗(xj +zj +1), 1 ≤ j ≤ t, zt+1 = l,

.

where .Q = Z2n , .(Q, ∗) is a quasigroup and + denotes addition modulo .2n . These transformations are not bijective mappings. One can create composite quasigroup

10 Applications of Quasigroups in Cryptography and Coding Theory Table 10.4 An algorithmic description of a quasigroup of order .2256 Quasigroup operation of order .2256 Input: .X = (X0 , X1 , . . . , X7 ) and .Y = (Y0 , Y1 , . . . , Y7 ), where .Xi and .Yi are 32-bit variables. Output: .Z = (Z0 , Z1 , . . . , Z7 ) where .Zi are 32-bit variables. Temporary 32-bit variables: .T0 , . . . , T15 . 0 . T0 ← ROT L (0xAAAAAAAA + X0 + X1 + X2 + X4 + X7 ); 4 . T1 ← ROT L (X0 + X1 + X3 + X4 + X7 ); 8 . T2 ← ROT L (X0 + X1 + X4 + X6 + X7 ); 13 .1. T3 ← ROT L (X2 + X3 + X5 + X6 + X7 ); . T4 ← ROT L17 (X1 + X2 + X3 + X5 + X6 ); . T5 ← ROT L22 (X0 + X2 + X3 + X4 + X5 ); 24 . T6 ← ROT L (X0 + X1 + X5 + X6 + X7 ); 29 . T7 ← ROT L (X2 + X3 + X4 + X5 + X6 ); T8 ← T3 ⊕ T5 ⊕ T6 ; T9 ← T2 ⊕ T5 ⊕ T6 ; . T10 ← T2 ⊕ T3 ⊕ T5 ; .2. T11 ← T0 ⊕ T1 ⊕ T4 ; . T12 ← T0 ⊕ T4 ⊕ T7 ; . T13 ← T1 ⊕ T6 ⊕ T7 ; . T14 ← T2 ⊕ T3 ⊕ T4 ; . T15 ← T0 ⊕ T1 ⊕ T7 ; . .

T0 . T1 . T2 .3. T3 . T4 . T5 . T6 . T7

← ROT L0 (0x55555555 + Y0 + Y1 + Y2 + Y5 + Y7 ); ← ROT L5 (Y0 + Y1 + Y3 + Y4 + Y6 ); ← ROT L9 (Y0 + Y1 + Y2 + Y3 + Y5 ); ← ROT L11 (Y2 + Y3 + Y4 + Y6 + Y7 ); ← ROT L15 (Y0 + Y1 + Y3 + Y4 + Y5 ); ← ROT L20 (Y2 + Y4 + Y5 + Y6 + Y7 ); ← ROT L25 (Y1 + Y2 + Y5 + Y6 + Y7 ); ← ROT L27 (Y0 + Y3 + Y4 + Y6 + Y7 );

Z5 Z6 . Z7 .4. Z0 . Z1 . Z2 . Z3 . Z4

← T8 + (T3 ⊕ T4 ⊕ T6 ); ← T9 + (T2 ⊕ T5 ⊕ T7 ); ← T10 + (T4 ⊕ T6 ⊕ T7 ); ← T11 + (T0 ⊕ T1 ⊕ T5 ); ← T12 + (T2 ⊕ T6 ⊕ T7 ); ← T13 + (T0 ⊕ T1 ⊕ T3 ); ← T14 + (T0 ⊕ T3 ⊕ T4 ); ← T15 + (T1 ⊕ T2 ⊕ T5 );

.

. .

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transformations MT by composition of different A and/or RA transformations with different leaders. As a candidate to the competition was sent NaSHA-.(m, 2, 6), where m = 224, 256, 384, 512.

.

Every round consists of one linear transformation obtained from an LFSR, followed by MT quasigroup string transformation, that is a composition of k alternate quasigroup string transformations A and RA. NaSHA uses novel design principle: the quasigroups used in every iteration in compression function are different, and depend on the processed message block. Even in one iteration, different quasigroups are used for two quasigroup transformations. Quasigroups in NaSHA are obtained by using Extended Feistel Networks as orthomorphisms and complete mappings on the groups .(Z216 , ⊕), (Z232 , ⊕) and .(Z264 , ⊕). NaSHA is of order .264 and is produced from known starting bijection of order .28 by using xoring, addition modulo .264 and table lookups. Extended Feistel networks .FA,B,C are defined in [40] as follows: Let .(G, +) be an abelian group, let .f : G → G be a mapping and let .A, B, C ∈ G be constants. The extended Feistel network .FA,B,C : G2 → G2 created by f is defined for every .(l, r) ∈ G2 as FA,B,C (l, r) = (r + A, l + B + f (r + C)).

.

When f is a bijection, .FA,B,C is an orthomorphism of the group .(G2 , +) (i.e., 2 .FA,B,C and .FA,B,C −I are permutations), so a quasigroup .(G , ∗F ) can be produced by Sade’s diagonal method: .X ∗F Y = FA,B,C (X − Y ) + Y.) GAGE GAGE [21] is a family of sponge-based hash functions with states between 232 and 576 bits and rates for injecting message blocks of 8, 16, 32 and 64 bits. The sponge permutation has an SPN structure with one very light .4 × 2-bits s-box and a cheap hardware wiring i.e., bit-shuffling layer. The round permutation, called QPERMUTATION is the main part of this primitive and consists of two parts: a nonlinear substitution part and a bit shuffling part. The nonlinear substitution part uses one .4 × 2-bit S-box that is applied in an interleaved way on the state of b bits (it is graphically represented in Fig. 10.5). Interleaved application means that the set of state bits is split in 2-bit subsets, and they enter the s-boxes in two different roles: as two left most bits and as two rightmost bits. The s-box is applied in parallel. This interleaved application makes the substitution layer as one big s-box with .(b+2)×bbits. For transforming the two left most bits of the state, a two bit round constant is used. Here, the .4 × 2-bit S-box, can be represented as a quasigroup .(Q, ∗) of order 4. There are 576 quasigroups of order 4, and if we sort all of them in a lexicographic order, in GAGE we are using the quasigroup number 173. Here are the criteria for choosing this particular quasigroup:

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Fig. 10.5 Graphical representation of the non-linear substitution part of QPERMUTATION, where .l00 and .l01 are two bits of the round constant, .s0 , . . . , sb−1 are bits from the old state, and ' ' .s0 , . . . , sb−1 are bits from the new state, obtained by applying .4 × 2-bit S-box, Q

• The quasigroup and its left parastrophe should be nonlinear Boolean functions; • The quasigroup should not have fixed points; • The quasigroup should give as little as possible cells with 100.% probability in its differential distribution table and in differential distribution tables obtained from its corresponding d-transformation. The algorithm for nonlinear substitution part using the algebraic structure, quasigroup .(Q, ∗) of order 4 is given bellow. For this, a state of b bits that is subject of transformation is represented as an array .A = a0 , ..., ab/2−1 of two bit elements. An algorithm for nonlinear substitution part of GAGE Input: .(Q, ∗) quasigroup of order 4, leader l and an array .A = a0 , ..., ab/2−1 of two bit elements. procedure D-TRANSFORMATION.(l, A) .ldr ← l for .i = 0 to .b/2 − 1 do nextldr .← ai .ai ← ldr∗ nextldr .ldr ← nextldr endfor

The complete permutation is a composition of alternating application of the substitution part and the bit-shuffling part for a number of rounds ROUNDS.

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For calling the substitution part we need to define an array of constants Leaders=.l0 , ..., lROU N DS−1 .

10.3.3 Message Authentication Codes Message Authentication Code (MAC) algorithms take a variable-size message and a secret key as an input, and output a tag or MAC value. The receiver who possess the secret key, by using the tag can verify data integrity of the message and authenticity of the message’s source. There are known ways to build a MAC scheme from block ciphers or hash functions. While several quasigroup based MAC schemes can be found in the literature [3, 8], here we will explain the QMAC [47]. QMAC Meyer [47] introduced a QMAC, a MAC algorithm based on publicly known quasigroup .(Q, ∗) (for better security, large “highly non-associative” quasigroup without any structure is preferable). The secret key K of length t is a parenthesis scheme on .m1 , . . . , mt (a choice of parenthesizing .m1 ∗ . . . ∗ mt ), along with a constant .c ∈ Q. The constant c is used to prevent an adaptive chosen-text attack on QMAC, by replacing every innermost multiplication .(mi ∗ mi+1 ) with .((mi ∗ c) ∗ mi+1 ). So, the tag for a message .M = m1 . . . mt with .mi ∈ Q, i ∈ {1, . . . , t}, denoted by .hK (M), is computed by application of quasigroup operation .∗ in the order specified by the key K, with respect to the previous rule about c for every innermost multiplication. Example 3 Let .(Q, ∗) be the quasigroup from Example 2, and let .M = 0223103. Let K of length 7 consists of the parenthesis scheme ((m1 ∗ m2 ) ∗ m3 ) ∗ (((m4 ∗ m5 ) ∗ m6 ) ∗ m7 )

.

and .c = 1. We have hK (M) = (((0 ∗ 1) ∗ 2) ∗ 2) ∗ ((((3 ∗ 1) ∗ 1) ∗ 0) ∗ 3)

.

.

= ((1 ∗ 2) ∗ 2) ∗ (((3 ∗ 1) ∗ 0) ∗ 3) .

= (1 ∗ 2) ∗ ((3 ∗ 0) ∗ 3) .

= 1 ∗ (0 ∗ 3) .

=1∗3 .

= 2.

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The author explains three different methods how to calculate the MAC value of longer messages, by breaking the message M in blocks, each with t elements of Q. If the last block is smaller than t, padding is applied. Let .M = M1 . . . MN with padding. For one of the methods the construction of the MAC value is given by the following equations: H0 = I V ∈ Q

.

Hi = Hi−1 ∗ hK (Mi ), 1 ≤ i ≤ N

.

QMACK (M) = HN

.

The author gives also a representation of the key of length t with permutation of t − 1 elements, and while this limits the size of the key space to .(t − 1)! · |Q|, the actual size is much smaller, because different permutations can rise into the same parenthesis scheme. Still, the size of the key space increases exponentially in the length of the key. For example, for the size of the key space of .2128 and .|Q| = 28 , one should need .t ≤ 66.

.

10.3.4 Pseudo Random Number Generators A truly random sequence can be obtained only in theory. Namely, if we take that a sequence is random only if it passes all of the statistical test for randomness, then we can never check if a sequence is random until all of the tests, infinitely many, are passed. So, sequences that look random are used in many applications were random sequences are needed. They are produced by some deterministic algorithms or physical phenomena and are called Pseudo Random Sequences (PRS). PRS have to pass all known approved battery of statistical tests for randomness (like Diehard, NIST, ...) The algorithms for producing PRS are called Pseudo Random Sequence Generator (PRSG), i.e., PRNG when we have number sequences. Many PRNG that are used in nowadays practice are biased, for example the next produced bit (or symbol) can be predictable with probability greater than 1/2. So, the obtained sequence of a good PRNG should be unbiased. By using quasigroup transformations several type of PRNG can be designed. Very simple PRNG, called Quasigroup PRNG (QPRNG), can be obtained by the procedure given on Table 10.5. QPRNG can produce pseudo random sequences from very biased sequences, even from periodical sequences as well. We emphasize that in QPRNG the choice of the quasigroup is very important, it should be shapeless and exponential with as higher period of growth as possible.

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Table 10.5 Algorithm for simple QPRNG

Quasigroup PRNG (QPRNG) Phase I. Initialization 1. Choose a positive integer .s ≥ 4; 2. Choose a quasigroup .(A, ∗) of order s; 3. Set a positive integer k; 4. Set a leader l, a fixed element of A such that .l ∗ l /= l; Phase II. Transformations of the random string .b0 b1 b2 b3 . . . , bj ∈ A 5. For .i = 1 to k do .Li ← l; 6. .j ← 0; 7. do .b ← bj ; .L1 ← L1 ∗ b; For .i = 2 to k do .Li ← Li ∗ Li−1 ; Output: .Lk ; .j ← j + 1; loop;

10.3.5 Block Ciphers Block cipher is an enciphering method that encrypt a block M of plaintext of length n into a block C of ciphertext of length n, by using a secret key K. It uses an encryption function .E : P × K → C and a decryption function .D : C × K → P, where .P, C and .K are the spaces of plaintext, ciphertext and keys; usually .P = C = {0, 1}n , .K = {0, 1}k . The functions .E(M, K) and .D(C, K) are permutation for fixed K and .D(E(M, K), K) = M, and there are no different keys .K1 , K2 such that E(M, K1 ) = E(M, K2 ).

.

Note that when .P = C = K = {0, 1}n , then E is a quasigroup operation with parastrophe .D. Besides the last property, there are no many block ciphers based on quasigroup. Here we show the design of the block cipher BCMPQ (Block Cipher Defined by Matrix Presentation of Quasigroups), [39]. The design of BCDMPQ uses matrix presentation of quasigroups of order 4. Thus, given a quasigroup .(Q, ∗) of order 4, for all .x, y ∈ Q, x = (x1 , x2 ), y = (y1 , y2 ), .xi , yi are bits: x ∗ y = mT + Ax T + By T + CAx T ◦ CBy T

.

(10.3)

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| | | | a11 a12 b11 b12 where .A = and .B = are nonsingular Boolean matrices, a21 a22 b21 b22 | | 11 .m = [m1 , m2 ] is a Boolean vector and .C = . The operation “.◦” denotes 11 the component wise product of two vectors. There are 144 quasigroups of form (10.3). Out of them, a list of 128 is chosen and stored in memory as follows: seq_num

.

m1 , m2 , a11 , a12 , a21 , a22 , b11 , b12 , b21 , b22

(10.4)

where .seq_num is a seven bit number (the number of the quasigroup in the list) while .m1 , m2 , a11 , a12 , a21 , a22 , .b11 , b12 , b21 , b22 are the bits appearing in the matrix form (10.3) of the quasigroup operation. (Note that a quasigroup of order 4 is given by using only ten bits, while 32 bits are needed for its Latin square.) The encryption and decryption algorithms use 16 quasigroups: .Q1 , Q2 , . . . , Q8 , .T1 , . . . , T8 in different steps. These matrices are determined by using the round key key, which is generated out of the secret key K and consists of 128 bits. The key length of 128 bits is distributed in the following way: • 16 bits for the leaders .l1 , l2 , ..., l8 (two bits per each leader) • 56 bits for the quasigroups .Q1 , Q2 , ..., Q8 (7 bits per each quasigroup, actually the value of .sequence_number) • 56 bits for the quasigroups .T1 , T2 , ..., T8 (7 bits per each quasigroup) The design of this block cipher is based on three algorithms: round key generation, encryption and decryption. Denote by K the secret symmetric key of 128 bits. In order to generate a round (working) key key out of the secret key, we first determine a fixed shapeless quasigroup Q and a fixed leader .l = 0 = [0, 0]. The round key is obtained by e-transformations. The procedure for generation a round key is described in the RoundKeyGeneration Algorithm . (There, and in the next two algorithms, auxiliary variables are used, tmp is two bits variable and .ltmp is one bit variable.) The message block length of BCDMPQ can be 8n for any n, but we take that .n = 8 , i.e., we consider the light version of the cipher. So, the plaintext message should be split into blocks of 64 bits. Afterwards, the Encryption Algorithm should be applied on each block. (If the message length is not divided by 64, a suitable padding will be applied). The encryption algorithm consists of two steps. In the first step we use the matrices .Q1 , Q2 , ..., Q8 and in the second the matrices .T1 , T2 , ..., T8 . Briefly, in the first step we split the 64 bit block into 8 smaller blocks (miniblocks) of 8 bits. We apply e-transformation on each of these mini-blocks with a different leader and a different quasigroup. Actually, we use the leader .li and the quasigroup .Qi for the i-th mini-blocks. The resulting string is used as input in the next step. In the second step, we apply e-transformations on each resulting string, repeating 8 times with alternately changing direction. In the i-th transformation we use the

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S. Markovski et al. RoundKeyGeneration Algorithm Input: The secret key .K = K1 K2 . . . K128 , Ki are bits. Output: The round key .key = k1 k2 . . . k128 , ki are bits. Initialization: .(Q, ∗) is a fixed matrix quasigroup of order 4 such that .a ∗ a /= a for each .a ∈ Q, .l = (0, 0) is a two bit leader. for .i = 1 to 128 do .ki ← Ki ; for .i = 1 to 4 do .ltmp ← l; for .j = 1 to 127 step 2 do .tmp ← (kj , kj +1 ); T T T T T .(kj , kj +1 ) = m + Altmp + Btmp + CAltmp ◦ CBtmp ; .ltmp ← (kj , kj +1 ); .ltmp ← l; for .j = 128 to 2 step 2 do .tmp ← (kj −1 , kj ); T T T T T .(kj −1 , kj ) = m + Altmp + Btmp + CAltmp ◦ CBtmp ; .ltmp ← (kj −1 , kj );

quasigroup .Ti and the leader .li . The detailed and formalized algorithm is presented in the Encryption Algorithm. For decryption purposes we use parastrophe .(Q, \) of quasigroup .(Q, ∗). So, what we actually need to do to decrypt is to start from the ciphertext and reverse the e-transformation, using the quasigroups .T8 , T7 , ..., T1 sequentially at first, and then reverse the e-transformations of the mini-blocks (from the encryption algorithm) using the quasigroups .Q8 , Q7 , ..., Q1 . This can be done using the inverse operation we mentioned shortly before. The decryption of a ciphertext .c1 c2 . . . c64 is done by the Decryption Algorithm. For the cipher BCDMPQ only preliminary security investigations were done. The avalanche effect and propagation of one bit and two bits changes were considered and satisfactory results were obtained. It is an open research problem to check the resistance on the other block cipher attacks.

10.3.6 Stream Ciphers Stream ciphers are classified mainly as synchronous (when the keystream is generated independently of plaintext and cyphertext) and asynchronous (when the keystream is generated by the key and a fixed number of previous ciphertext symbols). A synchronous stream cipher is binary additive when the alphabet consists of binary digits and the output function is the XORing of the keystream

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Encryption Algorithm Input: The round key .key = k1 k2 . . . k128 , ki are bits, the plaintext message .a = a1 a2 . . . a64 , ai are bits. Output: The ciphertext message .c = c1 c2 . . . c64 . Initialization: Put .li = (k2i−1 , k2i ) for .i = 1, 2, . . . , 8. Lookup the quasigroup .Qi using the sequence number binary presented as .(k7i−6 , k7i−5 , ..., k7i ) where .i = 1, 2, ..., 8. Initialize the matrices .AQi and .BQi , as well as the vector .mQi for .i = 1, 2, ..., 8. Lookup the quasigroup .Ti using the sequence number binary presented as .(k7(i+8)−6 , k7(i+8)−5 , ..., k7(i+8) ). Initialize the matrices .ATi and .BTi , as well as the vector .mTi for .i = 1, 2, ..., 8. for .i = 1 to 8 do .ltmp ← li ; for .j = 1 to 7 step 2 do .tmp ← (aj , aj +1 ); T T T .(cj , cj +1 ) = mQ + AQi ltmp + BQI tmp i T T .+CAQi ltmp ◦ CBQi tmp ; .ltmp ← (cj , cj +1 ); for .i = 1 to 4 do .ltmp ← li ; for .j = 1 to 63 step 2 do .tmp ← (cj , cj +1 ); T T T T T .(cj , cj +1 ) = mT + ATi ltmp + BTi tmp + CATi ltmp ◦ CBTi tmp ; i .ltmp ← (cj , cj +1 ); .ltmp ← li+4 ; for .j = 64 to 2 step 2 do .tmp ← (cj −1 , cj ); T + .A T T .(cj −1 , cj ) = mT Ti+4 ltmp + BTi+4 tmp + i+4 T

.CATi+4 ltmp .ltmp

◦ CBTi+4 tmp T ;

← (cj −1 , cj );

and the plaintext. There are several designs of stream cipher based on quasigroups, and here we will consider one of them. Edon80 Edon80 is a binary additive stream cipher that is unbroken eSTREAM finalist [17]. Schematic and behavioral description of Edon80 is given on Fig. 10.6. Edon80 works in three possible modes: (1) KeySetup, (2) IVSetup and (3) Keystream mode. For its proper work Edon80 beside the core (that will be described later) has the following additional resources: 1. One register Key of 80 bits to store the actual secret key, 2. One register I V of 80 bits to store padded initialization vector, 3. One internal 2-bit counter Counter as a feeder of Edon80 Core in Keystream mode, 4.

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S. Markovski et al. Decryption Algorithm Input: The round key .key = k1 k2 . . . k128 , ki are bits, the ciphertext message .c = c1 c2 . . . c64 , ci are bits. Output: The plaintext message .a = a1 a2 . . . a64 . Initialization: Put .li = (k2i−1 , k2i ) for .i = 1, 2, . . . , 8. Lookup the quasigroup .Qi using the sequence number binary presented as .(k7i−6 , k7i−5 , ..., k7i ) where .i = 1, 2, ..., 8. Initialize the matrices .AQi and .BQi , as well as the vector .mQi for .i = 1, ..., 8. Lookup the quasigroup .Ti using the sequence number binary presented as .(k7(i+8)−6 , k7(i+8)−5 , ..., k7(i+8) ). Initialize the matrices .ATi and .BTi , as well as the vector .mTi for .i = 1, 2, ..., 8. for .i = 1 to 64 do .ai ← ci ; for .i = 1 to 4 do .ltmp ← li+4 ; for .j = 64 to 2 step 2 do .tmp ← (aj −1 , aj ); −1 T + .(aj −1 , aj ) = BT mT + BT−1 (I + C)ATi+4 ltmp i+4 Ti+4 i+4 .BT

−1

T ) + B −1 tmp T + (CmTTi+4 ◦ CATi+4 ltmp Ti+4

.BT

T ◦ Ctmp T ); (CATi+4 ltmp

i+4

−1

i+4

← (aj −1 , aj ); ← li ; for .j = 1 to 63 step 2 do .tmp ← (aj , aj +1 ); −1 T −1 T .(aj −1 , aj ) = BT mT + BT (I + C)ATi ltmp + i i i −1 −1 T T T .BT (CmT ◦ CATi ltmp ) + BT tmp + i i i −1 T T .BT (CATi ltmp ◦ Ctmp ); i .ltmp ← (aj , aj +1 ); for .i = 1 to 8 do .ltmp ← li ; for .j = 1 to 7 step 2 do .tmp ← (aj , aj +1 ); −1 T −1 T .(aj −1 , aj ) = BQ mQ + BQ (I + C)AQi ltmp + i i i −1 −1 −1 T T T T T .BQ (CmQ ◦ CAQi ltmp ) + BQ tmp + BQ (CAQi ltmp ◦ Ctmp ); i i i i .ltmp ← (aj , aj +1 ); .ltmp

.ltmp

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Fig. 10.6 Edon80 components and their relations

Fig. 10.7 Quasigroups used in the design of Edon80

One 7 bit SetupCounter that is used in IVSetup mode, 5. One .4×4 = 16 bytes ROM bank where 4 quasigroups (i.e. Latin squares) of order 4, indexed from .(Q, •0 ) to .(Q, •3 ), are stored. Those 4 predefined quasigroups are described in Fig. 10.7. The structure of the Edon80 Core is described in the next two figures. The internal structure of Edon80 can be seen as pipelined architecture of 80 simple 2-bit transformers called e-transformers. The schematic view of a single e-transformer is shown on Fig. 10.8. The structure that performs the operation .∗i in e-transformers is a quasigroup operation of order 4. We refer an e-transformer by its quasigroup operation .∗i . So, in Edon80 we have 80 of this e-transformers, cascaded in a pipeline, one feeding another. Figure 10.9 shows the pipelined core of Edon80. We will not discuss in all details Edon80. What we want to emphasize is that the chosen quasigroups have enough big periods of growths. Thus, if any of the quasigroups is used k times in an e-transformations, the period of the obtained string will be correspondingly .2.66k , 2.48k , 2.43k , 2.37k . (Note that .2.4880 ≈ 2104.8 ). We have to state that 64 out of 576 quasigroups of order 4 have so big periods of growth, any 4 of them could be taken in the construction of Edon80. Edon80 shows that, when adequately designed, the quasigroups of very small order can produce crypto primitives of high quality.

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Fig. 10.8 Schematic representation of a single e-transformer of Edon80

Fig. 10.9 Edon80 core of 80 pipelined e-transformations

10.3.7 Authenticated Encryption Ciphers An authenticated encryption is a combination of encryption and authenticity and simultaneously assures the confidentiality and integrity of data. Its input is a key and a message; and its output is a ciphertext and a small tag. There is a case where the input space is extended, and some additional data has been authenticated without been encrypted. However, a variant of AE, called AEAD (Authenticated Encryption with associated data), allows a recipient to check the integrity of both the encrypted and non-encrypted information in the message. AEAD Algorithm-.π -Cipher π -Cipher is a new authenticated encryption cipher, that comes in four variants depending on the final purpose: .π 16-Cipher096 (the lightweight variant), .π 32Cipher128, .π 64-Cipher128 and .π 64-Cipher256. This cipher is one of the candidates that participate in the second round of the CAESAR competition for authenticated

.

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encryption ciphers [7]. The design of the cipher is online, parallel, incremental, provably secure, nonce-based with a support of associated data. It relies on the duplex sponge construction [5], where the core permutation function is built of the ARX operations. In the ARX approach, instead of using S-boxes for increasing the non-linearity of the input bits and permutation layer for their diffusion, operations of Addition, Rotation, and XOR are used. When mixed together in an appropriate way, these operations interact in a complex and non-linear way. In this survey we describe .π -Cipher at a level necessary to understand the main input of the quasigroups, and refer to [23, 24, 54] for the detailed and formal specification. The general permutation function .π consists of three main transformations μ, ν, σ : Z42ω → Z42ω , where .Z2ω is the set of all integers between 0 and .2ω − 1. These transformations do the work of diffusion and nonlinear mixing of the input. The following operations are applied:

.

• Addition + modulo .2ω ; • Rotate left (circular left shift) operation, .ROTLρ (X), where X is a .ω-bit word and .ρ is an integer with .0 ≤ ρ < ω; • Bitwise XOR operation .⊕ on .ω-bit words. Let .X = (X0 , X1 , X2 , X3 ), .Y = (Y0 , Y1 , Y2 , Y3 ) and .Z = (Z0 , Z1 , Z2 , Z3 ) be three 4-tuples of .ω-bit words. Further, let us denote by .∗ the following operation: Z = X ∗ Y ≡ σ (μ(X) O4 ν(Y))

.

(10.5)

( )4 where .O4 is the component-wise addition of two 4-dimensional vectors in . Z2ω . The operation .∗ defined as in (10.5) is a quasigroup operation of .(Qq , ∗), where q q .Qq = {0, 1} is a set of elements 0 and 1, with cardinality .2 and .q = 64, 128, 256. q Because of the fact that quasigroups of order .2 are used, also the quasigroup string transformations are mentioned in the definition of .π -function. The state of the .π -function is represented as a list of N , 4-tuples, each of length .ωbits (.b = N × 4 × ω). One round of the .π -function consists of two consecutive transformations e-transformation and d-transformation, shown in Fig. 10.10. AEAD Algorithm—InGAGE InGAGE [21] is a sponge-based family of authenticated ciphers with associated data built over the lightweight cryptographic hash function GAGE. InGAGE mode of operation is similar to Ascon family of authenticated ciphers [12]. Still, InGAGE has several differences with Ascon both in its mode of operation and in used parameters. The main building part of the encryption and decryption procedures of InGAGE is the round permutation called QPERMUTATION which is the same as GAGE hash function with only difference in the number of rounds. The lightweight hash function GAGE and the AEAD algorithm InGAGE [21] are both part of the first round of the standardization process organized by NIST for lightweight cryptographic (AEAD and hash) algorithms [51].

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Fig. 10.10 Graphical representation of one round of the .π -function, where .C ∈ Qq is a vector of q-bit constant values 64, 128 or 256 and .I = (I1 , . . . , IN ), J = (J1 , . . . , JN ) ∈ QN q are input and output of the function

10.3.8 Public Key Cryptography In public key cryptography (PKC) a pair of public and private key is generated per each participant in the communication. The public key, as its name suggests, is made public for everybody, and if somebody wants to send a secret message to its owner, it must encrypt the message with this public key. The owner by using his private key, known only to him, decrypts the message. So, each public key encryption scheme consists of: a key generation algorithm, an encryption algorithm and a decryption algorithm. Public key algorithms are usually much slower then symmetric algorithms, so, their primary application is to exchange the secret key between communicating parties, while the actual communication is encrypted with some symmetric cryptosystem. Public key algorithms are used also in digital signatures, password-authenticated key agreement, non-repudiation protocols, key escrow, etc. First attempt to design a quasigroup based public key scheme is made by Keedwell [28], by using crossed inverse quasigroups. Multivariate PKC is based on the NP-hard problem of solving multivariate quadratic (MQ) systems of equations over finite fields. For all MQ schemes it is common that the public key is constructed as .P = S ◦ F ◦ T, where .F is some easily invertible quadratic mapping, while .S and .T are bijective affine transformations. One important application of quasigroups (particularly multivariate quadratic quasigroups) that intrigued the crypto community a decade ago, belongs to the Multivariate PKC. The reasons were seeking for new post quantum solutions and in that time, the MQQ-SIG signature scheme was the fastest scheme in the ECRYPT benchmarking of cryptographic systems (eBACS).1 However, Faugère et al. [13] showed that MQQ based public key algorithms share a common algebraic structure that can be used for launching a successful polynomial time key-recovery

1 https://bench.cr.yp.to/results-sign.html.

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attack. Still, in the following subsections we will describe MQQ and MQQ-ENC cryptosystems, and MQQ-SIG signature scheme. MQQ and MQQ-ENC Cryptosystem MQQ cryptosystem was introduced in 2008, by Gligoroski et al. [25], with a modification given in [18]. Multivariate quadratic quasigroups are used for construction of multivariate quadratic polynomials over finite fields, that are used to build the trapdoor function .F. Description of the cryptosystem offers also a heuristic algorithm for finding MQQ of type .Quadd−k Link of order at most .25 . First, two sets with MQQs of type .Quad4 in1 and .Quad5 Lin0 are generated (each set with at least million elements). Then, two quasigroups (.∗1 , ∗2 ) are picked randomly from the first set and six from the second set (.∗3 , . . . , ∗8 ). Next, the system .F = (F1 (x1 , . . . , xn ), . . . , Fn (x1 , . . . , xn )) of quadratic polynomials over .F2 , where .n = 5k, k ≥ 28, is generated, by the following procedure: 1. Randomly generate n Boolean functions .F = (f1 , . . . , fn ) of n variables .x = (x1 , . . . , xn ), and represent the vector .F (x) as a string .X1 . . . Xk where .Xi are vectors of dimension 5. 2. Define a .(k − 1)−tuple .I = (i1 , . . . , ik−1 ), .ij ∈ {1, 2, . . . , 8}, as a set of indexes to determine which quasigroup will be used in the nonlinear transformation, and with the requirement that indexes 1 and 2 are repeated 8 times in I . 3. Compute .y = Y1 . . . Yk where: .Y1 = X1 , Yj +1 = Xj ∗ij Xj +1 , for .j = 1, 2, . . . , k − 1. 4. Set a 13-dimensional vector .Z = Y 1||Yr1 ,1 ||Yr2 ,1 || . . . ||Yr8 ,1 , where .Yrj ,1 means the first coordinate of the vector .Yrj . Transform Z by the bijection of Dobbertin: .W = Dob(Z). 5. Set .Y1 = (W1 , W2 , W3 , W4 , W5 ), Yr1 ,1 = W6 , Yr2 ,1 = W7 , . . . , Yr8 ,1 = W13 . 6. Return y as n multivariate quadratic polynomials .Fi (x1 , . . . , xn ), i = 1, 2, . . . n. .F is a bijective multivariate quadratic mapping. .S and .T are two nonsingular (n,n) linear transformations, selected uniformly at random from the set .F2 . The private key is the 10-tuple .(S, T, ∗1 , . . . , ∗8 ), while the map .P = S ◦ F ◦ T is the public key. The size of the public key is .n × (1 + n(n+1) 2 ) bits. Encryption of the message n .m = (m1 , . . . , mn ) ∈ F is done by computing 2

c = P(m),

.

while the decryption is done by m = P−1 (c) = T−1 ◦ F−1 ◦ S−1 (c)

.

MQQ-ENC encryption scheme is build on MQQ encryption scheme, with several differences [20]. It is designed to be a probabilistic encryption scheme with decryption errors, constructed over any small field .Fpk (.k ≥ 1 and p a

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prime number). The trapdoor function is using a minus modifier with fixed number of removed equations which destroys the bijectivity of .F, and in the decryption process a universal hash function helps for probability of an erroneous decryption to be negligible. The other important difference is deployment of the left MQQs (LMQQs). Also, the transformations .S and .T are constructed using a combination of two circulant matrices. MQQ-SIG Signature Scheme MQQ-SIG [19] signature scheme can be seen as .(1/2) truncation of the multivariate quadratic system .S ◦ F ◦ T : {0, 1}n → {0, 1}n (see Fig. 10.11). This means that .n/2 of the equations are removed in the public key, which in fact is an application of minus modifier. As quasigroup, MQQ of order .28 is used, obtained by the method from [22] (with size of 81B in memory). The affine transformations .S and .T are defined by circulant matrices, in the following way: n 16 +3

n

S

.

−1

=

16 O

Iσ 0 ⊕

O

Iσ 1 i

i

i=0

i=0

n n + 1} are .n × n where .Iσ 0 , i ∈ {0, 1, 2, . . . , 16 } and .Iσ 1 , i ∈ {0, 1, 2, . . . , 16 i

i

permutation matrices, while .σi0 and .σi1 are permutations on n elements that produce non-singular matrix .S −1 . For the last ones, .σ00 and .σ01 are random permutation on n k k .{1, 2, . . . , n}, while .σ = RotateLef t (σ i i−1 , 8), k = {0, 1}, i = {1, . . . 16 + k}. −1 −1 Transformation .S = (S ) , while .T is obtained from .S by the formula T=S·x+v

.

Fig. 10.11 Inside of the MQQ-SIG signature scheme

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Fig. 10.12 Signing and verification process with MQQ-SIG signature scheme [19]

where the vector .v = (v1 , . . . , vn ) is calculated from .σ01 = (s1 , . . . , sn ) by the following expression: vi =

.

(( (s

mod 16) × 16 1+L i−1 8 J (8−i) mod 8 2

) +

( s )) 65+L i−1 J 8

2(8−i)

mod 8

mod 2

MQQ-SIG scheme works in the following way. For generation of a pair of public and private key, first MQQ .∗ is generated, together with the .S and .T. Then, the central bijective multivariate quadratic mapping .F = (F1 (x1 , . . . , xn ), . . . , Fn (x1 , . . . , xn )), where .n = 32 × k, k ∈ {5, 6, 7, 8}, is generated, by the following procedure: 1. Randomly generate n Boolean functions of n variables. Represent a given vector .x = (f1 , . . . , fn ) as a string .x = X1 . . . X n where .Xi are vectors of dimension 8 8. 2. Compute .y = Y1 . . . Y n8 where: .Y1 = X1 , Yj +1 = Xj ∗ Xj +1 , for even .j = 2, 4, . . . , and .Yj +1 = Xj +1 ∗ Xj , for odd .j = 3, 5, . . . ,. 3. Return .y as n multivariate quadratic polynomials .Fi (x1 , . . . , xn ), i = 1, 2, . . . n. Next, compute .y = S ◦ F ◦ T(x).The public key is .y as . n2 multivariate quadratic polynomials .Fi (x1 , . . . , xn ), i = 1 + n2 , . . . n.n, and the private key is the tuple 0 1 .(σ , σ , ∗). 0 0 In order to reduce the size of the public key, the designers decided to split the hash result of the message M in two parts and sign it using twice the same trapdoor function (see Fig. 10.12).

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10.4 Error-Correcting Codes Based on Quasigroups In this part we will describe Random Codes Based on Quasigroups (RCBQ). A usual way to obtain error-correcting codes resistant to an intruder attack consists in application of some of the known ciphers on the codewords, before sending them through an insecure channel. Then two algorithms are used, one for error-correcting codes and another for obtaining information security. The concept of cryptcoding combines the processes of encoding and encryption in one algorithm. RCBQs are crypt-codes. Namely, they are defined by using a cryptographic algorithm during the encoding/decoding process and they allow not only correction of certain amount of errors in data transmitted through noisy channel, but they also provide an information security, all build in one algorithm. If the data are encoded with these codes, then the recipient can decode the original data only if s/he knows exactly which parameters are used in the process of encoding, even the communication channel is without noise. In the sequel, we are considering RCBQs as errorcorrecting codes, and consequently we do not analyze its cryptographic properties. The initial idea for applying quasigroups in random codes is proposed in [15] and [16]. We will denote these coding/decoding algorithms as Standard algorithm for RCBQ. RCBQs use several parameters (redundancy pattern, initial keys, quasigroups) in their design. Their error-correction performances and decoding speed depend on these parameters. The decoding process is a list decoding, and the size of the lists (called decoding candidate sets) has an impact on the decoding speed and probability of correct decoding. Therefore, several modifications of coding/decoding algorithms (Cut-Decoding algorithms, 4-Sets-Cut-Decoding algorithms) are proposed in [57, 61]. In these algorithms, using intersections of decoding candidate sets obtained in parallel decoding processes, a significant speed-up of decoding process is obtained. For improving the performances of RCBQ for transmission through a burst channel, two new algorithms called Burst-Cut-Decoding and Burst-4-Sets-Cut-Decoding algorithm are proposed in [43]. Additionally, to provide faster and more efficient decoding, particularly for transmission over a low-noise channels in [58, 59] we consider new modifications (Fast-Cut-Decoding, Fast-4-Sets-Cut-Decoding, FastB-Cut-Decoding and FastB-4-Sets-Cut-Decoding algorithms) of previously mentioned coding/decoding algorithms for RCBQs. Also, in several papers [2, 44–46, 55] we investigate performances of these algorithms for transmission of messages, images and audio files trough a different noisy channels (binary-symmetric, Gaussian and burst channel). Here, we will consider only Standard, Cut-Decoding and 4-Sets-Cut-Decoding algorithms and their performances for transmission trough a binary-symmetric channel. Also, we will present some results of application of 4-Sets-Cut-Decoding algorithm in transmission of images and in audio signal processing. RCBQs are designed using algorithms for encryption/decryption from the implementation of TASC (Totally Asynchronous Stream Ciphers) by quasigroup string transformations [15]. These cryptographic algorithms use the alphabet Q and a quasigroup operation .∗ on Q together with its parastrophe .\. In the code design

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authors use the alphabet of nibbles .Q = {0, 1, . . . , 9, a, b, c, d, e, f }. But, here we will give the encoding and decoding algorithms for a more general case where we use a-bit letters instead of nibbles.

10.4.1 Description of Standard Coding/Decoding Algorithms for RCBQ Standard coding algorithm is the following one. Divide the sequence of bits, obtained from the information source, into messages (blocks) of .Nblock bits. Let .M = m1 m2 ...ml be a block of .Nblock = la bits, where .mi ∈ Q and Q is the alphabet of all a-bit symbols. A redundancy of v a-bit zero symbols is added and it is produced a block .L = L(1) L(2) ...L(s) = L1 L2 ...Lm of N bits, where .L(i) are sub-blocks of r symbols from Q and .Li ∈ Q (r is a fixed positive integer). After erasing the redundant zeros from each .L(i) , the message L will produce the original message M. In this way an .(Nblock , N) code with rate .R = Nblock /N is obtained. The codeword is produced after applying the encryption algorithm of TASC (given in Table 10.6) on the block L. For that aim, previously, a key .k = k1 k2 . . . kn ∈ Qn should be chosen. The obtained codeword of M is .C = C1 C2 ...Cm , where .Ci ∈ Q. After transmission through a noisy channel the codeword C will be transformed to a received message .D = D (1) D (2) ...D (s) = D1 D2 ...Dm , where .D (i) are subblocks of r symbols from Q and .Di ∈ Q. The decoding process consists of four steps: .(i) procedure for generating the sets with predefined Hamming distance; .(ii) inverse coding algorithm; .(iii) procedure for generating decoding candidate sets and .(iv) decoding rule. (i) Procedure for Generating the Sets with Predefined Hamming Distance The probability that maximum t bits in .D (i) are not correctly transmitted is P (p; t) =

.

t ( ) E ra k p (1 − p)ra−k , k k=0

where p is the probability of bit-error in the channel. Let .Bmax be a given integer that denotes the assumed maximum number of bit errors that occur in a block during transmission. Generate the sets .Hi = {α|α ∈ Qr , H (D (i) , α) ≤ Bmax }, for (i) , α) is a Hamming distance between .D (i) and .α. The .i = 1, 2, . . . , s, where .H (D cardinality of the sets .Hi is Bchecks

.

) ( ) ( ) ( ra ra ra =1+ + + ... + 1 2 Bmax

and the number .Bchecks determines the complexity of the decoding procedure. Namely, decoding with this algorithm is a list decoding, i.e. decoding speed and the probability of accurate decoding depend on the size of the lists of potential

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candidates for the decoded message. .Bchecks is actually the length of the list. Clearly, for efficient decoding, the number of checks .Bchecks has to be reduced as much as it is possible. (ii) Inverse Coding Algorithm The inverse coding algorithm (ICA) is the decrypting algorithm of TASC given in Table 10.6. (iii) Procedure for Generating Decoding Candidate Sets The decoding candidate sets .S0 , .S1 ,. . . , .Ss are defined iterative. Let .S0 = (k1 . . . kn ; λ), where .λ is the empty sequence. Let .Si−1 be defined for .i ≥ 1. Then .Si is the set of all pairs .(δ, w1 w2 . . . wrai ) obtained by using the sets .Si−1 and .Hi as follows (.wj are bits). For each .(β, w1 w2 . . . .wra(i−1) ) ∈ Si−1 and for each element .α ∈ Hi , we apply the inverse coding algorithm (i.e. algorithm for decryption given in Table 10.6) with input .(α, β). If the output is the pair .(γ , δ) and if both sequences .γ and .L(i) have the redundant zeros in the same positions, then the pair .(δ, w1 w2 . . . wra(i−1) c1 c2 . . . cra ) ≡ (δ, w1 w2 . . . wrai ) is an element of .Si . (iv) Decoding Rule The decoding of the received message D is given by the following rule. If the set .Ss contains only one element .(d1 . . . dn , w1 . . . wras ) then .L = w1 . . . wras is the decoded (redundant) message. In this case, we say that we have a successful decoding. If the decoded message is not the correct one then we have an uncorrected-error. In the case when the set .Ss contains more than one element, we Table 10.6 TASC algorithms for encryption and decryption Encryption Input: Key .k = k1 k2 . . . kn and .L = L1 L2 . . . Lm Output: codeword .C = C1 C2 ...Cm For .j = 1 to m . X ← Lj ; . T ← 0; For .i = 1 to n . X ← ki ∗ X; . T ← T ⊕ X; . ki ← X; . kn ← T Output: .Cj ← X

Decryption Input: The pair .(a1 a2 . . . as , k1 k2 . . . kn ) Output: The pair .(c1 c2 . . . cs , K1 K2 . . . Kn ) For .i = 1 to n . Ki ← k i ; For .j = 0 to .s − 1 . X, T ← aj +1 ; . temp ← Kn ; For .i = n to 2 . X ← temp \ X; . T ← T ⊕ X; . temp ← Ki−1 ; . Ki−1 ← X; . X ← temp \ X; . Kn ← T ; . cj +1 ← X; Output: .(c1 c2 . . . cs , K1 K2 . . . Kn )

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say that the decoding of D is unsuccessful (of type more-candidate-error). In the case when .Sj = ∅ for some .j ∈ {1, . . . , s}, the process will be stopped (we say that error of type null-error appears). We conclude that for some .i ≤ j, D (i) contains more than .Bmax errors, resulting in .Ci ∈ / Hi . Theorem 3 ([16]) The packet-error probability of RCBQ is .P ERtheory = 1 − (1 − qB )s , where .qB ≥ 1−P (p; Bmax ) (probability that more than .Bmax bit errors occur in a block). From the proof of this theorem given in [16], it is clear that in the probability P ERtheory the more-candidate-errors are not provided.

.

10.4.2 Choosing Parameters for Optimal RCBQ RCBQ have several parameters, and in [56] the influence of the code parameters on the code performances is investigated. By making several experiments, it is pointed out how • the pattern of the redundancy • the length of the initial key • the chosen quasigroup affect the code performances. In the experiments, the alphabet is of nibbles, the quasigroup (and its parastrophe) is given in Fig. 10.13, and the initial key is .k = 01234. The experiments were maden for different values of bit-error probability p of binary symmetric channel and .Bmax = 3 and .Bmax = 4 until .BERs < p. (For .Bmax > 4, the experiments do not terminate in real time.) In the experiments, the number of incorrectly decoded bits when the decoding process finishes with more-candidate-error or null-error, is calculated as follows. When null-error appears, i.e., .Si = ∅, all the elements (without redundant symbols) from the set .Si−1 are taken and their maximal common prefix substring is found. If this substring has k bits and the length of the sent message is m bits (.k ≤ m), then it

Fig. 10.13 Quasigroup of order 16 and its parastrophe used in the experiments

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Table 10.7 Patterns of redundancy patt.1 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

patt.2 1100 1100 0000 1100 1100 0000 1100 1100 0000 1100 1100 0000 1100 0000 0000 0000 0000 0000

patt.3 1100 1100 1000 0000 1100 1000 1000 0000 1100 1100 1000 0000 1100 1000 1000 0000 0000 0000

patt.4 1100 1100 1100 0000 0000 1100 1100 1100 0000 0000 1100 1100 1100 0000 0000 0000 0000 0000

patt.5 1100 1000 0000 1100 1000 0000 1100 1000 0000 1100 1000 0000 1100 1000 0000 1100 1000 0000

patt.6 1100 1100 1000 0000 1100 1100 1000 0000 1100 1100 1000 0000 1000 1000 1000 0000 0000 0000

Fig. 10.14 (a) .P ERs and (b) .BERs for all six patterns and .Bmax = 3

is compared this substring with the first k bits of the sent message. If they differ in s bits, then the number of incorrectly decoded bits is .m − k + s. If a more-candidateserror appears all the elements from the set .Ss are taken and their maximal common prefix substring is found. The number of incorrectly decoded bits is computed as previously. Redundancy Pattern In order to check the influence of the redundancy pattern to the code performances, experiments for 6 different patterns for redundant zero nibbles for code (72,288), with rate .R = 1/4, were realized. The 6 patterns are given in Table 10.7. In the patterns, with 1 is denoted the place of a message (information) symbol, and with 0 the redundant zero symbol. From the experimental results for all six proposed patterns (presented in Fig. 10.14 for .Bmax = 3 and Fig. 10.15 for .Bmax = 4), it can be concluded that the best results for P ER and BER are obtained for the third pattern, especially for .Bmax = 4. Let notice that in Fig. 10.15 some lines are not complete since the experiments were made until value of BER that is larger than bit-error probability p of the channel is obtained. When .BER > p the using of coding does not have a sense.

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Fig. 10.15 (a) .P ERs and (b) .BERs for all six patterns and .Bmax = 4

Key Length The theoretical probability of packet-error given in Theorem 3 is determined under the assumption that the code is perfectly random (i.e., the r-tuples are uniformly distributed in each codeword with length N , .r ≤ N). Therefore, in that theorem the more-candidates-errors are not provided. In [34] it is proven that if t quasigroup transformations on a string are applied, a string where n-tuples of letters are uniformly distributed for .n ≤ t is obtained. In the design of these codes, the length of the key k determines how many quasigroup transformations will be applied in forming of a codeword. Therefore, a longer key of the code gives a “more random” code. This means that the results of experimental P ER will be closer to the theoretical values for P ER, i.e., the number of more-candidates-errors will be reduced. So, experiments with the third pattern (which give the best results) with key length of 10 were made. From the results it can be seen that in some experiments more-candidates-errors have not appeared, and if they appear, their number is very small. It can be concluded that when a longer key is used, better results for P ER with almost the same duration of the decoding process can be obtained. On the other hand, the key length is not a unique parameter which has influence on the P ER. By an experiment with key length of 10 and the first pattern, the number of more-candidates-errors was not smaller than the previous case with the shorter key. Hence, it can be concluded that each parameter in this code design has great influence over the performances, i.e., the parameters are mutually dependent. Choosing of a Quasigroup Several experiments with different quasigroups showed that the choice of the quasigroup does not affect only the values of P ER and BER, but it has a great influence on the speed of decoding as well. Experiments with the cyclic quasigroup of order 16 and a quasigroup of order 16, obtained as a direct product of quasigroups of order 2, and the key length of 10 were done. Decoding for the third pattern was too slow. The cyclic group and the direct product of quasigroups of order 2 are examples of fractal quasigroup. The

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quasigroup in Fig. 10.13 is an example of non-fractal quasigroups, and the results obtained using this quasigroup were quite satisfactory. From the experiments, it can be concluded that the best results for RCBQ of rate (72,288) were obtained by the third pattern, the key length of 10, the quasigroup given in Fig. 10.13 (together with its parastrophe) and .Bmax = 4.

10.4.3 Method for Decreasing the Number of Null-Errors From all experiments with different patterns, key lengths, and quasigroups it can seen how these parameters affect on the number of more-candidates-errors. But, changing these parameters does not have a great influence on the number of nullerrors. Their number is determined with the theoretical probability .P ERtheory (given in Theorem 3), and this probability does not depend on these parameters. Unsuccessful decoding with null-error occurs when in some of the sub-blocks of the encoded message, more than predicted .Bmax bit errors appear during transmission. Therefore, it is clear that some of these errors can be eliminated if a few iterations of the decoding process are canceled and all of them or part of them with a larger value of .Bmax are reprocessed. With this procedure, only part of these unsuccessfully decoded messages will be eliminated since it cannot be known exactly in which iteration the correct sub-block does not enter in the set of candidates for decoding and exactly how many transmission errors (.Bmax + 1, .Bmax + 2, or more) occur in this sub-block. Moreover, the cancellation of the iterations slows down the decoding, and the number of elements in sets .Si can become too large, leading to unsuccessful decoding of type more-candidates-error. To show this, in [56] authors proposed the following modification of the decoding process. If an empty set is obtained in some iteration (for example ith), the two previous iterations .((i − 1)th and .(i − 2))th are canceled. After that, the .(i − 2)th iteration are reprocessed with .Bmax = Bmax + 1, and the next iterations continue with the old value of .Bmax . If an empty set appears again in the same iteration, then the process is stopped and decoding ends unsuccessfully. But, if an empty set is obtained in a next iteration .((i + 1)th, .(i + 2)th, ...) then the above procedure is repeated for that iteration again. From the results given in [56] we can conclude that this modification gives a significant elimination of null-errors and the decoding process is only slightly slower.

10.4.4 Cut-Decoding Algorithm From the experiments with RCBQ it is concluded that the speed of the decoding process is one of the biggest problem for these codes. In order to improve the decoding speed, in [57] authors define a new coding/decoding algorithm called Cut-

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Decoding algorithm. The decoding of RCBQ is a list decoding, so the length of the list must be reduced as much as it is possible. Therefore, in the Cut-Decoding algorithm, modifications are made in order to reduce the number of decoding candidates in all iterations of the decoding process. In such a way the decoding process is 4.5 times faster than the original one for code (72,288). Also, some other modifications of the decoding algorithm for decreasing the packet-error probability (P ER) and the bit-error probability (BER) are proposed.

10.4.4.1

Coding with Cut-Decoding Algorithm

In Cut-Decoding algorithm, instead of using a .(Nblock , N) code with rate R, two (Nblock , N/2) codes with rate 2R are used, that encode/decode a same message of .Nblock bits. First, redundant zero symbols are added in the message .M = m1 m2 . . . ml (in the same way as in the standard coding algorithm) and a redundant message .L = L(1) L(2) . . . L(s/2) = L1 L2 . . . Lm/2 of .N/2 bits is produced. For coding, the encryption algorithm (given in Table 10.6) is applied twice on the same redundant message L using different parameters (different keys or quasigroups). The codeword of the message is a concatenation of the two codewords of .N/2 bits.

.

10.4.4.2

Decoding with Cut-Decoding Algorithm

After transmitting through a binary symmetric channel, the outgoing message D = D (1) D (2) . . . D (s) , where .D (i) are sub-blocks of r symbols from the alphabet Q, is divided into two messages .D (1) = D (1) D (2) . . . D (s/2) and .D (2) = D (s/2+1) D (s/2+2) . . . D (s) with equal lengths and they are decoded parallel with the corresponding parameters. In Cut-Decoding algorithm, there is a modification in part .(iii) of the decoding process, i.e., in the procedure for generating decoding candidate sets. In this algorithm, the decoding candidate sets are generated in the following way.

.

(1)

(1)

(1)

(2)

(2)

(2)

Step 1. Let .S0 = (k1 . . . kn ; λ) and .S0 = (k1 . . . kn ; λ) where .λ is (1) (1) (2) (2) the empty sequence, .k1 . . . kn and .k1 . . . kn are the initials keys used for obtaining the two codewords. (1) (2) Step 2. Let .Si−1 and .Si−1 be defined for .i ≥ 1. (1)

(2)

Step 3. Let two decoding candidate sets .Si and .Si be obtained in both decoding processes, in the same way as in the standard RCBQ. (1) Step 4. Let .V1 = {w1 w2 . . . wrai |(δ, w1 w2 . . . wrai ) ∈ Si }, (2) .V2 = {w1 w2 . . . wrai .|(δ, w1 w2 . . . wrai ) ∈ S i } and .V = V1 ∩ V2 . (1) Step 5. For each .(δ, w1 w2 . . . wrai ) ∈ Si , if .w1 w2 . . . wrai ∈ / V then (1) (1) (2) .S ← Si \ {(δ, w1 w2 . . . wrai )}. Also, for each .(δ, w1 w2 . . . wrai ) ∈ Si , if i (2) (2) .w1 w2 . . . wrai ∈ / V then .Si ← Si \ {(δ, w1 w2 . . . wrai )}.

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(Actually, we eliminate from .Si all elements whose second part does not match (2) with the second part of an element in the .Si , and vice versa. In the next iteration, (1) (2) both processes use the corresponding reduced sets .Si and .Si .) Step 6. If .i < s/2 then increase i and go back to Step 3. The decoding rule in the Cut-Decoding algorithm is defined in the following (1) (2) way. If, after the last iteration, the reduced sets .Ss/2 and .Ss/2 have only one element with same second component .w1 . . . wras/2 , then .L = w1 . . . wras/2 is the decoded redundant message. In this case, we say that we have a successful decoding. If, after (1) (2) the last iteration, the reduced sets .Ss/2 and .Ss/2 have more than one element we have more-candidate-error. If it is obtained .Si(1) = ∅, Si(2) /= ∅ or .Si(2) = ∅, Si(1) /= ∅ in some iteration, then the decoding of the message continues only with the nonempty (2) (1) set .Si or .Si , correspondingly, by using the standard RCBQ decoding algorithm. (1) (2) In the case when .Si = Si = ∅ in some iteration, then the process will be stopped (null-error appears). In experiments with this method of decoding, a significant reduction in the number of elements in the sets S is noticed and great improvement in the speed of the decoding process is achieved. Namely, this algorithm of decoding is 4.5 times faster than Standard algorithm for code (72,288). The problem in Cut-Decoding algorithm is that for obtaining code with rate R we need a pattern for code with twice larger rate. But, it is hard to make a good pattern for larger rates, since the number of redundant zeros in these patterns is smaller. Therefore, with this decoding method results in the number of unsuccessful decodings of type more-candidate-error are worse, but the number of unsuccessful decodings with null-error is smaller. To resolve the problem of the greater number of more-candidate-errors one heuristic in the decoding rule for elimination of this type of error is proposed. Namely, in the experiments with RCBQ when the decoding process ends with more elements in the reduced decoding candidate sets in the last iteration, almost always the correct message is in these sets (as a second part of an element in both sets). So, in this case, we can randomly select a message from one of the reduced sets in the last iteration and it can be taken as the decoded message. If the selected message is the correct one, then the bit-error is 0, so BER will also be reduced. In the experiments with this modification in around half of the cases, the correct message is selected.

10.4.5 Comparison of Standard and Cut-Decoding Algorithm for Rate R = 1/4 Here, the best results obtained with Standard algorithm with suitable results for Cut-Decoding algorithm are compared. Experiments are made using Cut-Decoding algorithm for codes (72,288) over the alphabet of nibbles and transmission through

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Fig. 10.16 Comparison of (a) P ER and (b) BER

a binary symmetric channel. In order to find the best code parameters, 17 different patterns of redundancy for code (72, 144) of rate .R = 1/2, different lengths of the initial keys, and different quasigroups of order 16 are considered. It is clear that the pattern of redundancy should be the same in both coding processes, due to the reduction of the elements in the decoding candidate sets (introduced in the Step 4 and the Step 5). First, experiments using only different keys in the two processes of coding/decoding and the same quasigroup are made. The best results were obtained for the following parameters: the pattern of redundancy 1100 1110 1100 1100 1110 1100 1100 1100 0000, two different keys .k1 = 01234 and .k2 = 56789 of 5 nibbles. and the quasigroup given in Fig. 10.13. The results with keys of length of 10 nibbles are similar. Let .P ERc be the probability of packet-error and .BERc be the probability of bit-error obtained with Cut-Decoding algorithm. As previous, .P ERs and .BERs are suitable probabilities for Standard RCBQ. The results for .P ERs and .P ERc (.BERs and .BERc ), for different values of bit-error probability p of binary symmetric channel and .Bmax = 4, are presented in Fig. 10.16. From the results, it can be concluded that the results for the values of P ER and BER with both decoding algorithms are approximately the same. Moreover, it is noted that for .p > 0.05 the values of P ER with Cut-Decoding algorithm are slightly better. Also, experiments for a code (72,288) using different keys and different quasigroups are made. From the experiments made using the quasigroup given in Fig. 10.13 and the quasigroup, with high coefficient of period growth, given at the end of the paper [9] it is concluded that the values of P ER and BER are similar to the previous values obtained using the same quasigroup in both processes. Also, the results obtained using two different quasigroups and the same key in the two coding/decoding processes are not better. Nevertheless, if the cyclic group of order 16 is used in one of the processes then the results for P ER and BER are much worse.

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10.4.6 Method for Reducing the Null-Errors in Cut-Decoding Algorithm Experiments using the method for decreasing the number of null-error (defined in Part 10.4.3) in the Cut-Decoding algorithm when a null-error appears (i.e. both reduced sets are empty) are made using different variants of this idea: returning two or three iterations back and using .Bmax + 1 or .Bmax + 2 in the first of canceled iterations (the remaining iterations use the previous value of .Bmax ). The best results are obtained with two iterations back and .Bmax + 2 in the first of the canceled iterations. It is concluded that using this method in Cut-Decoding algorithm, for larger values of p, gives a greater improvement of P ER. Also, approximately twice better values of BER for all p are achieved. This happens since in the cases when the null-error will not be eliminated by backtracking, the empty reduced sets can occur in some higher (later) iteration, so the bigger part of the message will be decoded and the value of the bit-error will be smaller. Also, experiments are made using this modification in the case when only one of (1) (2) the sets .Si or .Si is empty in some iteration. But in these experiments, the results are not better than in the previous case when one decoding process continues with the standard algorithm if the other set is empty. In that case, the decoding often ends successfully, especially when the empty set appears in some of the last iterations.

10.4.7 Method for Reducing the More-Candidate-Errors In [60], authors propose a similar modification with backtracking for decreasing the number of more-candidate-errors. If the decoding process ends with more-candidate-error, then in order to obtain one candidate for the decoded message, some iterations are reprocessed with a smaller value of .Bmax . Namely, when decoding ends with more elements in the last reduced decoding candidate sets, then a few iterations are canceled, and the first of canceled iterations is reprocessed using a smaller value of .Bmax . The next iterations use the previous value of .Bmax . Experiments using this modification in Cut-Decoding algorithm for the code (72,288) with the code parameters that give the best results for this algorithm are made. The improvements of the probabilities for successful decoding obtained by this method using different numbers of canceled iterations and using .Bmax − 1 or .Bmax − 2 in the first canceled iteration are considered. The best results are obtained if the last two iterations are canceled and .Bmax −1 is used in the first of the canceled iterations. From the results, it is concluded that this modification for reducing the number of more-candidate-errors improves P ER and BER, for all values of p. The improvement of packet-error and bit-error probabilities obtained with both methods for reducing errors by backtracking gives an idea to use a combination of these two methods. So, experiments are made using both methods with back-

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tracking, for null-errors and for more-candidate-errors. In these experiments, when null-error appears (i.e., empty sets in some iteration of the decoding process), two iterations are canceled and the first of canceled iterations is reprocessed using .Bmax + 2. On the other hand, if the decoding process ends with more elements in the last sets, then the process goes two iterations back and the .(s/2 − 1)-th iteration is reprocessed with .Bmax − 1. In one decoding process, only one backtracking for null-error is made, since with more than one too large cardinality of the sets S in some iteration is obtained. But, if after the backtracking for null-error there are more candidates in the last iteration then one more backtracking for more-candidateerror is made. From the experimental results, it is concluded that the proposed combination of the methods by backtracking for both types of errors gives the best improvement for the values of P ER and BER for all p (the values of BER are more than twice smaller).

10.4.8 Experiments with Quasigroups of Order 4 and Order 256 In [60] authors investigate the performances of the random codes based on quasigroups when quasigroups of order 4 and order 256 are used in coding/decoding processes. Then the messages and the codewords are strings of 2-bit letters or 8-bit letters (bytes), correspondingly. Several experiments with Standard and CutDecoding algorithms for a binary symmetric channel with different patterns for redundancy, different keys, different lengths of the blocks in the decoding process and several quasigroups of order 4 and order 256 are made. Experiments with Quasigroups of Order 4 Using image pattern, a classification of quasigroups of order 4 as fractal and non-fractal quasigroups is obtained. In [14] the authors give a classification of these quasigroups as linear and nonlinear by Boolean representation. The performances of RCBQ with messages of 2bit symbols using quasigroups from different classes of these classifications are investigated. Experiments are made using fractal quasigroups; non-fractal and weak non-linear; and non-fractal and pure non-linear quasigroups. The worst results are obtained using fractal quasigroups. Namely, if a fractal quasigroup is used in the algorithms for encryption/decryption (given in Table 10.6), then many unsuccessful decodings with more-candidate-error, even for .Bmax = 3 are obtained. On the other hand, in the experiments with non-fractal and weak non-linear quasigroups or non-fractal and pure non-linear quasigroups the values of packet-error probability and bit-error probability are similar, with slightly better results for a non-fractal and weak non-linear quasigroup. For .Bmax = 3 the results, in the experiments with 2-bit symbols, are better compared with experiments with nibbles. But, for larger values of .Bmax , there is a lot of unsuccessful decodings with more-candidate-error in all experiments for codes with 2-bit symbols (with the Standard algorithm and with the Cut-Decoding algorithm).

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Experiments with Quasigroups of Order 256 Also, experiments with the alphabet of bytes (8-bit symbols) using different patterns, keys, and quasigroups of order 256 are made. In these experiments, both decoding algorithms (Standard and CutDecoding) values for P ER and BER are almost the same as for codes with the alphabet of nibbles.

10.4.9 4-Set-Cut-Decoding Algorithms For obtaining a short decoding list and a faster decoding process, in [61] another coding/decoding algorithm is defined. It is called the 4-Sets-Cut-Decoding algorithm. This new algorithm gives a greater improvement of the decoding speed. Also, for improving the packet-error and bit-error probabilities several methods for generating reduced decoding candidate sets are defined.

10.4.9.1

Coding with 4-Sets-Cut-Decoding Algorithms

In this modification of Cut-Decoding algorithm instead of .(Nblock , N) code with rate R, four .(Nblock , N/4) codes with rate 4R, that encode/decode a same message of .Nblock bits are used. So, in the process of coding the encryption algorithm, given in Table 10.6, is applied on the same redundant message L four times using different parameters (different keys or quasigroups) and the codeword of the message is a concatenation of the four codewords of .N/4 bits.

10.4.9.2

Decoding with 4-Sets-Cut-Decoding Algorithms

After transmitting through a binary symmetric channel, the outgoing message D = D (1) D (2) . . . D (s) is divide in four messages .D 1 = D (1) D (2) . . . D (s/4) , 2 (s/4+1) D (s/4+2) . . . D (s/2) , .D 3 = D (s/2+1) D (s/2+2) . . . D (3s/4) and .D 4 = .D = D (3s/4+1) D (3s/4+2) . . . D (s) with equal lengths and they are decoded parallel with D the corresponding parameters. Similarly, as in Cut-Decoding algorithm, in each iteration of the decoding process the decoding candidate sets obtained in the four decoding processes are reduced. The authors propose four algorithms for reducing the length of the list. .

4-Sets-Cut-Decoding Algorithm#1 In the 4-Sets-Cut-Decoding algorithm.#1, the decoding algorithm consists of the following 6 steps.

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(1)

(1)

(4)

(4)

459 (4)

Step 1. Let .S0 = (k1 . . . kn ; λ), . . . , .S0 = (k1 . . . kn ; λ), where .λ is the (1) (1) (4) (4) empty sequence, and .k1 . . . kn , . . . , .k1 . . . kn are the initials keys used for obtaining the four codewords, respectively. (1) (4) Step 2. Let .Si−1 , . . . , .Si−1 be defined for .i ≥ 1. (1)

(4)

Step 3. Let four decoding candidate sets .Si , . . . , .Si be obtained in the four decoding processes, in the same way as in the standard RCBQ. Step 4. Let .V1 = {w1 w2 . . . wr·a·i |(δ, w1 w2 . . . wr·a·i ) ∈ Si(1) }, . . . , .V4 = {w1 w2 . . . wr·a·i .|(δ, w1 w2 . . . wr·a·i ) ∈ Si(4) } and .V = V1 ∩ V2 ∩ V3 ∩ V4 . (j ) Step 5. For each .j = 1, 2, 3, 4 and for each .(δ, w1 w2 . . . wr·a·i ) ∈ Si , if (j ) (j ) .w1 w2 . . . wr·a·i ∈ / V then .Si ← Si \ {(δ, w1 w2 . . . wr·a·i )}. (j ) (* Actually, for each .j = 1, 2, 3, 4, from .Si all elements whose second part does not match with the second part of an element in all other three sets are eliminated. In the next iteration, the four processes use the corresponding reduced (1) (2) (3) (4) sets .Si , .Si , .Si , .Si *). Step 6. If .i < s/4 then increase i and go back to Step 3. The decoding rule in all 4-Sets-Cut-Decoding algorithms is similar as in the Cut-Decoding algorithm. The difference is that now there are 4 sets S instead of two and if only one (or two) empty decoding candidate sets are empty then the decoding continues with the three (or two) nonempty sets (the set V in Step 4 is the intersection of the non-empty sets only). If in some iteration, only one set is nonempty, then the decoding continues with the nonempty set using the Standard decoding algorithm of RCBQ.

4-Sets-Cut-Decoding Algorithm#2 In the experiments with 4-Sets-Cut-Decoding algorithm.#1 has been noted that when the decoding process ends with null-error, i.e., when all four reduced sets are empty, very often the correct message is in three of the four non-reduced sets. Therefore, in 4-Sets-Cut-Decoding algorithm.#2 the following modification in Step 4 of the procedure for generating decoding candidate sets is made. (1)

Step .4#2 Let .V1 = {w1 w2 . . . wr·a·i |(δ, w1 w2 . . . wr·a·i ) ∈ Si }, . . . , .V4 = (4) {w1 w2 . . . wr·a·i .|(δ, w1 w2 . . . wr·a·i ) ∈ Si } and .V = V1 ∩ V2 ∩ V3 ∩ V4 . 1. 2. 3. 4.

If .V = ∅ then .V ' = V1 ∩ V2 ∩ V3 and .V = V ' . If .V ' = ∅ then .V '' = V1 ∩ V2 ∩ V4 and .V = V '' . If .V '' = ∅ then .V ''' = V1 ∩ V3 ∩ V4 and .V = V ''' . If .V ''' = ∅ then .V iv = V2 ∩ V3 ∩ V4 and .V = V iv .

In this way, a great improvement in the packet-error and bit-error probabilities is obtained, without decreasing the decoding speed. But, for greater improvement of the performances, the authors [61] consider two more modifications of Step 4 when the intersection of all four sets is an empty set.

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4-Sets-Cut-Decoding Algorithm#3 In the experiments with 4-Sets-Cut-Decoding algorithm.#2, better results for P ER and BER for all values of bit-error probability p of a binary symmetric channel are obtained. But, analyzing the experiments with null-error obtained with this algorithm, it can be notice the following situation. Namely, in some experiments ' /= ∅, but the correct message is not in .V ' and it is in .V '' or .V ''' or .V iv . .V Similarly, if (.V ' = ∅ and .V '' /= ∅) or (.V ' = ∅, .V '' = ∅ and .V ''' /= ∅) and the correct message is in some of the next intersections (which are not considered if a previous intersection is not empty). Therefore, the following modification of Step 4 is considered. (1)

Step .4#3 Let .V1 = {w1 w2 . . . wr·a·i |(δ, w1 w2 . . . wr·a·i ) ∈ Si }, . . . , .V4 = (4) {w1 w2 . . . wr·a·i .|(δ, w1 w2 . . . wr·a·i ) ∈ Si } and .V = V1 ∩ V2 ∩ V3 ∩ V4 . If .V = ∅ then .V = (V1 ∩V2 ∩V3 )∪(V1 ∩V2 ∩V4 )∪(V1 ∩V3 ∩V4 )∪(V2 ∩V3 ∩V4 ). With this modification, the improvement of the probabilities for packet-error and bit-error is better than with the 4-Set-Cut-Decoding algorithm.#2. Also, this modification does not decrease the speed of the decoding.

4-Sets-Cut-Decoding Algorithm#4 In 4-Sets-Cut-Decoding algorithm.#3, if new .V = ∅ then we have unsuccessful decoding with null-error. In order, to reduce these errors, another modification in the algorithm is made. Step .4#4 Let .V1 = {w1 w2 . . . wr·a·i |(δ, w1 w2 . . . wr·a·i ) ∈ Si(1) }, . . . , .V4 = {w1 w2 . . . wr·a·i .|(δ, w1 w2 . . . wr·a·i ) ∈ Si(4) } and .V = V1 ∩ V2 ∩ V3 ∩ V4 . If .V = ∅ then .V = (V1 ∩V2 ∩V3 )∪(V1 ∩V2 ∩V4 )∪(V1 ∩V3 ∩V4 )∪(V2 ∩V3 ∩V4 ). If .V = ∅ then .V = (V1 ∩ V2 ) ∪ (V1 ∩ V3 ) ∪ (V1 ∩ V4 ) ∪ (V2 ∩ V3 ) ∪ (V2 ∩ V4 ) ∪ (V3 ∩ V4 ). With this modification, only for .Bmax = 4 the results are better. For .Bmax = 5 a good percentage of eliminated null-errors is obtained, but a larger number of morecandidate-errors.

10.4.10 Comparison of the Algorithms for Rate R = 1/8 In this section, experimental results for the probabilities for packet-error (P ER) and bit-error (BER) obtained by using the 4-Sets-Cut-Decoding algorithms are presented and they are compared with the results obtained by the Standard decoding algorithm and Cut-Decoding algorithm. For obtaining a code .(Nblock , N) with rate R in the 4-Sets-Cut-Decoding algorithms, four .(Nblock , N/4) codes with rate 4R

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are used. Therefore, in these algorithms for a code with rate .1/4 there is not any redundancy (if .R = 1/4 then .N/4 = Nblock , i.e., the length of the codeword is equal to the length of the message). So, the comparison is for experiments with code (72,576) of rate .R = 1/8. Experimental results for packet-error probabilities for .Bmax = 4 and different values of bit-error probability p of a binary symmetric channel are presented in Fig. 10.17a. There, .P ERs are the packet-error probabilities obtained by Standard algorithm and .P ERc by Cut-Decoding algorithm. Also, by .P ERc4.1 , .P ERc4.2 , .P ERc4.3 and .P ERc4.4 the packet-error probabilities obtained using 4-Sets-CutDecoding#1, 4-Sets-Cut-Decoding#2, 4-Sets-Cut-Decoding#3 and 4-Sets-CutDecoding#4 are denoted, correspondingly. In Fig. 10.17b are presented the values for bit-error probability (BER) from the same experiments and we use the same labels for values obtained using different algorithms. For all considered algorithms experiments are made until .BER > p is obtained. From the results obtained for P ER, it can be derived the following conclusions. Using Cut-Decoding algorithm instead of Standard algorithm a great improvement of the probabilities for packet-error is obtained (for .p > 0.04, .P ERc are approximately twice smaller than .P ERs ). Also, Cut-Decoding algorithm is more than twice faster than the Standard algorithm. With 4-Sets-Cut-Decoding algorithm#1 we obtain worse results for P ER than those with Cut-Decoding algorithm. But this algorithm is more than 3 times faster than Standard algorithm and about 1.4 times faster than Cut-Decoding algorithm. From the values for .P ERc4.2 we can see that with this modification the results for P ER for all values of p are better (.P ERc4.2 are from 2 to 3.5 times smaller than .P ERc ) and the decoding speed is almost the same as with 4-Sets-Cut-Decoding algorithm#1 (more than 3 times faster than the Standard algorithm). From the results for .P ERc4.3 we can see that with 4-Sets-Cut-Decoding algorithm#3 a large number of null-errors are eliminated (for .p ≥ 0.04, .P ERc4.3 are from 1.2 to 5.2 times smaller than .P ERc4.2 ), again without decreasing the decoding speed.

Fig. 10.17 Comparison of (a) P ER and (b) BER for .Bmax = 4

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Fig. 10.18 Comparison of (a) P ER and (b) BER for .Bmax = 5

And with the last modification, i.e., 4-Sets-Cut-Decoding algorithm#4, for .p > 0.05 the results for packet-error probabilities are better, with almost the same decoding speed. Moreover, for .p > 0.09, the values of .P ERc4.4 are approximately 3 times smaller than .P ERc4.3 . From the results for BER we can derive the same conclusions as for P ER (for all algorithms and for all p, BER is approximately .P ER/2). In Fig. 10.18, we present the results for packet-error and bit-error probabilities for .Bmax = 5. From the results for .Bmax = 5, we can conclude that using Cut-Decoding algorithm we obtain better results than with Standard algorithm and for .p < 0.06 the experiments with Cut-Decoding algorithm are 5.2 times faster than with the standard one. For .p ≥ 0.06, in some experiments with Cut-Decoding algorithm, the cardinality of the decoding candidate sets is very large (after an iteration), so the decoding speed decreases (but it is still better than the speed obtained with Standard algorithm). Using the 4-Sets-Cut-Decoding algorithm#1 the results for P ER and BER are worse than those with Cut-Decoding algorithm. But, this algorithm is 6.3 times faster than the Standard algorithm and from 1.2 to 6.2 times faster than CutDecoding algorithm. From the results obtained by 4-Sets-Cut-Decoding algorithm#2, we can see that the values for P ER and BER are better with a significantly small decrease of the decoding speed (the maximum difference is 0.84 s per message). Also, with 4-Sets-Cut-Decoding algorithm#3, the values for P ER and BER are almost twice better than with 4-Sets-Cut-Decoding algorithm#2. Again, the speed is almost unchanged. The results for .P ERc4.3 and .P ERc4.4 are almost identical, but the ratios of the number of null-errors and more-candidate-errors obtained by these two modifications of the algorithm are very different. Namely, with 4Sets-Cut-Decoding algorithm#3, we obtained only few unsuccessful decodings with more-candidate-error, and with 4-Sets-Cut-Decoding algorithm#4, we eliminated many of null-errors, but we obtained a much larger number of more-candidateerrors (especially for .p ≥ 0.08). The obtained results for .BERc4.4 are smaller than .BERc4.3 .

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10.4.11 Experiments with Methods for Reducing the Number of Errors Here, we present results from the experiments with the 4-Sets-Cut-Decoding algorithms using the following combination of both proposed methods with backtracking. If null-error appears, i.e., empty reduced sets in some iteration of the decoding process then two (for .Bmax = 5 or one for .Bmax = 4) iterations are canceled and the first of canceled iterations is reprocessed using .Bmax + 2. If the decoding process ends with more elements in the sets after the last iteration, then the process goes two iterations back and the penultimate iteration is reprocessed with .Bmax − 1. Also, if after the backtracking for null-error there are more candidates in the last iteration then a backtracking for more-candidate-error is made. We will analyze the results obtained using the above combination of the methods for reducing the number of unsuccessful decodings, with 4-Sets-Cut-Decoding algorithm.#3 and 4-Sets-Cut-Decoding algorithm.#4 (the best-obtained results). In Fig. 10.19a (for .Bmax = 4) and Fig. 10.20a (for .Bmax = 5) we compare the packet-error probabilities (.P ERc4.3_back , .P ERc4.4_back ) obtained using the above combination of both methods with backtracking with the probabilities (.P ERc4.3 , .P ERc4.4 ) obtained by 4-Sets-Cut-Decoding algorithm.#3 and 4-SetsCut-Decoding algorithm.#4 without backtracking. Also, in Fig. 10.19b (.Bmax = 4) and Fig. 10.20b (.Bmax = 5) we compare the suitable bit-error probabilities (.BERc4.3_back , .BERc4.4_back , .BERc4.3 , .BERc4.4 ) obtained in the same experiments. Experiments are made until .BER > p is obtained. From the results for .Bmax = 4, we can conclude that with the proposed backtracking (in both algorithms) P ER and BER for all p are improved. For 4Sets-Cut-Decoding algorithm#4, the values of BER obtained by backtracking are approximately twice smaller than suitable values without backtracking. Also, from the results for .Bmax = 5 we can see that with the proposed backtracking, a greater improvement of P ER and BER is obtained using 4Sets-Cut-Decoding algorithm.#3. Even more, for all values of p, .P ERc4.3_back are smaller than .P ERc4.4_back , although these probabilities in the experiments without

Fig. 10.19 (a) P ER and (b) BER without and with backtracking for .Bmax = 4

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Fig. 10.20 (a) P ER and (b) BER without and with backtracking for .Bmax = 5 Fig. 10.21 The original image

backtracking are almost identical. The reason for that is the increased cardinality of the decoding candidate sets in 4-Sets-Cut-Decoding algorithm.#4.

10.4.12 Application of RCBQ for Decoding Images In [61] performances of the 4-Sets-Cut-Decoding algorithm.#3 (using the combination of both methods with backtracking mentioned before) for coding/decoding images transmitted through a binary symmetric channel are investigated. For that aim experiments with the picture of Lenna (Fig. 10.21) were made. In these experiments for coding/decoding images transmitted through a binary symmetric channel, the code (72,576) with rate .R = 1/8 and the same parameters (as above) for 4-Sets-Cut-Decoding algorithms, are used. Experiments are made using .Bmax = 5 in the decoding process and the following values of bit-error probability in the channel: .p = 0.05, .p = 0.10, and .p = 0.15. In all decoding algorithms for RCBQ, when null-error appears then the decoding process ends early and only a part of the message is decoded. Therefore, in the experiments with images, the following solution is used. In the cases when null-

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error appears, i.e., all reduced sets .Si , . . . , .Si are empty, the strings without (1) (4) redundant symbols from all elements in the sets .Si−1 , . . . , .Si−1 are taken, and their maximal common prefix substring is found. If this substring has k symbols then, in order to obtain decoded message of l symbols, these k symbols are taken and .l − k zero symbols are added at the end of the message. In the experiments with images, it is noticed that this type of error makes the most visible changes in the decoded images. Images obtained for the considered values of bit-error probability p in the binarysymmetric channel are presented in Figs. 10.22, 10.23, and 10.24. In these figures, the images in (a) are obtained after transmission through the channel without using any error-correcting code. In (b) we give the images obtained using RCBQ with

Fig. 10.22 Images for .p = 0.05

Fig. 10.23 Images for .p = 0.10

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Fig. 10.24 Images for .p = 0.15

4-Sets-Cut-Decoding algorithm.#3 with the proposed combination of both methods with backtracking. As it is explained above, in the experiments with RCBQ we put zero symbols in the place of the non-decoded part of the message when the decoding process ends with null-error. In fact, these zero symbols are the horizontal lines that can be seen in Figs. 10.22b–10.24b. On the other hand, the images obtained without using errorcorrecting codes (Figs. 10.22a–10.24a) do not have these lines, but the entire images have points that are incorrectly transmitted symbols. From the figures, it can be seen that using RCBQ with 4-Sets-Cut-Decoding algorithm.#3 (with backtracking) the obtained images are relatively clear even for large values of p. Therefore, these codes and the proposed algorithms can be applied for coding/decoding data transmitted through a channel with a large biterror probability. Moreover, due to their cryptographic properties, these codes will provide information security as well.

10.4.13 Application of RCBQ for Decoding Audio Files In [55] authors investigate performances of RCBQ for transmission of audio files through a binary-symmetric channel. Here, we will present and analyze several experimental results obtained using Cut-Decoding and 4-Sets-Cut-Decoding algorithm.#3 for code (72, 576) with rate .1/8 and .Bmax = 5. In both algorithms, the proposed methods with backtracking for reducing the number of errors are used. In all experiments, authors use the audio signal that is consisted of one 16-bit channel with a sampling rate of 44,100 Hz and it is a part of Beethoven’s “Ode to joy” with a total length of approximately 4.3 s.

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The experimental results for bit-error probabilities .p = 0.05, .p = 0.08 and p = 0.11, using Cut-Decoding (in a) and 4-Sets-Cut-Decoding algorithm.#3 (in b) are presented in Figs. 10.25, 10.26, and 10.27. In all figures, the difference between the audio signals (original and decoded after transmission through the channel) is presented. There, the number of the sample in the sequence of samples in the audio signal is on the x-axis and the value of the sample is on the y-axis. The original audio samples are colored in red, and the decoded audio samples are colored in blue. It is evident from Figs. 10.25, 10.26, and 10.27, that for these probabilities 4-SetsCut-Decoding algorithm.#3 gives better results than Cut-Decoding algorithm. Also, from Fig. 10.27a we can see that the signal decoded using Cut-Decoding algorithm is with a lot of noise. In the experiments with Cut-Decoding algorithm for bit-error probabilities greater than 0.11, BER for decoded audios is greater than the bit-error probability in the channel. Therefore, there is no sense to make experiments with this algorithm for channels with .p ≥ 0.11.

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In Fig. 10.28 we present the difference between the original and decoded signal using the 4-Sets-Cut-Decoding algorithm.#3 for bit-error probability .p = 0.14 (in a) and .p = 0.17 (in b). From these results, for transmission of audio files through a binary-symmetric channel, it can be concluded that for all values of p, the 4-Sets-Cut-Decoding algorithm.#3 gives better results than the Cut-Decoding algorithm. Also, 4-Sets-CutDecoding algorithm.#3 is from 1.2 to 6.2 times faster than Cut-Decoding algorithm. All audio files from these experiments, original and decoded, can be found on the link https://www.dropbox.com/sh/mt36x7rq1u5czqu/AAC0zcKiODy4fYOWoTNx 6cmGa?dl=0. If one listens to these (decoded) audio files, s/he can notice the following: as p increases, the noise increases too, but the original melody is heard completely in the background.

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10.4.14 Some Theoretical Results for the New Algorithms of RCBQ The formula for theoretical packet-error probability for Standard algorithm is given in Theorem 3. But, from the proof of this theorem, given in [16], it is clear that in this formula more-candidate-errors are not provided. The number of these errors strongly depends on the chosen quasigroup, the pattern for adding redundancy and the length of the initial key. Therefore, it is very difficult to predict the number of more-candidate-errors. In [61] it is proven that the formula for .P ERtheory in Theorem 3 (for same code rate, length of the blocks, and .Bmax in the decoding process) gives an upper bound for P ER obtained by Cut-Decoding and 4-Sets-Cut-Decoding algorithms. For the same reasons as above, in the following theorem and its proof only the probability for null-errors and uncorrected-errors (decoding that completes successfully, but the decoded message is incorrect) is considered. Theorem 4 The packet-error probability .(without more-candidate-errors.) obtained by Standard decoding algorithm of RCBQ is an upper bound for the packet-error probability obtained using Cut-Decoding and 4-Sets-Cut-Decoding algorithms .(for same code rate, length of the blocks and .Bmax in the decoding processes.). Proof The probability that maximum t bits in one block .D (i) of r symbols are not correctly transmitted is ) t ( E r ·a k p (1 − p)r·a−k , .P (p; t) = k k=0

where p is the probability of bit-error in a binary symmetric channel. The events .Ai : “maximum .Bmax errors appear in a block .D (i) ”, for all i are independent and the probability that the correct block .C (i) is contained in the set .Hi is .P (p; Bmax ) ≥ 1 − qB . Since, for a code with rate R in Cut-Decoding algorithm, two codes with rate 2R are used, the number of iterations in the decoding process is twice smaller. So, if the decoding process with Standard algorithm has s iterations for a given code, then for the same code the number of iterations in the two parallel processes of decoding in Cut-Decoding algorithm is .s/2. When Cut-Decoding algorithm is used, then the correct block .L(i) , .i = 1, 2, ..., s/2 is contained (as the second part of an element) in the reduced (1) (2) sets .Si and .Si if in both processes the correct blocks .C (i) and .C (s/2+i) (from (1) (2) the two codewords) are contained in the corresponding sets .Hi and .Hi . The probability for that event is .(1 − qB )(1 − qB ) for each i, since both decoding processes are independent. Therefore, the probability each correct block .L(i) to (1) (2) be contained in the reduced sets .Si and .Si for .i = 1, 2, ..., s/2 is at least

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((1 − qB )(1 − qB ))s/2 = (1 − qB )s . So, the probability for packet-error is at most s .1 − (1 − qB ) . But, according to the decoding rule of Cut-Decoding algorithm, when one of the decoding candidate sets is empty, decoding of the message continues using the Standard algorithm. In this case, the process will end successfully if the correct block is contained in the non-empty set. Therefore, probability .1 − (1 − qB )s (which is probability .P ERtheory in Theorem 3) remains an upper bound for the packet-error probability (without more-candidate-errors) obtained by Cut-Decoding algorithm. Similarly, the same upper bound for the 4-Sets-Cut-Decoding algorithm is obtained. In this algorithm, there are four decoding processes, and for a code with rate R, four codes with rate 4R are used. Therefore, in 4-Sets-Cut-Decoding algorithm, the number of iterations is we have four times smaller (.s/4). In all iterations, the correct block .L(i) , .i = 1, 2, ..., s/4 is contained (as the second part of an element) in the reduced sets .Si(1) , .Si(2) , .Si(3) and .Si(4) if the correct blocks .C (i) , (s/4+i) , .C (s/2+i) and .C (3s/4+i) (from the four codewords) are contained in the .C (1) (2) (3) (4) corresponding sets .Hi , .Hi , .Hi and .Hi . As previously, the upper bound for packet-error probability is .1−((1−qB )(1−qB )(1−qB )(1−qB ))s/4 = 1−(1−qB )s . Again this probability is an upper bound for P ER (without more-candidate-errors) since the decoding of the message also continues when at least one of the decoding candidate sets is not empty. .

With this theorem, it is proved that with the Cut-Decoding and 4-Sets-CutDecoding algorithms improvements are obtained not only in the decoding speed but also in decreasing the number of unsuccessful decodings (when the upper bound is not reached). In [16] the authors give an approximate formula for calculating the cardinality .|Si | of the decoding candidate sets obtained in the Standard algorithm of RCBQ. They obtained an approximation for .|Si | in the following way. Consider .(Nblock , N) code with rate R and let .L = L(1) L(2) ...L(s) be a redundant message. Let .Ni be the number of information nibbles in the sub-block .L(i) of r nibbles. Let .Bchecks be the cardinality of the set .Hi , for .i ≥ 1. If we suppose that the strings obtained as the output of the inverse coding algorithm are almost random, then the probability for obtaining a string with .r − Ni zero nibbles (as output) is 1 . . So, the approximate formula for the (expected) cardinality of the first set 4(r−N i) 2 .S1 is: .

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|Si | ≈ |Si−1 |

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These approximate formulas for the cardinality of decoding candidate sets give better approximation if for calculation of .|Si | we use experimentally obtained values of .|Si−1 | instead of the approximate one (obtained in the previous step of approximation). Similar approximate formulas for the cardinality of the reduced decoding candidate sets obtained by Cut-Decoding and 4-Sets-Cut-Decoding algorithms are derived. These approximations are obtained using the same assumption, as above, that the process of forming the decoding candidate sets is almost random. (1) (2) Let .S1 and .S1 be the sets obtained in the first iteration| of |Cut-Decoding | (1) | (1) algorithm before the reduction in Step 4 and Step 5 and let .|S1 | = n1 and | | | (2) | (2) .|S 1 | = n1 . As we mentioned above it is assumed that the decoding candidate sets are random. If .N1 is the number of information nibbles in the sub-block .L(1) of (j ) r nibbles, then the strings (second parts of the elements) in the sets .S1 , j = 1, 2 can be considered as strings of .4N1 bits. For any string k of .4N1 bits, the probability (1) .p1 that k is a second part of some element in the set .S 1 , i.e., .k ∈ V1 is: ( .p1 = P {k ∈ V1 } = 1 − 1 −

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(and .Si ) obtained from each element in the corresponding decoding candidate set in the previous iteration is approximately equal. This means that the number of (1) (2) elements in .Si (and .Si ) with strings that have an equal prefix is approximately equal. The experimental results show that this is (approximately) true. Let .mi−1 be the number of elements candidate sets obtained in the | in| the reduced |decoding | | (1) | | (2) | (1) (2) .(i − 1)th iteration and .|S i | = ni and .|Si | = ni . Then by using the above (j )

assumption, it is obtained that in the set .Si , i.e., sets .Vj (in Step 4), for .j = (j ) ni 1, 2, there are .mi−1 classes of . elements that have strings in the second part mi−1 with an equal prefix of .r(i − 1) nibbles. So, in each of these classes the elements differ only in the last r nibbles, i.e., more precisely in .Ni nibbles, where .Ni is the number of information nibbles in the sub-block .L(i) of the redundant message. The intersection .V = V1 ∩ V2 (in Step 4) will be non-empty only for the elements from the corresponding classes (of elements with the same prefix) in the sets .V1 and .V2 . Using the same method, as for the sets in the first iteration, it is obtained that the cardinality of the intersection of the corresponding classes from the sets .V1 and .V2 is approximately: ( 4Ni

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1 where .q = 1 − 4N . Since, there are .mi−1 classes of elements with the same 2 i prefix of .r(i − 1) nibbles, the approximate formula for the cardinality of the reduced decoding candidate sets in the ith iteration is: ( ( (1) ) ( (2) )) ni ni | | | (j ) | 4Ni . |S 1 − q mi−1 , j = 1, 2. 1 − q mi−1 i | ≈ mi−1 2 Similarly, approximate formulas for the cardinality of the reduced decoding candidate sets in 4-Sets-Cut-Decoding algorithm.#1 are derived. The only difference is the number of sets in this algorithm. Here, there are 4 sets instead of 2. Using the same notations as above and in Sect. 10.4.9 the following approximate formulas are obtained: • for the cardinality of the reduced decoding candidate sets in the first iteration | | | (j ) | E |S1 | ≈ k P {k ∈ V1 ∩ V2 ∩ V3 ∩ V4 } .

( )( )( )( ) (1) (2) (3) (4) 1 − q n1 1 − q n1 1 − q n1 , = 24N1 1 − q n1 for j = 1, 2, 3, 4;

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• for the cardinality of the reduced decoding candidate sets in the ith iteration, .i > 1 (1) )( (2) )( (3) )( (4) )) ( ( ni ni ni ni | | | (j ) | 4Ni mi−1 mi−1 mi−1 mi−1 . |S 1−q 1−q 1−q , 1−q i | ≈ mi−1 2

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With these approximate formulas, a good approximation for the cardinality of the decoding candidate sets after the reduction in the Cut-Decoding and 4-Sets-CutDecoding algorithms is obtained. The same as previous, the derived formulas give better approximation if for calculation of .|Si | we use experimentally obtained values of .|Si−1 | instead of an approximate one. The precision is almost the same as in the formulas given in [16] for the Standard algorithm (the maximum deviation that is obtained is 10). Using these approximations for the cardinality of the decoding candidate sets, it can be checked whether some pattern for redundancy is a good one. Also, the speed of decoding and the number of more-candidate-errors depend on the number of elements in the sets S. From these formulas, it is clear that the reduction of the decoding candidate sets, in Cut-Decoding and 4-Sets-Cut-Decoding algorithms, significantly decreases the number of elements in the sets.

10.5 Error-Detecting Codes Based on Quasigroups The error-detecting codes are the codes used for detecting the error in the received data bitstream. Error-detecting codes encode the message before sending it over the noisy channels. Actually, error-detecting codes add some redundant symbols to a given original message to help us detect if any error has occurred during transmission of this message. The encoding scheme is performed in such a way that the decoder can find the errors easily in the receiving data with a higher chance of success. In this section, error-detecting codes based on quasigroups are considered [1, 27, 31, 32].

10.5.1 Designing of Error-Detecting Codes Based on Quasigroups Let A be an arbitrary finite set called alphabet and .(A, ∗) be a given quasigroup. Let consider an input message a1 a2 . . . an an+1 an+2 . . . a2n a2n+1 . . . ,

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which will be transmitted through a noisy channel. Since of the noise, the received message can be different of the sent one. The goal is designing a code which will detect the errors during transmission such that the probability of undetected errors will be as small as possible. For that reason, some redundancy to the message (i.e., some control bits) has to be added. Let’s divide the input message to blocks with length n: a1 a2 . . . an , an+1 an+2 . . . a2n , . . .

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where .k ≤ n. The rate of this code is .1/2. At first, each letter from the extended block .a1 a2 . . . an b1 b2 . . . bn will be presented in 2-base system. After that the obtained binary block will be transmitted through the binary symmetrical channel with probability of bit error p (.0 < p < 0.5). The presence of noise in the channel leads to incorrect transmission of some bits. Let .ai be transmitted as .ai' , .bi as .bi' , .i ∈ {1, 2, . . . , n}. If the character transmission is correct than .ai' will have the same value as .ai . Otherwise, .ai' will be different than .ai (.ai' = 0, .ai = 1, or opposite). So, the output message is ' ' ' ' ' ' .a a . . . an b b . . . bn . To check if there are any errors during transmission, the 1 2 1 2 receiver of the message checks if b1' b2' . ... bn'

= a1' ∗ a2' ∗ · · · ∗ ak' ' = a2' ∗ a3' ∗ · · · ∗ ak+1 ... .................. ' = an' ∗ a1' ∗ · · · ∗ ak−1

If any of these equalities are not satisfied, the receiver concludes that some errors occurred during the block transmission and it asks from the sender to send that block once again. But, some equality can be satisfied although some characters in that equality are incorrectly transmitted. In that case, incorrect transmission (error in transmission) will not be detected. We will consider two special cases of the proposed code. For the first one, we choose .A = {0, 1} and .k = 4 and for the second one, .A = {0, 1, 2, 3} and .k = 2. Our goal is finding approximately the probability of undetected errors and make that probability as small as possible. In the both codes, each redundant symbol .bi , defined in (10.6), includes the same number of bits, i.e., 4 bits from the input message, so it is reasonably to compare the obtained probabilities of undetected errors.

10 Applications of Quasigroups in Cryptography and Coding Theory

10.5.1.1

475

An Error-Detecting Code Based on Quasigroup of Order 2 and k=4

Let consider the binary set .A = {0, 1}. There are only two quasigroup operations on the set A, and here we took .(A, ∗) to be defined by the table

.

∗01 001 110

Denote that same results will be obtained if another quasigroup is used. Each block .a1 a2 . . . an (.ai ∈ A) is extended to a block a1 a2 . . . an b1 b2 . . . bn ,

.

where .bi = ai ∗ ari+1 ∗ ari+2 ∗ ari+3 . Here { rj =

.

j, j ≤n j mod n, j > n

for .j = i + 1, i + 2, i + 3. Let introduce the following notation: g(x1 , x2 , . . . , xn ) = x1 ∗ x2 ∗ · · · ∗ xn ,

.

where .xi ∈ {0, 1}, .i = 1, 2, . . . , n. In order to determine the probability of undetected errors, the following proposition can be used. Its proof is obvious. Proposition 1 If odd number of .x1 , x2 , . . . , xn .(xi ∈ {0, 1}) change their values then .g(x1 , x2 , . . . , xn ) will change its value, too. If even number of .x1 , x2 , . . . , xn change their values then the value of .g(x1 , x2 , . . . , xn ) will be unchanged. Using the previous proposition and some combinatorics the following theorem can be proved. Theorem 5 Let .f2 (n, p) be the probability function of undetected errors in a transmitted block with length n through the binary symmetric channel, where p is the probability of incorrect transmission of a bit. Then .f2 (n, p) is given by the following formulas:

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f2 (4, p) = 6p2 (1 − p)6 + p4 (1 − p)4 + 4p5 (1 − p)3 + 4p7 (1 − p) f2 (5, p) = 10p4 (1 − p)6 + 16p5 (1 − p)5 + 5p8 (1 − p)2 f2 (6, p) = 2p3 (1 − p)9 + 6p4 (1 − p)8 + 18p5 (1 − p)7 +16p6 (1 − p)6 + 6p7 (1 − p)5 + 9p8 (1 − p)4 + O(p9 ) f2 (7, p) = 7p4 (1 − p)10 + 21p5 (1 − p)9 + 21p6 (1 − p)8 + 29p7 (1 − p)7 +28p8 (1 − p)6 + O(p9 ) f2 (8, p) = 14p4 (1 − p)12 + 8p5 (1 − p)11 + 24p6 (1 − p)10 + 56p7 (1 − p)9 +49p8 (1 − p)8 + O(p9 ) f2 (9, p) = 9p4 (1 − p)14 + 9p5 (1 − p)13 + 36p6 (1 − p)12 + 81p7 (1 − p)11 +63p8 (1 − p)10 + O(p9 ) 4 16 + 12p 5 (1 − p)15 + 20p 6 (1 − p)14 + 100p 7 (1 − p)13 . f2 (10, p) = 10p (1 − p) 8 +120p (1 − p)12 + O(p9 ) f2 (11, p) = 11p 4 (1 − p)18 + 11p5 (1 − p)17 + 22p6 (1 − p)16 + 99p7 (1 − p)15 +132p8 (1 − p)14 + O(p9 ) f2 (12, p) = 12p 4 (1 − p)20 + 12p5 (1 − p)19 + 30p6 (1 − p)18 + 72p7 (1 − p)17 +162p8 (1 − p)16 + O(p9 ) f2 (13, p) = 13p 4 (1 − p)22 + 13p5 (1 − p)21 + 26p6 (1 − p)20 + 78p7 (1 − p)19 +182p8 (1 − p)18 + O(p9 ) f2 (n, p) = np4 (1 − p)2n−4 + np5 (1 − p)2n−5 + 2np6 (1 − p)2n−6 +6np7 (1 − p)2n−7 + Ap8 (1 − p)2n−8 + Bpn/2 (1 − p)3n/2 + O(p9 ), for n ≥ 14 where ⎧ (n + 9)n ⎪ ⎨ , n = 15, 17, 19, . . . 2 .A = (n + 8)n ⎪ ⎩ , n = 14, 16, 18, . . . 2

⎧ ⎨ 0, n odd B = 2, n even, but 4 | n ⎩ 6, 4|n

The remainder .O(p9 ) denotes that the coefficients are exactly determined in terms which contain .pi , .i < 9. To obtain exactly the probability of undetected errors, i.e., to obtain exactly .O(p9 ), one has to make much complicated combinatorial calculations. In Fig. 10.29, we can see that for small values of n, all functions have maximum in .p = 0, 5. When the block length n increases, the maximum becomes smaller, it goes to the left and the sequence of maximums converges to 0.

10.5.1.2

An Error-Detecting Code Based on Quasigroup of Order 4 and k=2

Consider the set .A = {0, 1, 2, 3} and let .∗ be an arbitrary quasigroup operation on A. According to (10.6), we extend each block .a1 a2 . . . an (.ai ∈ A) to a block .a1 a2 . . . an b1 b2 . . . bn , where .bi = ai ∗ a(i mod n)+1 , .i = 1, 2, . . . , n. The extended

10 Applications of Quasigroups in Cryptography and Coding Theory

477

Fig. 10.29 The probability functions of undetected errors

message is transmitted through the binary symmetrical channel again. As previous, the probability of undetected errors has to be calculated. There are 576 quasigroups of order 4. One can find that for some quasigroups, the probability of undetected errors depends on the distribution of letters in the input message. So, let filter the quasigroups such that this formula is independent from the distribution of the input message. After filtering, from the 576 quasigroups of order 4, only 160 quasigroups remain. All of them are fractal quasigroups, but not all fractal quasigroups are in these 160 quasigroups. Let .f4 (n, p) be the probability of undetected errors in a transmitted block with length n through the binary symmetric channel where p is the probability of incorrect transmission of a bit. In [1], .f4 (n, p) is determined for all 160 filtered quasigroups. But, these 160 quasigroups do not define 160 different functions for the probability of undetected errors, but only 7. The graphs of these seven function for .n = 7 is given on Fig. 10.30. Second index of .f4,j (n, p), for .j = 1, . . . , 7 denotes the serial number of the function. From Fig. 10.30, we can see that the function .f4,1 (n, p) is the best one, it gives the smallest probability of undetected errors. But the function .f4,2 (n, p) is very closed to the .f4,1 (n, p). Their plots almost overlap each other. The function .f4,1 (n, p) is given by the following formulas:

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Fig. 10.30 Seven different functions of probability of undetected errors

f4,1 (2, p) = p 4 (15p 4 − 56p 3 + 84p 2 − 56p + 14) f4,1 (3, p) = p 4 (63p 8 − 372p 7 + 990p 6 − 1540p 5 + 1545p 4 − 1032p 3 + 452p 2 − 120p + 15) f4,1 (4, p) = p 4 (255p 12 − 2032p 11 + 7560p 10 + 27556p 8 − 32112p 7 − 17360p 9 + 28440p 6 − 19440p 5 + 10206p 4 − 4000p 3 + 1104p 2 − 192p + 16) 4 2(2n−8) × .f 4,1 (n, p) = np [ (1 − p) × 4 − 48p + 274p 2 − 980p 3 + (8n + 2431)p 4 − − − −

8(8n + 547)p 5 + 2(130n + 2853)p 6 4(166n + 1259)p 7 + (9n2 + 1078n + 2297)p 8 4(9n2 + 270n − 139)p 9 + (81n2 + 371n − 890)p 10 11 + (3/8)(9n3 2(45n2 − 165n + 194)p ]

− 42n2 + 75n − 34)p 12 + O(p 7 ),

for n ≥ 5.

The function .f4,1 (n, p) without the remainder .O(p7 ) gives the probability that at most 4 characters of the input message are incorrectly transmitted and the errors are not detected. As previous, to obtain the probability of undetected errors exactly, one has to calculate the probability that more than 4 characters are incorrectly transmitted and the errors are not detected, which is much complicated combinatorial problem. The shape of the probability functions of undetected errors is similar as in the previous case. When the block length n increases the maximum of

10 Applications of Quasigroups in Cryptography and Coding Theory

479

Fig. 10.31 The probability functions of undetected errors

these functions becomes smaller, it goes to the left and the sequence of maximums converges to 0 (Fig. 10.31). These seven probability functions of undetected errors (presented on Fig. 10.30) enable to classify 160 filtered quasigroups in seven different sets according to goodness for designing of error-detecting codes. These classes are given in [1] where each quasigroup is presented by a number according to lexicographic ordering of the set of quasigroups of order 4. Repeat that all of these 160 quasigroups are fractal. The sets 1–6 contain only linear fractal quasigroups. The set 7 contains two subsets such that the subset .7' contains linear fractal quasigroups too, but the subset .7'' contains 16 nonlinear fractal quasigroups with nonlinear part .x1 x3 +x2 x3 + x1 x4 +x2 x4 . Also, one can check that there is not quasigroup in set with the smallest probability of undetecting errors (determined by the function .f4,1 (n, p)) which is a group.

10.5.1.3

Controlling of Undetected Errors and Comparing

The main goal for these codes is to control the probability of undetected errors, actually to make that probability smaller than some previous given value .ε. So, one has to find for which values of n the maximum of the function .f (n, p) (.f (n, p) can be .f2 (n, p) or .f4,1 (n, p)) is smaller then .ε. Since the sequence of maximums of the functions .f (n, p) is strictly decreasing and converges to 0 when .n → ∞, there will be .n0 ∈ N, such that the maximum of the function .f (n, p) will be smaller than .ε, for all .n ≥ n0 and the maximum of the function .f (n, p) will be greater than .ε, for all .n < n0 . We choose .n = n0 (see Fig. 10.32). Now, let’s separate the message in blocks with length n and we code every block individually. From all values of n which satisfies the condition .f (n, p) < ε, the smallest one is chosen since in this case the transmission will be fastest. Namely, if the receiver detects errors in the received block, it asks for repeated transmission, so it is better the block length to be as small as possible.

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Fig. 10.32 Choosing of .n0 Table 10.8 The maximums of the probability functions

n 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Quasigroups of order 2 × 10−4 −4 .5.29529 × 10 −4 .3.52349 × 10 −4 .2.48784 × 10 −4 .1.86131 × 10 −4 .1.43616 × 10 −4 .1.13480 × 10 −5 .9.13489 × 10 −5 .7.47017 × 10 −5 .6.19084 × 10 −5 .5.19030 × 10 −5 .4.39585 × 10 −5 .3.75666 × 10 −5 .3.23631 × 10 −5 .2.80827 × 10 −5 .2.45283 × 10 −5 .2.15517 × 10 .9.75609

Quasigroups of order 4 × 10−5 −5 .6.82458 × 10 −5 .5.14707 × 10 −5 .3.97896 × 10 −5 .3.14013 × 10 −5 .2.52198 × 10 −5 .2.05631 × 10 −5 .1.69878 × 10 −5 .1.41968 × 10 −5 .1.19860 × 10 −5 .1.02120 × 10 −6 .8.77182 × 10 −6 .7.59050 × 10 −6 .6.61231 × 10 −6 .5.79537 × 10 −6 .5.10775 × 10 −6 .4.52483 × 10 .9.35406

In Table 10.8, we give the maximums of the probability functions of undetected errors for the first and the second proposed code. From this table, one can conclude that the maximums of the functions of undetected errors are smaller when the quasigroups of order 4 are used. It suggest that by using a quasigroup of order 4 better and more efficiently codes will be obtained.

10 Applications of Quasigroups in Cryptography and Coding Theory

481

10.5.2 Error-Detecting Codes Based on Quasigroups with Cyclically Defined Redundancy In paper [32], the authors investigate some modification of the error-detecting codes proposed in previous subsection. The question is, if some other combinations of symbols in the definition of redundant bits (10.6) are used, then the probability of undetected error will be smaller. There, error-detecting codes based on quasigroup operation .∗ on the set .A = {0, 1} are considered. The operation .∗ is given by the multiplication table

.

∗01 010 101

The codes from the previous subsection are generalized on the following way. The input message .a1 a2 . . . an is extended to .a1 a2 . . . an d1 d2 . . . dn , where d1 d2 . ... di

= aσ (1) ∗ aσ (2) ∗ · · · ∗ aσ (k) = aσ (1)+1 ∗ aσ (2)+1 ∗ · · · ∗ aσ (k)+1 ... .................. = aσ (1)+i−1 ∗ aσ (2)+i−1 ∗ · · · ∗ aσ (k)+i−1

(10.7)

and .σ (1)σ (2) . . . σ (k) is an arbitrary combination of the set .{1, 2, . . . , n}. If some index .σ (r) + s is greater than n, the index .(σ (r) + s) mod n is taken. The authors looked for the combination that gives the best function of undetected errors, i.e., the smallest probability for undetected errors. Let note that for that aim it is not necessary to consider all combinations. For example, the combinations 123, 234, 345 of the set .{1, 2, 3, 4, 5} give the same bits of the redundancy extension in different order, but the order do not change the probability function of undetected errors (by Proposition 1). Therefore, it is enough to consider combinations that contain 1. The operation .∗ is associative and this gives possibility to reduce the number of combinations more. For example, if .n = 5 and the combination 124 is considered, the obtained block is .a1 . . . a5 d1 . . . d5 , where d1 d2 . d3 d4 d5

= a1 ∗ a2 ∗ a4 = a2 ∗ a3 ∗ a5 = a3 ∗ a4 ∗ a1 = a4 ∗ a5 ∗ a2 = a5 ∗ a1 ∗ a3

So, the combinations 235, 134, 245, 135 can be excluded for consideration. In that way, the number of combinations can be reduced more. Here are some particular results.

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Table 10.9 Probability functions for combinations of 3 elements of .{1, 2, . . . , 8} Combinations 123, 136 124, 127 125, 126 135

Probability function of undetected errors = 16p 4 (1 − p)12 + 48p 6 (1 − p)10 + 126p 8 (1 − p)8 + 48p 10 (1 − p)6 12 4 16 .+16p (1 − p) + p 4 12 6 10 8 8 10 6 .s2 (p) = 8p (1 − p) + 80p (1 − p) + 78p (1 − p) + 80p (1 − p) 12 4 16 .+8p (1 − p) + p 4 12 6 10 8 8 10 6 .s3 (p) = 20p (1 − p) + 32p (1 − p) + 150p (1 − p) + 32p (1 − p) 12 4 16 .+20p (1 − p) + p 4 12 8 8 12 4 16 .s4 (p) = 28p (1 − p) + 198p (1 − p) + 28p (1 − p) + p .s1 (p)

Fig. 10.33 Probability functions of undetected errors for some combinations of 3 elements of the set .{1, 2, . . . , 8}

10.5.2.1

Combinations of the Set {1, 2, . . . , 8}

According to previous, the set of all combinations of 3 elements of the set {1, 2, . . . , 8} can be reduced to the set .{123, 124, 125, 126, 127, 135, 136}. The probability functions of undetected errors for these seven combinations are given in Table 10.9. The plots of the functions .s1 (p), .s2 (p), .s3 (p) and .s4 (p) are given on Fig. 10.33 and .s2 (p) is the best result for .n = 8 and .k = 3. The results of the probability functions for combinations of 4 elements of the set .{1, 2, . . . , 8} are given in Table 10.10. The plots of the functions .s5 (p), .s6 (p) and .s7 (p) are given on Fig. 10.34. The functions .s8 (p) and .s9 (p) are not presented on this figure since these functions are much worse than the other tree functions. Analyzing the functions on Fig. 10.34, one can conclude that the function .s6 (p) is a best one, i.e. it gives the smallest probability of undetected errors. The comparison of .s2 (p) and .s6 (p) (as the best in their classes) shows that .s6 (p) is a little bit better than .s2 (p). The investigation of all other combinations for .k = 5, 6, 7 show that .s6 (p) is the best result for .n = 8.

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483

Table 10.10 Probability functions for combinations of 4 elements of .{1, 2, . . . , 8} Combinations 1234, 1247

Probability function of undetected errors = 14p 4 (1−p)12 +8p5 (1−p)11 +24p6 (1−p)10 +56p7 (1−p)9 +49p8 (1−p)8

.s5 (p)

.+56p

1235, 1237, 1246, 1257 1236, 1245

1256

9 (1−p)7 +28p 10 (1−p)6 +8p 11 (1−p)5 +8p 12 (1−p)4 +4p 14 (1−p)2

= 24p5 (1 − p)11 + 44p 6 (1 − p)10 + 40p 7 (1 − p)9 + 45p 8 (1 − p)8 9 7 10 6 11 5 12 4 .+40p (1 − p) + 28p (1 − p) + 24p (1 − p) + 10p (1 − p) .s7 (p) = 2p4 (1−p)12 +24p5 (1−p)11 +36p6 (1−p)10 +40p7 (1−p)9 +57p8 (1−p)8 9 7 10 6 11 5 12 4 .+40p (1 − p) + 20p (1 − p) + 24p (1 − p) + 12p (1 − p) .s8 (p) = 4p2 (1−p)14 +22p 4 (1−p)12 +8p 5 (1−p)11 +20p 6 (1−p)10 +56p 7 (1−p)9 .s6 (p)

.+33p

8 (1−p)8 +56p 9 (1−p)7 +24p 10 (1−p)6 +8p 11 (1−p)5 +16p 12 (1−p)4 .+8p

1357

14 (1 − p)2

= 12p 2 (1−p)14 +38p 4 (1−p)12 +8p 5 (1−p)11 +12p 6 (1−p)10 +56p 7 (1−p)9

.s9 (p)

.+p

8 (1−p)8 +56p 9 (1−p)7 +16p 10 (1−p)6 +8p 11 (1−p)5 +32p 12 (1−p)4 .+16p

14 (1 − p)2

Fig. 10.34 Probability functions of undetected errors for some combinations of 4 elements of the set .{1, 2, . . . , 8}

10.5.2.2

Combinations of the Set {1, 2, . . . , 9}

For length .n = 9 of the input message the combinations of 2, 3, 4, 5, 6, 7 and 8 elements are considered. The best results are obtained for combinations of 4 and 5 elements and they are presented in Tables 10.11 and 10.12. The graphs of the functions .h2 (p), .h5 (p) and .h6 (p) (as the best of all functions in Tables 10.11 and 10.12) are presented in Fig. 10.35, where the function .s6 (p) (the best result for .n = 8) is added for comparison reasons. It can be seen the function .h6 (p) is the best one and that, up to .n = 9, the least probability of undetected errors is obtained for the block .a1 a2 . . . a9 d1 d2 . . . d9 ,

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Table 10.11 Probability functions for combinations of 4 elements of .{1, 2, . . . , 9} Combinations 1234, 1256, 1357

Probability function of undetected errors = 9p 4 (1 − p)14 + 9p5 (1 − p)13 + 36p6 (1 − p)12 + 81p7 (1 − p)11

.h1 (p)

.+63p

8 (1 − p)10

+ 94p 9 (1 − p)9 + 108p 10 (1 − p)8 + 54p 11 (1 − p)7 + 9p 13 (1 − p)5 + 9p 15 (1 − p)3 5 13 6 12 7 11 8 10 .h2 (p) = 18p (1 − p) + 45p (1 − p) + 72p (1 − p) + 81p (1 − p) 12 6 .+39p (1 − p)

1235, 1236, 1237, 1238,

.+76p

1246,1268 1245, 1248, 1257

9 (1 − p)9

+ 90p 10 (1 − p)8 + 72p 11 (1 − p)7 + 30p 12 (1 − p)6 + 9p 14 (1 − p)4 3 15 5 13 6 12 7 11 .h3 (p) = 3p (1 − p) + 18p (1 − p) + 48p (1 − p) + 54p (1 − p) 13 5 .+18p (1 − p)

.+81p

8 (1 − p)10

+ 100p 9 (1 − p)9 + 72p 10 (1 − p)8 + 63p 11 (1 − p)7 + 18p 13 (1 − p)5 4 14 5 13 8 10 9 9 .h4 (p) = 9p (1 − p) + 36p (1 − p) + 171p (1 − p) + 184p (1 − p) 12 6 13 5 .+75p (1 − p) + 36p (1 − p) 12 6 .+54p (1 − p)

1247, 1258

Table 10.12 Probability functions for combinations of 5 elements of .{1, 2, . . . , 9} Combinations 12345, 12346, 12348, 12357, 12367, 12368,

.h5 (p)

12378, 12457, 12458, 12468 12347, 12358, 12467

.h6 (p)

Probability function of undetected errors = 9p4 (1 − p)14 + 75p 6 (1 − p)12 + 171p 8 (1 − p)10 + 171p 10 (1 − p)8 12 6 14 4 18 .+75p (1 − p) + 9p (1 − p) + p = 102p6 (1 − p)12 + 153p 8 (1 − p)10 + 153p 10 (1 − p)8 12 6 18 .+102p (1 − p) + p

Fig. 10.35 The best cases for .n = 9 and .s6 (p)

where

10 Applications of Quasigroups in Cryptography and Coding Theory

d1 d2 d3 d4 . d5 d6 d7 d8 d9

= = = = = = = = =

485

a1 ∗ a2 ∗ a3 ∗ a4 ∗ a7 a2 ∗ a3 ∗ a4 ∗ a5 ∗ a8 a3 ∗ a4 ∗ a5 ∗ a6 ∗ a9 a4 ∗ a5 ∗ a6 ∗ a7 ∗ a1 a5 ∗ a6 ∗ a7 ∗ a8 ∗ a2 a6 ∗ a7 ∗ a8 ∗ a9 ∗ a3 a7 ∗ a8 ∗ a9 ∗ a1 ∗ a4 a8 ∗ a9 ∗ a1 ∗ a2 ∗ a5 a9 ∗ a1 ∗ a2 ∗ a3 ∗ a6 .

10.6 Conclusion The early interest in quasigroups and their application in cryptography dates long ago. The motivation for using the theory of quasigroups in cryptography is based on the fact that quasigroups (as non-associative algebraic structures) are, in a way, generalized permutations, and the number of quasigroups of order n is greater than .n!(n − 1)! . . . 2!1!. Sometimes, in cryptographic designs, non-associativity can be a preferable feature compared to similar but associative algebraic structures. In this short survey, we presented several applications of quasigroups and their transformations in designing crypto primitives. We decided to put an accent on the primitives that took part in secret-key cryptography competitions with a long tradition. However, our intentions could be enlarged by many other constructions. We emphasize that a more comprehensive survey of using quasigroups for designing crypto primitives is given in [42]. We also consider in this survey some applications of quasigroups in coding theory. First, we present several coding/decoding algorithms for Random Codes Based on Quasigroups (RCBQ). These random error-correcting codes are defined by using a cryptographic algorithm during the encoding/decoding process, i.e., they are cryptocodes. RCBQs have several parameters and we analyze the code parameters’ influence and the messages’ length on the code performance. From the experiments, we can conclude that the speed of the decoding process is one of the biggest problems for these codes. Therefore, new coding/decoding algorithms called Cut-Decoding and 4-Sets-Cut-Decoding algorithms are proposed. Also, for improving the packet-error and bit-error probabilities several methods for reducing the unsuccessful decoding are defined. Here, we analyze and compare the performances of these coding/decoding algorithms for transmission through a binary-symmetric channel. Also, we present some results for applying these codes for decoding images and audio files. At the end of this part of the survey, we give a theoretical upper bound for the packet-error probability obtained by the considered algorithms and approximate formulas for the cardinality of the reduced decoding candidate sets. With the derived formulas, it is proved that Cut-Decoding and 4Sets-Cut-Decoding algorithms improved the performances of RCBQ.

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At the end of this survey, applications of quasigroups for defining error-detecting codes are also given. Codes of rate 1/2 are considered and for them probability functions of undetected errors are determined in the case when the transmission is through a binary symmetric channel. It should be emphasized that the probability of undetected errors decreased to zero when the lengths of the code blocks increase.

References 1. Bakeva, V., Ilievska,N.: A probabilistic model of error-detecting codes based on quasigroups, Quasigroups and Related Systems 17 (2009), pp. 135–148 2. Bakeva, V., Popovska-Mitrovikj, A., Mechkaroska, D., Dimitrova, V., Jakimovski, B., Ilievski, V.: Gaussian channel transmission of images and audio files using cryptcoding, IET Communications, Institution of Engineering and Technology (2019) I.F. 1.443. https://doi.org/10.1049/ iet-com.2018.5636 3. Bakhtiari, S., Safavi-Naini, R., Pieprzyk, J.: A Message Authentication Code based on Latin Square. In V. Varadharajan, J. Pieprzyk, and Y. Mu (Eds.) Proceedings of ACISP’97, LNCS 1270, pp. 194–203), 1997. Springer Berlin Heidelberg. 4. Belousov, V.D.: n-ary kvazigrup. Shtiintsa, Kishinev, 1972. 5. Bertoni, G., Daemen, J., Peeters, M., and Assche, G.: Duplexing the Sponge: Single-Pass Authenticated Encryption and Other Applications. In Selected Areas in Cryptography: 18th International Workshop, SAC 2011, Toronto, ON, Canada, August 11-12, 2011, Revised Selected Papers, pp. 320–337, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. 6. Bogdanov, A., Knudsen, L. R., Le, G., Paar, C., Poschmann, A., Robshaw, M. J. B., Seurin, Y., Vikkelsoe, C.: PRESENT: An Ultra-Lightweight Block Cipher. In: The Proceedings of CHES 2007, Springer-Verlag, pp. 450–466 7. CAESAR: Competition for Authenticated Encryption: Security, Applicability, and Robustness, 2013, https://competitions.cr.yp.to/caesar.html. Cited 19 Aug 2022 8. Dénes, J., Keedwell, A. D.: A new authentication scheme based on Latin squares. Discrete Mathematics 106/107, pp. 157–161, 1992. 9. Dimitrova V., Markovski J.: On Quasigroup Pseudo Random Sequence Generators, Proceedings of the 1st Balkan Conference in Informatics, Thessaloniki, Greece, (2003) pp. 393–401. 10. Dimitrova V., Markovski S.: Classification of quasigroups by image patterns, Proc. of the Fifth International Conference for Informatics and Information Technology, Macedonia, (2007) pp. 152–160. 11. Dimitrova, V., Bakeva, V., Popovska-Mitrovikj, A., Krapež, A. : Cryptographic Properties of Parastrophic Quasigroup Transformation. In S. Markovski and M. Gusev (Eds.), Advances in Intelligent Systems and Computing - ICT Innovations 2012, pp. 235–243, 2013. Springer Berlin Heidelberg. 12. Dobraunig, Ch., Eichlseder, M., Mendel, F. and Schläffer, M.: Ascon v1.2, CAESAR web page, 2016. https://competitions.cr.yp.to/round3/asconv12.pdf. Cited 20 Aug 2022. 13. Faugère, J. C., Gligoroski, D., Perret, L., Samardjiska, S., Thomae, E.: A Polynomial-Time Key-Recovery Attack on MQQ Cryptosystems. In: Katz, J. (eds) Public-Key Cryptography – PKC 2015. LNCS 9020, 2015 Springer, Berlin, Heidelberg. 14. Gligoroski, D., Dimitrova, V., Markovski, S.: Quasigroups as Boolean functions, their equation systems and Groebner bases. In M. Sala, T. Mora, L. Perret, S. Sakata, C. Traverso (Eds.): Groebner Bases, Coding, and Cryptography. Springer Berlin, 2009. 15. Gligoroski D., Markovski S., Kocarev Lj.: Totally asynchronous stream ciphers + Redundancy = Cryptcoding, S. Aissi, H.R. Arabnia (Eds.): Proc. Internat. Confer. Security and management, SAM 2007, Las Vegas, CSREA Press (2007) pp. 446–451.

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16. Gligoroski D., Markovski S., Kocarev Lj.: Error-correcting codes based on quasigroups, Proc. 16th Intern. Confer. Computer Communications and Networks (2007), pp. 165–172. 17. Gligoroski, D., Markovski, S., Knapskog, S. J.: The Stream Cipher Edon80. In: New Stream Cipher Designs: The eSTREAM Finalists, LNCS 4986, pp. 152–169, 2008. Springer-Verlag. 18. Gligoroski, D., Markovski, S., Knapskog, S. J.: A Public Key Block Cipher based on Multivariate Quadratic Quasigrops. IACR Cryptology ePrint Archive, Report 2008: 320, 2008. 19. Gligoroski, D., Odegard, R.S., Jensen, R.E., Perret, L., Faugere, J.C., Knapskog, S.J., Markovski, S.: MQQ-SIG - An Ultra-Fast and Provably CMA Resistant Digital Signature Scheme. In: INTRUST. LNCS 7222, pp.184–203. Springer, 2011. 20. Gligoroski, D., Samardjiska, S.: The Multivariate Probabilistic Encryption Scheme MQQENC. SCC, 2012. 21. Gligoroski, D, Mihajloska, H, Otte, D.: GAGE and InGAGE v1.0. Submission to the NIST’s Lightweight Cryptography Standardization Process, 2019. 22. Chen, Y., Knapskog, S.J., Gligoroski, D.: Multivariate quadratic quasigroups (MQQs): Construction, bounds and complexity. In: Inscrypt, 6th International Conference on Information Security and Cryptology. Science Press of China, 2010. 23. Gligoroski, D, Mihajloska, H, Samardjiska, S, Jacobsen, H, Jensen, R.E, El-Hadedy, M: π Cipher: Authenticated Encryption for Big Data. In: Bernsmed, K, Fischer-Hübner, S (eds) Secure IT Systems. NordSec 2014. LNCS 8788. Springer, Cham. 24. Gligoroski, D, Mihajloska, H, Samardjiska, S, Jacobsen, H, Jensen, R.E, El-Hadedy, M: π Cipher v2.0. Submission to the CAESAR competition for Authenticated Encryption: Security, Applicability, and Robustness, 2014. 25. Gligoroski, D., Markovski, S., Knapskog, S.J.: Multivariate Quadratic Trapdoor Functions based on Multivariate Quadratic Quasigroups. In: Proceedings of the American Conference on Applied Mathematics. MATH, World Scientific and Engineering Academy and Society (WSEAS), pp. 44–49, 2008. 26. Gligorovski, D., Ødegård, R.S., Mihova, M., Knapskog, S.J., Kocarev, Lj., Drapal, A., Klima, V.: Cryptographic Hash Function EDON- R. Submission to NIST, First Round SHA-3 Candidate. 27. Ilievska, N. ,Bakeva,V.: A Model of error-detecting codes based on quasigroups of order 4, Proc. Sixth International Conference on Informatics and Information Technology, Bitola, Republic of Macedonia (2008), pp.7–11. 28. Keedwell, A. D.: Crossed inverse quasigroups with long inverse cycles and applications to cryptography. Australasian Journal of Combinatorics 20, pp. 241–250, 1999. 29. Krapež, A.: An application of quasigroups in cryptology. Math. Maced. Vol.8 (2010) 47-52. 30. Leander, G., and Poschmnan, A.: On the Classification of 4 Bit S-Boxes, In Proceedings of the 1st International Workshop on Arithmetic of Finite Fields. Springer-Verlag, 2007, pp. 159–176. 31. Markovski, S., Bakeva, V.: A Class of Error-Detecting Codes Based on a Quasigroup Operations, Proceedings of 4th International Conference of Informatics and Information Technology, Bitola (2003), pp. 400–405 32. Markovski, S., Bakeva, V.: Error-Detecting Codes with Cyclically Defined Redundancy, Proceedings of 3th Congress of mathematicians and computer scientists in Macedonia, Struga, R.Macedonia (2006), pp. 485–492. 33. Markovski, S., Gligoroski, D., Andova, S.: Using quasigroups for one-one secure encoding. In Proceedings of VIII Conf. Logic and Computer Science LIRA97, Novi Sad, Serbia, pp. 157–162, 1997. 34. Markovski S., Gligoroski D., Bakeva V.: Quasigroup string processing: Part 1, Contributions, Sec. Math. Tech. Sci., MANU, Vol. XX 1-2 (1999) pp. 13–28. 35. Markovski, S., Gligoroski, D., Bakeva, V.: Quasigroup String Processing - Part 1. Contributions, Sec. Math. Tech. Sci., MANU XX, 1-2, pp. 13–28, 1999. 36. Markovski, S., Gligoroski, D., Kocarev, Lj.: Unbiased Random Sequences from Quasigroup String Transformations. In H. Gilbert, H. Handschuh (Eds.) FSE 2005, LNCS 3557, pp. 163– 180, 2005. Springer.

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37. Markovski, S., Kusakatov, V.: Quasigroup String Processing - Part 2. Contributions, Sec. Math. Tech. Sci., MANU XXI, 1-2, 2000, pp. 15–32. 38. Markovski, S., Kusakatov, V.: Quasigroup String Processing - Part 3. Contributions, Sec. Math. Tech. Sci., MANU XXIII-XXIV, 1-2, 2002-2003, pp. 7–27. 39. Markovski, S., Dimitrova, V., Trajcheska, Z., Petkovska, M., Kostadinoski, M., Buhov, D.: Block Cipher Defined by Matrix Presentation of Quasigroups. Proceedings of the 11th Conference on Informatics and Information Technology, CIIT 2014, Bitola 40. Markovski, S., Mileva, A.: Generating huge quasigroups from small non-linear bijections via extended Feistel function. Quasigroups and Related Systems 17, 2009, pp. 91–106. 41. Markovski, S., Mileva A.: NaSHA. Submission to NIST, First Round SHA-3 Candidate. Retrieved January 15, 2013 from http://csrc.nist.gov/groups/ST/hash/sha-3/Round1/ documents/NaSHAUpdate.zip, 2008. 42. Markovski, S.: Design of crypto primitives based on quasigroups. Quasigroups and Related Systems. 23 (2015) No. 1. pp. 41–90 43. Mechkaroska, D., Popovska-Mitrovikj, A., Bakeva, V.: New Cryptcodes for Burst Channels, ´ c, M. Droste, JÉ. Pin (Ed.), Algebraic Informatics. CAI 2019. Lecture Notes in In M. Ciri´ Computer Science, vol 11545 (2019) pp. 202–212 Springer, Cham. 44. Mechkaroska D., Popovska-Mitrovikj, A., Bakeva, V.: Cryptcodes Based on Quasigroups in Gaussian channel, Quasigroups and Related Systems, vol. 24 (2) (2016) pp. 249–268. 45. Mechkaroska D., Popovska-Mitrovikj, A., Bakeva, V.: Performances of Fast Algorithms for Random Codes Based on Quasigroups for Transmission of Audio Files in Gaussian Channel, In: Kalajdziski, S., Ackovska, N. (eds): ICT Innovations 2018. Engineering and Life Sciences, Springer International Publishing (2018) pp. 286–296. 46. Mechkaroska D., Popovska-Mitrovikj, A., Bakeva, V.: Cryptcoding of Images for Transmission Trough a Burst Channels, Journal of Engineering Science and Technology Review (JESTR), Special Issue on Telecommunications, Informatics, Energy and Management 2019 (2020) pp.65–69. 47. Meyer, K. A.: A new message authentication code based on the non-associativity of quasigroups. Doctoral dissertation. Iowa State University, 2006 48. Mihajloska, H., Gligoroski, D.: Construction of Optimal 4-bit S-boxes by Quasigroups of Order 4. In Proceedings of SECURWARE 2012, Rome, Italy, 2012, pp. 163–168. 49. Mileva, A., Markovski, S.: Shapeless quasigroups derived by Feistel orthomorphisms. Glasnik Matematicki 47(2), 2012, pp. 333–349. 50. Mileva, A., Stojanova, A., Bikov, D., Xu, Y.: Investigation of Some Cryptographic Properties of the 8x8 S-boxes Created by Quasigroups. Computer Science Journal of Moldova, 28 (3 (84)). pp. 346–372, 2020. 51. NIST Lightweight Cryptography Standardization Process, 2019 https://csrc.nist.gov/Projects/ lightweight-cryptography. Cited 19 Aug 2022 52. NIST SHA-3 Cryptographic Hash Algorithm Competition, 2007 https://csrc.nist.gov/Projects/ Hash-Functions/SHA-3-Project. Cited 19 Aug 2022 53. Petrescu, A.: n-quasigroup Cryptographic Primitives: Stream Ciphers. Studia Univ. Babes Bolyai, Informatica, LV (2), 2010, pp. 27–34. 54. π -Cipher - Authenticated Encryption Cipher with Associated Data. http://pi-cipher.org/. Cited 19 Aug 2022 55. Popovska-Mitrovikj, A., Ilievski, V., Bakeva, V.: Performances of Random Codes Based on Quasigroups for transmission of audio files, Proceedings of 23rd Telecommunications Forum (TELFOR) 2015, Belgrade, Serbia, pp. 327–330. 56. Popovska-Mitrovikj A., Markovski S., Bakeva V.: Performances of error-correcting codes based on quasigroups, D. Davcev, J.M. Gomez (Eds.): ICT-Innovations 2009, Springer (2009), pp. 377–389. 57. Popovska-Mitrovikj A., Markovski S., Bakeva V.: Increasing the decoding speed of random codes based on quasigroups, S. Markovski, M. Gusev (Eds.): ICT Innovations 2012, Web proceedings, ISSN 1857-7288, pp. 93–102.

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58. Popovska-Mitrovikj, A., Bakeva, V., & Mechkaroska, D.: New decoding algorithm for cryptcodes based on quasigroups for transmission through a low noise channel. In. D. Trajanov, V. Bakeva, (Ed.) ICT Innovations 2017. CCIS, vol. 778, (pp. 196–204). Springer, Cham. https:// doi.org/10.1007/978-3-319-67597-819 59. Popovska-Mitrovikj A., Bakeva V., & Mechkaroska D.: Fast Decoding with Cryptcodes for Burst Errors. In V. Dimitrova, I. Dimitrovski (Ed.), ICT Innovations 2020. Machine Learning and Applications. ICT Innovations 2020. Communications in Computer and Information Science, vol 1316. Springer, Cham https://doi.org/10.1007/978-3-030-62098-1_14 60. Popovska-Mitrovikj A., Markovski S., Bakeva V.: Some new results for random codes based on quasigroups, Proceedings of the 10th Conference on Informatics and Information Technology with International Participants, Bitola, Macedonia, April 2013, ISBN 978-608-4699-01-9, pp. 178–181. 61. Popovska-Mitrovikj, A., Markovski, S., Bakeva, V.: 4-Sets-cut-decoding algorithms for random codes based on quasigroups, International Journal of Electronics and Communications (AEU), 69(10), 1417–1428. 62. Simjanovska, M., Mihova, M., Markovski, S.: Matrix Presentation of Quasigroups of Order 4. In: Mishkovski, I. and Ristov, S. (Eds.) Proceedings of the Tenth International Conference CIIT 2013, Bitola, 2013, pp. 192–196.

Chapter 11

Generalized Quadratic Quasigroup Functional Equations Aleksandar Krapež

11.1 Functional Equations—An Introduction Functional equations on quasigroups are the main topic of this chapter. They are: • . . .

functional generalized quadratic on quasigroups.

Let us see what that means. The term ‘functional’ implies the type of logic in which we formulate our results. In this case it is the second order logic so that we have functional variables (i.e. variables for operations), in addition to more common object variables, which are the only variables used in the first order logics. Thus, we can formulate equations which define properties the unknown operations are required to satisfy. Such equations are equalities which have at least one functional variable. Solving them means finding functions on some set S which, upon replacement of functional variables by corresponding functions on S, turn equations into true statements about the model on S of the theory in which equations are formulated. The term functional equation will be the only fixed notion in this chapter. All others will be defined as fairly narrow, only to be generalized later in a sequence of various domains, with the ambition to reach the boundaries defining presently solved cases, or at least feasible to formulate the problems which are not solved yet or which are not proved to be unsolvable.

A. Krapež (O) Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_11

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Most of the equations will also be generalized, which means that no functional variable repeats in the given equation. Equations are of the form .t1 = t2 . We can think of .t1 and .t2 as the trees of related terms. It will be shown that properties of quasigroups that satisfy this equation depend on the position of related operation symbol in one (or both) of the trees. It is much simpler situation if a symbol, say A, appears only once in the equation then if it appears more often. Think of it as if these several positions in .t1 and .t2 are sending different, sometimes conflicting messages about the properties that A must satisfy. On the other hand, even these simple properties reduce the number of appropriate models enormously. All equations considered will be quadratic. This means that every object variable will appear in the related equation exactly twice. There are two other cases that are excluded from consideration by choosing only quadratic equations. . Assume that we have an equation, say .t1 = t2 and that there is a variable x in it which appears only once. Using inverse operations if necessary, we can solve for x to get an equivalent equation of the form .x = t, for an appropriate term t. We now fix all variables appearing in t to get some constant a. Therefore equation .x = a is a consequence of .t1 = t2 . But this means that there is only one element in S. Consequently all quasigroups on S are identical i.e. reduce to .A(a, a) = a. Such solutions are assumed to be trivial and we don’t bother with them. . All equations with at least one object variable appearing more than twice in it are considered as too difficult and left alone. If you are unhappy with the last item, you are welcome to try your hand with (generalized) distributive quasigroup equation. These algebras are (roughly) between 50 and 100 years old and we still do not know their exact description. On the other hand, if you think that what was left to solve is just too easy, you might try to work out how exactly all solutions of (generalized) associativity equation look like. Structure(s) are called groups and are defined almost 200 years ago by Évariste Galois (1811–1832) and despite all this time, we also have to work hard to discover all types of groups and their properties. You can check the daily output of announced papers on groups in ResearchGate (https://www.researchgate. net/) to appreciate the effort to work out this one. We emphasize two important special cases of quadratic equations. The first is the case of balanced equations i.e. equations .t1 = t2 where sets of object variables of .t1 and .t2 are exactly the same. It follows that the sequence of variables of .t2 is a permutation of analogous sequence for .t1 . If the permutation happen to be identical, the equation .t1 = t2 is said to be of the first kind. Finally, equations are said to be on quasigroups. This promise we shall broke very often and for various reasons, mostly declared at the scene of the crime. In most cases the reason would be to state a more general result—in structures which, in some way, generalize quasigroups.

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11.2 Quasigroups Most of our work will be done in quasigroups. So, we start the Chapter defining them. It turns out that it is not entirely trivial matter. Quasigroups are more general than groups because they need not be associative. But just like groups they can be defined in several ways. The problem is that, even if these definitions are equivalent, defined algebras have properties that may depend on the language they are defined by. Here is the first definition where quasigroup is considered a particular type of groupoid: Definition 1 We say that groupoid .(S; ·) is a quasigroup if for all .a, b ∈ S there are unique solutions .x, y ∈ S of equations .a · x = b and .y · a = b. Loop is a quasigroup with unit e (usually called the identity or neutral element) such that .e · x = x · e = x for all .x ∈ S. Quasigroup .(S; ·) is a group if it is associative: x · (y · z) = (x · y) · z

.

(11.1)

in which case it necessarily contains a unit. Quasigroup (loop, group) is commutative if we have: x · y = y · x.

.

(11.2)

Commutative group is often called an Abelian group. It is often convenient to say that the operation .· itself is a quasigroup. Also, whenever unambiguous, a term like .x · y is shortened to xy. Officially, quasigroups were defined in 1935 in the paper [34] by R. Moufang, but the first results about them were published in a series of papers by E. Schroeder in the second half of the nineteenth century (see S.G. Ibragimov [17]). Quasigroups are important algebraic structures arising in various areas of mathematics. We mention just a few of their applications: . . . . . .

in geometry (as nets/webs, see V.D. Belousov [9]) in combinatorics (as Latin squares, see J. Denes and A.D. Keedwell [15]) in economy and engineering (see J. Aczél [1, 2]) in statistics (see R.A. Fisher [16]) in special theory of relativity, (see Ungar [44]) and in particular, in cryptography (see [15]).

In the second definition quasigroup operation .· is considered together with its inverse operations: left .(\) and right .(/) division. Inverse operations are defined by: xy = z

.

iff x\z = y

iff z/y = x .

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Both inverse operations are quasigroup operations as well. However, an inverse operation of a loop (group) operation need not be a loop (group). Definition 2 Triple groupoid .(S; ·, \, /) is an equational quasigroup (also known as equasigroup or primitive quasigroup) if it satisfies the following axioms: .

x\xy = y x(x\y) = y

xy/y = x (x/y)y = x

(11.3)

If it further satisfies .x\x = y/y (i.e. if the operation .· is a loop operation), we have an equational loop. As hinted above, the systems of quasigroups (loops) and equational quasigroups (loops) are equivalent, but the advantage of the latter is that it defines a variety. We can also define equational groups as associative equational quasigroups, but the usual definition of a group as a universal algebra is the following: Definition 3 Algebra .(S; ·, e, −1 ) with constant e and unary operation .−1 is a group if the operation .· is associative, the element e is the unit and the operation .−1 satisfies: x · x −1 = x −1 · x = e.

(11.4)

.

But in this case we have to prove that the group defined as above is a quasigroup as given in Definition 2: Theorem 1 Algebra .(S; ·, e, −1 ) is a group iff the algebra .(S; ·, \, /) is an associative equational quasigroup. We use definitions .x\y = x −1 · y and .x/y = x · y −1 in one direction. For the converse, we first prove .x\x = y/y, then define .e = x\x and prove .x\e = e/x and −1 = x\e. .x Definition 4 Dual operations of .·, \, / are: x ∗ y = yx,

.

x//y = y/x,

x\\y = y\x.

These are also quasigroup operations, and the six operations .·, \, /, ∗, \\, // are said to be the parastrophes of each other. We use quasigroups in the context of functional equations and usually there are several different quasigroups in every equation. In such cases we use alternative notation—capital Latin letters for quasigroup operations. If, for example, quasigroup operation is denoted by A, we have a quasigroup .(S; A, A−1 , A−2 ). All six parastrophes are defined by: .

A(x, y) = z iff A−1 (z, y) = x iff A−2 (x, z) = y iff A∗ (y, x) = z iff A=1 (y, z) = x iff A=2 (z, x) = y.

(11.5)

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Quasigroup axioms are: .

A(A−1 (x, y), y) = x A−1 (A(x, y), y) = x

A(x, A−2 (x, y)) = y A−2 (x, A(x, y)) = y

(11.6)

and we see that .A−1 , A−2 are respectively right and left division operations of −1 , A−2 ). The reader should note that this notation is different (and even .(S; A, A contradictory) to what is usually used in quasigroup literature. However, there is also a universally accepted notation for parastrophes of quasigroups:

.

A(x1 , x2 ) = x3 iff A(1) (x1 , x2 )=x3 iff A(12) (x2 , x1 )=x3 iff A(13) (x3 , x2 )=x1 iff A(23) (x1 , x3 )=x2 iff A(123) (x2 , x3 )=x1 iff A(132) (x3 , x1 )=x2 .

In general .A(x1 , x2 ) = x3 iff .Aσ (xσ (1) , xσ (2) ) = xσ (3) for .σ ∈ S3 (symmetric group in three elements). Here is yet another definition of quasigroups, related to the notion of parastrophe: Definition 5 Let .(S; E) be an algebra with the set .E of binary operations on S. We say that .(S; E) is an algebra of quasigroups if every .A ∈ E is a quasigroup operation and if the set .E contains all parastrophes of A. When .E consists of all six parastrophes of one particular quasigroup operation A, we also say that the algebra .(S; E) is a quasigroup. Definition 6 If .(S; ·) and .(T ; ×) are quasigroups and .α, β, γ : S → T are bijections such that .α(xy) = β(x) × γ (y), then we say that .(S; ·) and .(T ; ×) are isotopic and that .(α, β, γ ) is an isotopy. Isotopy is principal if .α = Id. Then .S = T and I d is the identity permutation on S. If .α(xy) = γ (y) × β(x), we call the triple .(α, β, γ ) a dual isotopy. Diisotopy is a triple of bijections which is either isotopy or dual isotopy. Isotopy is a generalization of isomorphism. Isotopic image of a quasigroup is again a quasigroup. Every quasigroup is isotopic to some loop. If two quasigroups are isotopic, so are their corresponding parastrophes. We also have: Theorem 2 (A.A. Albert [4]) If a loop is isotopic to a group then it is also isomorphic to it. In particular, if two groups are isotopic they are also isomorphic. Definition 7 Two quasigroups are isostrophic if one of them is isotopic to a parastrophe of the other. All these relations are equivalences between quasigroups. Isomorphism is finer relation than isotopy, which in turn is finer than isostrophy. Diisotopy may be placed between isotopy and isostrophy. However, in groups the distinction between parastrophies and isotopies vanishes (we use .Ix = x −1 ): Theorem 3 In groups:

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x\y = Ix · y . x\ \y = I(Ix · y)

x/y = x · Iy x//y = I(x · Iy)

(11.7)

x ∗ y = I(Ix · Iy) The proof is trivial. Definition 8 Let A be a quasigroup operation on a set S and .a1 , a2 two arbitrary fixed elements from S. We define: .A1 (x) = A(x, a2 ) A2 (y) = A(a1 , y) and .a = A(a1 , a2 ). Operations .A1 and .A2 are (right, left) translations of A (also called unary retracts). Element a is a nullary retract of A. It is easy to see that .A1 and .A2 are bijections. Note also that operations .A1 , A2 and element a depend on the choice of .a1 , a2 , so fixing these elements enables us to simplify notation we use. We have the following theorem: Theorem 4 Let .(S; A) be a quasigroup and .a1 , a2 elements of S. If we define −1 a = A(a1 , a2 ), .A1 x = A(x, a2 ), .A2 y = A(a1 , y) and .L(x, y) = A(A−1 1 x, A2 ) Then .(S; L) is a loop with unit a.

.

This result will be used in almost every proof in this Chapter. For more on quasigroups, see [8, 15, 34] and [40].

11.3 Balanced Equations on Quasigroups We start our investigation of functional equations on balanced equations with all operations representing (binary) quasigroups. Most of the equations are generalized, i.e. there is a single appearance of every functional variable in them. To make it easier for the reader who is not familiar with the subject we first solve some important special cases, mostly generalized versions of familiar identities of associativity, mediality etc. There are also a few less familiar equations which help us to make clear differences between facts which are just accidental and those which are important.

11.3.1 Trivial Balanced Equations There are two such equations: .

A(x, y) = B(x, y)

(11.8)

A(x, y) = B(y, x)

(11.9)

.

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They have trivial solutions: .B = A in the case of (11.8) and .B = A∗ in case of (11.9). But, in order to fit the chosen form for results we state: Theorem 5 General solution of the equation (11.8) (on .S = / ∅) is given by: .P (x, y) = π L(π1 x, π2 y) (for .P ∈ o = {A, B}), where: . L is an arbitrary loop with unit a, and . .π1 , π2 , π(P ∈ o) are arbitrary permutations of S, such that:

.

α = β = Id α1 = β1 α2 = β2 .

(11.10)

Theorem 6 General solution of the equation (11.9) (on .S = / ∅) is given by: .A(x, y) = αL(α1 x, α2 y) .B(x, y) = βL(β2 y, β1 x), where: . L is an arbitrary loop with unit a, and . .α1 , α2 , α, β1 , β2 , β are arbitrary permutations of S, such that: α = β = Id . α1 = β2 α2 = β1 .

(11.11)

11.3.2 Generalized Associativity Equation This is the equation: A(x, B(y, z)) = C(D(x, y), z))

.

(11.12)

—a generalization of the associativity equation (11.1) which served the purpose to single out groups among quasigroups. Equation (11.12) was solved in the paper [3] which still remains one of the most cited in the field of quasigroup functional equations. We assume that .A, B, C, D are unknown and independent quasigroups defined on the same nonempty set S. We target a general solution i.e. a common form which generates all solutions, irrespectively of the set S. We state the solution from [3] using somewhat different notation, making it easier to formulate future generalizations. Theorem 7 (J. Aczél, V. D. Belousov and M. Hosszú [3]) General solution of the generalized associativity equation (11.12) (on a set .S /= ∅) is given by:

498

A. Krapež

P (x, y) = π(π1 x · π2 y)

.

(for .P ∈ o = {A, B, C, D}),

where: . .· is an arbitrary group on S (with unit a), and . .π, π1 , π2 are arbitrary permutation of S, such that:

.

α = γ = Id α2 β = Id γ1 δ = Id

α1 = δ1 β1 = δ2 β2 = γ2 .

(11.13)

Proof Above formulas actually define a solution of (11.12): A(x, B(y, z)) = α(α1 x · α2 β(β1 y · β2 z)) = Id(α1 x · Id(β1 y · β2 z))

.

= α1 x · (β1 y · β2 z) = (α1 x · β1 y) · β2 z = (δ1 x · δ2 y) · γ2 z = Id(Id(δ1 x · δ2 y) · γ2 z) = γ (γ1 δ(δ1 x · δ2 y) · γ2 z) = C(D(x, y), z). To prove that our solution is general, assume that the quadruple .(A, B, C, D) of quasigroups on S is a particular solution of (11.12). Choose .a1 , b1 , b2 ∈ S and define: b = B(b1 , b2 ) . a2 = b a = A(a1 , a2 )

d1 = a1 d2 = b1 d = D(d1 , d2 )

c1 = d c2 = b2 c = C(c1 , c2 )

i.e. .p = P (p1 , p2 ) for .P ∈ {A, B, C, D} and appropriate .p1 , p2 , p ∈ S. It follows that .a = c. Also, .P1 x = P (x, p2 ) and .P2 y = P (p1 , y). Then:

.

A1 = C1 D1 A2 B1 = C1 D2 A2 B2 = C2 .

(11.14)

Replacing systematically one of the variables in Eq. (11.12) by the appropriate constant we confirm that all operations .A, B, C, D are mutually isotopic:

.

A(x, B1 y) = C1 D(x, y) A(x, B2 z) = C(D1 x, z) A2 B(y, z) = C(D2 y, z).

(11.15)

−1 Define now a new binary operation .· on S by .u · v = A(A−1 1 u, A2 v). By Theorem 4, operation .· is a loop with the unit element a. Also: .A(x, y) = A1 x ·A2 y. By the isotopy relation of operations A and C and relations (11.14), we have:

C(u, z) = A(D1−1 u, B2 z) = A1 D1−1 u · A2 B2 z = C1 D1 D1−1 u · C2 z = C1 u · C2 z.

.

11 Generalized Quadratic Quasigroup Functional Equations

499

−1 Analogously .B(y, z) = A−1 2 (A2 B1 y·A2 B2 z) and .D(x, y) = C1 (C1 D1 x·C1 D2 y). If we replace all operations in Eq. (11.12) by their values in terms of .·, we get: −1 A1 x · A2 A−1 2 (A2 B1 y · A2 B2 z) = C1 C1 (C1 D1 x, C1 D2 y) · C2 z

.

which is by (11.14) equivalent to: .u · (v · w) = (u · v) · w. Therefore, it follows that · is a group operation (with a as a unit). If we define:

.

.

α1 = A1 α2 = A2 α = Id

β1 = A2 B1 β2 = A2 B2 β = A−1 2

γ1 = C1 γ2 = C2 γ = Id

δ1 = C 1 D 1 δ2 = C 1 D 2 δ = C1−1

(11.16)

then expressions for all operations .A, B, C, D and relations (11.14) become (11.13) as required in the statement of the theorem. u n Before we proceed, let me answer the obvious question: why so many unnecessary permutations are defined in the proof of the Theorem 7. In other words, why introduce eight translations .A1 , A2 , B1 , B2 , C1 , C2 , D1 , D2 and then, immediately reduce it (by (11.14)) to five independent ones? And then do it again, only this time introducing 12 new permutations .α, . . . , δ2 by (11.16) and then supply them with the system (11.13) to reduce it to five necessary ones? The extreme objection can be made—that we didn’t solve the equation (11.12), we just replaced it by the system of permutation equations, either (11.14) or (11.13). To my defence of the method, which I jokingly call creative rewriting, I give two arguments: . Both systems (11.14) and (11.13) are easy to solve. . More to the point, these systems give important informations about the equation (11.12). The first system (11.14) replaces one equation of two trees of the left and right hand side of the equation (i.e. it linearizes the 2-dimensional equation). The second system makes possible to point out just how the isotopies of the operations to the resolving group .· replace the branch system (11.14) by the neighbour system (11.13) for the components of these isotopies. The whole process can be described as transforming original equation between operations by the formulas: (i) Original equation (ii) Common group .·, isotopies, the branch system for permutations (iii) Common group .·, independent isotopies, the neighbour system for permutations. This gives an overview of the method of solving equations and makes us ready to solve new, more complicated equations. Let us fix this plan of the proof of the Theorem 7 because similar but usually more complicated plan will be behind the proof of every solution of the (system of) equation(s) that we shall solve in this Chapter.

500

A. Krapež

(

) x y z (i) Choose valuation . . a1 b1 b2 (ii) Gradually define other necessary elements of S: d1 = a1 , d2 = b1 , c2 = b2 b = B(b1 , b2 ), d = D(d1 , d2 ) .a2 = b, c1 = d .a = A(a1 , a2 ), c = C(c1 , c2 ). . .

(iii) Infer 0-consequence: .a = c. (iv) Define unary retracts .P1 x = P (x, a2 ), P2 y = P (a1 , y) for all .P ∈ o. (v) Infer all 1-consequences: {x}-consequence: .A1 = C1 D1 {y}-consequence: .A2 B1 = C1 D2 .{z}-consequence: .A2 B2 = C2 i.e. system (11.14). . .

(vi) Infer all 2-consequences: {x, y}-consequence: .A(x, B1 y) = C1 D(x, y) {x, z}-consequence: .A(x, B2 z) = C(D1 x, z) .{y, z}-consequence: .A2 B(y, z) = C(D2 y, z) i.e. system (11.15). . .

(vii) (viii) (ix) (x) (xi) (xii)

Deduce: .B ∼ C ∼ A ∼ D (All operations are mutually isotopic). Define: .π, π1 , π2 for all .P ∈ o. Transform (11.14) into (11.13). Define operation .· in terms of A. Define all operations .P ∈ o in terms of .·. Prove that .· is a group operation. ∗

.

Let us consider now the equation: A(x, B(y, z)) = C(x, D(y, z))

.

(11.17)

quite similar to generalized associativity (11.12). But similar is not the same, therefore we should be careful and define/prove our every step while solving (11.17). Choose .a1 , b1 , b2 ∈ S and define:

.

b = B(b1 , b2 ) a2 = b a = A(a1 , a2 )

d1 = b1 d2 = b2 d = D(d1 , d2 )

c1 = a1 c2 = d c = C(c1 , c2 ).

11 Generalized Quadratic Quasigroup Functional Equations

501

It follows that .a = c. We also define: .P1 x = P (x, p2 ) and .P2 y = P (p1 , y) just as in Theorem 7. But this time, resulting identities (analogous to (11.14)) are slightly different:

.

A1 = C1 A2 B1 = C2 D1 A2 B2 = C2 D2 .

(11.18)

But the most striking differences are in the resulting isotopies: A(x, B1 y) = C(x, D1 y) . A(x, B2 z) = C(x, D2 z) A2 B(x, y) = C2 D(x, y).

(11.19)

We do not have the proof that all four operations .A, B, C, D are mutually isotopic! −1 Let us define the operation .· anyway: .u · v = A(A−1 1 u, A2 v). But let also be: −1 −1 .L(u, v) = A2 B((A2 B1 ) u, (A2 B2 ) v). By Theorem 4, both operations are loops with the common unit a. Also .A(x, u) = A1 x · A2 u and C(x, D2 z) = A(x, B2 z) = A1 x · A2 B2 z = C1 x · C2 D2 z

.

i.e. .C(x, u) = C1 x · C2 u. Similarly, it follows that: .B(y, z) = A−1 2 L(A2 B1 y, A2 B2 z) and .D(y, z) = −1 C2 L(C2 D1 y, C2 D2 z). If we now replace all operations in Eq. (11.17), we get: −1 A1 x · A2 A−1 2 L(A2 B1 y, A2 B2 z) = C1 x · C2 C2 L(C2 D1 x, C2 D2 y)

.

which is by (11.18) equivalent to: .u · L(v, w) = u · L(v, w) and this is identically true. In other words, there is no way to infer associativity of either .· or L. The only connections between loops .· and L are the domain S, the common unit a and nothing else. If we define:

.

α1 = A1 α2 = A2 α = Id

β1 = A2 B1 β2 = A2 B2 β = A−1 2

γ1 = C1 γ2 = C2 γ = Id

δ1 = C 2 D 1 δ2 = C 2 D 2 δ = C2−1

(11.20)

then relations (11.18) become:

.

α = γ = Id α2 β = Id γ2 δ = Id

α1 = γ1 β1 = δ1 β2 = δ2 .

(11.21)

502

A. Krapež

Therefore we proved: Theorem 8 General solution of the equation (11.17) (on a set .S = / ∅) is given by: P (x, y) = π(π1 x · π2 y) (for .P ∈ {A, C}), Q(x, y) = ωL(ω1 x, ω2 y) (for .Q ∈ {B, D}),

. .

where: . .· and L are arbitrary loops on S (with the common unit a), and . .π1 , π2 , π, ω1 , ω2 , ω are arbitrary permutations of S (related to P , Q) and such that (11.21) is true. What seems to be the reason of such striking difference in solutions? Both equations are generalized, balanced of the first kind and the only difference seems to be in the pair of variables where the first application of some of the unknown functions happens. In (11.17) we have the triple of variables .(x, y, z) on both sides of the equality sign and functions B and D both apply to pair .(y, z). On the contrary, in (11.12) B applies to .(y, z), while D applies to .(x, y). ∗

.

Let us check our conclusion on two similar equations, but with different order of variables on the right hand side of the equality sign as compared to the standard order .(x, y, z) on the left side. The equations are: .

A(x, B(y, z)) = C(D(z, y), x)

(11.22)

A(x, B(y, z)) = C(D(z, x), y)

(11.23)

.

The first equation (11.22) can be transformed into equivalent equation: A(x, B(y, z)) = C ∗ (x, D ∗ (y, z))

.

(11.24)

with a general solution: A(x, y) B(x, y) . C(x, y) D(x, y)

= α(α1 x · α2 y) = βL(β1 x, β2 y) = γ (γ2 y · γ1 x) = δL(δ2 y, δ1 x)

where: . .· and L are arbitrary loops on S with the common unit a, and . .π1 , π2 , π are appropriate permutations of S, related to .P ∈ {A, B, C, D}, such that:

11 Generalized Quadratic Quasigroup Functional Equations

α = γ = Id . α2 β = Id γ1 δ = Id

α1 = γ2 β1 = δ2 β2 = δ1 .

503

(11.25)

The second equation may be transformed into: A(x, B ∗ (z, y)) = C(D ∗ (x, z), y)

.

(11.26)

with a general solution: P (x, y) = π(π1 x · π2 y) Q(x, y) = ω(ω2 y · ω1 x)

. .

(for .P ∈ {A, C}), (for .Q ∈ {B, D}),

where: . .· is an arbitrary group on S with unit a, and . .π1 , π2 , π are appropriate permutations of S, related to P and Q and such that:

.

α = γ = Id α2 β = Id γ1 δ = Id

α1 = δ2 β1 = γ2 β2 = δ1 .

(11.27)

Conclusion is that the grouping of variables in subterms is more important than their order, since this grouping determines whether general solution depend on group(s), rather then just (almost independent) loops. The independence itself is a result of the diisotopy equivalence relation among the operations from the equation. For the list of all possible generalized balanced equations with three variables, check [24] where all such quadratic equations are treated in detail.

11.3.3 Generalized Mediality Equation No functional equation has more names than the one that we treat next. We choose to call it medial, but it is also known as: abelian, entropic, bisymmetric, bicommutative etc. See [1] for even more names. Theorem 9 (J. Aczél, V. D. Belousov, M. Hosszú [3]) General solution of the generalized mediality equation: A(B(x, y), C(u, v)) = E(D(x, u), F (y, v)))

.

(on a set .S = / ∅) is given by: P (x, y) = π(π1 x + π2 y)

.

(for .P ∈ o = {A, B, C, D, E, F }),

(11.28)

504

A. Krapež

where: . .+ is an arbitrary Abelian group on S (with the unit e), and . .π1 , π2 , π are arbitrary permutations of S, related to P and such that: α = Id α1 β = Id . α2 γ = Id

β1 β2 γ1 γ2

= δ1 = ϕ1 = δ2 = ϕ2

ε = Id ε1 δ = Id ε2 ϕ = Id

(11.29)

Proof Trivially, above formulas actually define a solution of (11.28). To prove that our solution is general, assume that the sixtuple .(A, B, C, D, E, F ) of quasigroups on S is a particular solution of (11.28). Choose .b1 , b2 , c1 , c2 ∈ S and define: b c a1 . a2 a e1 e2

= B(b1 , b2 ) = C(c1 , c2 ) =b =c = A(a1 , a2 ) =d =f

d1 d2 d f1 f2 f e

= b1 = c1 = D(d1 , d2 ) = b2 = c2 = F (f1 , f2 ) = E(e1 , e2 ).

It follows that .a = e. Also, for all .P ∈ o define .P1 x = P (x, p2 ) and .P2 y = P (p1 , y). We now have: .

A1 B1 = E1 D1 A2 B2 = E2 F1

A2 C1 = E1 D2 A2 C2 = E2 F2

(11.30)

For .y = b2 = f1 we get .A(B1 x, C(u, v)) = E(D(x, u), F2 v). This is an example of generalized associativity equation and we conclude that all operations .A, C, E, D are isotopic to a group operation .+, which is defined by: .z + w = −1 A(A−1 1 z, A2 w). Consequently .A(x, y) = A1 x + A2 y and similarly: C(u, v) = A−1 2 (A2 C1 u + A2 C2 v) . E(y, z) = E1 y + E2 z D(x, u) = E1−1 (E1 D1 x + E1 D2 u). Also: −1 −1 B(x, y) = A−1 1 E(D1 x, F1 y) = A1 (E1 D1 x + E2 F1 y) = A1 (A1 B1 x + A1 B2 y)

.

and:

11 Generalized Quadratic Quasigroup Functional Equations

505

F (y, v) = E2−1 A(B2 y, C2 v) = E2−1 (A1 B2 y + A2 C2 v) = E2−1 (E2 F1 x + E2 F2 y).

.

Therefore, all operations .A, B, C, D, E, F are mutually isotopic (and consequently isotopic to .+). But for .x = b1 and .v = c2 we also have: .A(B2 y, C1 u) = E(D2 u, F1 y) i.e. quasigroups A and E are also dually isotopic. Let us see what this implies. Rewrite Eq. (11.28) in terms of the group operation .+: .

−1 A1 A−1 1 (A1 B1 x + A1 B2 y) + A2 A2 (A2 C1 u + A2 C2 v) = = E1 E1−1 (E1 D1 x + E1 D2 u) + E2 E2−1 (E2 F1 y + E2 F2 v).

Since .+ is a group and (11.30) is true, we have: A1 B1 x + A1 B2 y + A2 C1 u + A2 C2 v = A1 B1 x + A2 C1 u + A1 B2 y + A2 C2 v.

.

After cancellation from both left and right, and the introduction of new variables z = A1 B2 y and .w = A2 C1 u, we conclude that .z + w = w + z, i.e. .+ is also commutative. This also resolves the problem of dual isotopy between A and E. Finally, let us define:

.

α α1 α2 β . β1 β2 γ γ1 γ2

= = = = = = = = =

Id A1 A2 A−1 1 A1 B1 A1 B2 A−1 2 A2 C1 A2 C2

ε ε1 ε2 δ δ1 δ2 ϕ ϕ1 ϕ2

= = = = = = = = =

Id E1 E2 E1−1 E1 D 1 E1 D 2 E2 E2 F 1 E2 F 2 .

This turns (11.30) into (11.29). Consequently .P (x, y) = π(π1 x + π2 y) for .P ∈ o and appropriate .π1 , π2 , π such that (11.29). This proves that the chosen solution is u n of the required form. ∗

.

In a similar way, we can solve the generalized paramedial equation (see [18]): A(B(x, y), C(u, v)) = E(D(v, y), F (u, x)).

.

(11.31)

But there is a faster way. Equation (11.31) can be transformed into: A(B ∗ (y, x), C ∗ (v, u)) = E(D ∗ (y, v), F ∗ (x, u)).

.

(11.32)

506

A. Krapež

and this is a generalized medial equation. By the Theorem 9, we have: Theorem 10 (J. Ježek, T. Kepka [18]) General solution of the generalized paramediality equation: (11.31) (on a set .S = / ∅) is given by: P (x, y) = π(π1 x + π2 y)

.

(for .P ∈ o),

where: . .+ is an arbitrary Abelian group on S (with the unit e), and . .π1 , π2 , π are arbitrary permutations of S, related to P and such that: α = Id α1 β = Id . α2 γ = Id

β1 β2 γ1 γ2

= = = =

ϕ2 δ2 ϕ1 δ1 .

ε = Id ε1 δ = Id ε2 ϕ = Id

(11.33)

11.3.4 The System of Generalized Cyclic Associativity on Quasigroups The system of generalized cyclic associativity is: A(x, B(y, z)) = C(y, D(z, x)) = E(z, F (x, y)).

.

(11.34)

There is a possible missunderstanding related to (11.34). Let me try to explain it. It is important to do it since we shall have more examples of the same type later— systems of generalized associativity (11.59) of n-ary quasigroups being the most important one. The system (11.34) was declared to be a system of equations. Mathematical logicians would probably call it a syntactical error instead. Admittedly, there are no mathematical theories with the mathematical formulas of the form .t1 = t2 = t3 . Logicians would require that we write: .

t1 = t2 t1 = t3 .

But then it doesn’t fit our definition of generalized as anything occurring in .t1 appears twice in the system. If we give new names to operations in .t1 , we get the system: t1 = t2 t1 = t3

. '

11 Generalized Quadratic Quasigroup Functional Equations

507

which really is not a system, but just two independent equations. So, we do what is usually done in mathematics. We fail formal logic a little to stick with what is important to us—and this is the property of being generalized. Therefore, we accept .t1 = t2 = t3 as a shortcut of .t1 = t2 ∧ t1 = t3 which makes being generalized more logical. The next theorem gives a general solution of the system (11.34): Theorem 11 (A. Krapež [19], part II) A general solution of the system of generalized cyclic associativity (11.34) on quasigroups is given by: P (x, y) = π(π1 x + π2 y)

.

for .P ∈ o = {A, B, C, D, E, F },

where: . .+ is an arbitrary Abelian group on S (with the unit e), and . .π1 , π2 , π are arbitrary permutations of S, related to P and such that: α = γ = ε = Id α1 = δ2 = ϕ1 . β1 = γ1 = ϕ2 β2 = δ1 = ε1

α2 β = Id γ2 δ = Id ε2 ϕ = Id.

(11.35)

Proof To check that quasigroups .A, . . . , F satisfying conditions of the theorem also satisfy Eq. (11.34) is trivial. Let us assume that we do have a particular solution .A, . . . , F of (11.34) and try to prove that it is of required form. Assume that .a1 , b1 , b2 are three arbitrary elements of S and let us define other elements that we use in our proof. b d1 . f1 c1 e1

= B(b1 , b2 ) = b2 = a1 = b1 = b2

a2 d2 f2 c2 e2

=b = a1 = b1 =d =f

a d f c e

= A(a1 , a2 ) = D(d1 , d2 ) = F (f1 , f2 ) = C(c1 , c2 ) = E(e1 , e2 ).

It follows that .a = c = e. We also define .P1 x = P (x, p2 ) and .P2 y {A, B, C, D, E, F }, with the consequence that:

.

=

P (p1 , x) for .P

A1 = C2 D2 = E2 F1 A2 B1 = C1 = E2 F2 A2 B2 = C2 D1 = E1

Equation .A(x, B(y, z)) = E(z, F (x, y)) may be written as

∈

(11.36)

508

A. Krapež

A(x, B(y, z)) = E ∗ (F (x, y), z)

.

which proves, by Theorem 7, that all operations .A, B, F are isotopic to the same group .+, while the operation E is dually isotopic to it. Group .+ is, by Theorem 7 −1 defined by .u + v = A(A−1 1 x, A2 v) and consequently E(z, F1 x) = A(x, B2 z) = A1 x + A2 B2 z = E2 F1 x + E1 z,

.

therefore .E(z, u) = E2 u + E1 z. Also .B(y, z) = A−1 2 (A1 B1 y + A1 B2 z) and −1 .F (x, y) = E (E F x + E F y). 2 1 2 2 2 Operations C and D are also diisotopic to the first four, namely .C(y, D1 z) = A2 B(y, z) and .C2 D(z, x) = A(x, B2 z) and consequently to .+ as well: .C(y, v) = C1 y + C2 v and .D(z, x) = C2−1 (C2 D2 x + C2 D1 z). If we replace all these operations into original system, we get: .

−1 A1 x + A2 A−1 2 (A2 B1 y + A2 B2 z) = C1 y + C2 C2 (C2 D2 x + C2 D1 z) = E2 E2−1 (E2 F1 x + E2 F2 y) + E2 z. (11.37)

Using associativity of .+ and equalities (11.36) in the system (11.37) we reduce it to u + v + w = v + u + w = u + v + w. Dropping third term and cancelling w, we get .u + v = v + u proving that the group operation .+ is also commutative. Define:

.

α1 α2 . β β1 β2

= = = = =

A1 A2 A−1 2 A2 B1 A2 B2

α=γ γ1 γ2 δ δ1 δ2

= = = = = =

ε = Id C1 C2 C2−1 C2 D 1 C2 D 2

ε1 ε2 ϕ ϕ1 ϕ2

= = = = =

E1 E2 E2−1 E2 F 1 E2 F 2 .

(11.38)

The system (11.38) transforms (11.36) into (11.35) and gives us the possibility to define all operations from (11.34) using just one formula .P (x, z) = π(π1 x + π2 y) for all .P ∈ o. u n

11.3.5 Arbitrary Balanced Equations We learned from previous examples that one of the most important details about solution is the information on diisotopies among operations appearing in the given equation. So, let us try to do it for the more complex equation (11.39): A(B(C(x, y), D(z, z' )), F (u, H (v, w)))

.

= E(J (K(v, u), w), M(N(z' , y), R(x, z)))

(11.39)

11 Generalized Quadratic Quasigroup Functional Equations

509

In this case, the set of all unknown functions is .o = {A, B, C, D, E, F, H, J, K, M, N, R}. Let us follow the procedure we fixed in Theorem 7. ( ) x y z z' u v w We chose valuation . and define missing constants c1 c2 d1 d2 f1 h1 h2 among .p1 , p2 , p as well as all translations .π1 , π2 , π for all .P ∈ o. For example, .f2 = h and .f = F (f1 , f2 ), while .F1 x = F (x, f2 ) and .F2 y = F (f1 , y). It follows that .a = e. 1-consequences imply: A1 B1 C1 = E2 M2 R1 A2 F1 = E1 J1 K2 A1 B1 C2 = E2 M1 N2 A2 F2 H1 = E1 J1 K1 . A1 B2 D1 = E2 M2 R2 A2 F2 H2 = E1 J2 A1 B2 D2 = E2 M1 N1 .

(11.40)

All seven 1-consequences are included in (11.40), however we do not need all 2-consequences. {x, u}-consequence: .A(B1 C1 x, F1 u) = E(J1 K2 u, M2 R1 x) implies: .A ∼ E (i.e. A and E are diisotopic), .{x, y}-consequence: .A1 B1 C(x, y) = E2 M(N2 y, R1 x) implies: .C ∼ M, ' ' ' .{z, z }-consequence: .A1 B2 D(z, z ) = E2 M(N1 z , R2 z) implies: .D ∼ M, .{x, z}-consequence: .A1 B(C1 x, D1 z) = E2 M2 R(x, z) implies: .B ∼ R .{y, z}-consequence: .A1 B(C2 y, D1 z) = E2 M(N2 y, R2 z) implies: .B ∼ M, ' ' ' .{y, z }-consequence: .A1 B(C2 y, D2 z ) = E2 M1 N(z , y) implies: .B ∼ N, .

Also: {v, w}-consequence: {u, w}-consequence: .{u, v}-consequence: . .

A2 F2 H (v, w) = E1 J (K2 u, w) implies: .H ∼ J , A2 F (u, H2 w) = E1 J (K1 v, w) implies: .F ∼ J , .A2 F (u, H1 v) = E1 J1 K(v, u) implies: .F ∼ K. .

.

The conclusion is that we have three diisotopy classes: .{A, E}, .{B, C, D, M, N, R} and .{F, H, J, K}. Therefore we can make three independent consequences of Eq. (11.39): {x, u}-consequence: .A(B1 C1 x, F1 u) = E(J1 K2 u, M2 R1 x) as above. {x, y, z, z' }-consequence: .A1 B(C(x, y), D(z, z' )) = E2 M(N(z' , y), R(x, z)), and .{u, v, w}-consequence: .A2 F (u, H (v, w)) = E1 J (K(v, u), w). . .

We can define three loops L, .+ and .· corresponding to three diisotopy classes. If we rewrite the second equation as: A1 B(C(x, y), D(z, z' )) = E2 M(N(z' , y), R ∗ (z, x))

.

we realize that it is a paramedial equation and according to Theorem 10 operation + is an abelian group. Analogously, the third equation is equivalent to:

.

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A. Krapež

A2 F (u, H (v, w)) = E1 J (K ∗ (u, v), w)

.

proving that .· is a group. Therefore: Theorem 12 A general solution of the equation (11.39) on quasigroups is given by: A(x, y) = αL(α1 x, α2 y), E(x, y) = εL(ε2 y, α1 x), .P (x, y) = π(π1 x + π2 y) for .P ∈ {B, C, D, M, N, R}, .Q(x, y) = ω(ω1 x · ω2 y) for .Q ∈ {F, H, J }, .K(x, y) = κ(κ2 y · κ1 x), . .

where: . . . .

L is an arbitrary loop on S with the unit e, + is an arbitrary Abelian group on S (with the unit e), .· is an arbitrary group (with the unit e) .π1 , π2 , π are arbitrary permutations of S, related to .P ∈ {A, B, C, D, E, F, H, J, K, M, N, R} and such that: .

α α1 β . β1 γ β2 δ α2 ϕ ϕ2 χ

= = = = = =

Id Id Id Id Id Id

γ1 γ2 δ1 δ2 ϕ1 χ1 χ2

= ρ1 = ν2 = ρ2 = ν1 = κ2 = κ1 = ι2

ε= ε1 ι = ι1 κ = ε2 μ = μ1 ν = μ2 ρ =

Id Id Id Id Id Id.

(11.41)

The missing details in the proof of Theorem 12 may be filled easily—just stick to the general plan. The new part was handling division of the set .o of all operations into diisotopy classes. The following definition describes the general procedure. Definition 9 Let .s = t be a balanced equation on quasigroups with the set .o of functional variables and the set .V = {x1 , . . . , xn } of object variables. . If .W ⊆ V and we replace all variables from .V \ W by some fixed elements from the base set, we get a W -consequence of .s = t. . If .|W | = m we also say that the W -consequence is (one of) m-consequence(s) of .s = t. xy . We write .F ∼ G◦ if .αF (x, y) = G◦ (βx, γ y) is .{x, y}-consequence of .s = t (where .G◦ is either G (even case) or .G∗ (odd case)). . The relation .∼ is the smallest equivalence relation on .o generated by the set of all 2-consequences of .s = t.

11 Generalized Quadratic Quasigroup Functional Equations

511

xy

. Moreover, .≈ (↔) is generated using .∼ sequences with even (odd) number of relations .↔. Relation .∼, being an equivalence relation, divides the set .o into disjoint .∼-classes: o1 , . . . , or . If F is an operation symbol from the equation .s = t and .F ∈ oi we can define .o0i to be the .≈-class of F and .o1i to be the .≈-class of .F ∗ . In most cases 0 1 .o will be nonempty and therefore the complement of .o in .oi . But there are two i i 0 exceptional cases when .oi = oi . One possibility is that all operations in .oi are mutually isotopic and consequently .o1i is empty set. But there is other possibility that the difference between isotopy and dual isotopy vanishes and therefore .o1i = o0i . In all cases .oi = o0i ∪ o1i . Also, in all cases except the last: .o0i ∩ o1i = φ. In the last case we have: .oi = o0i = o1i . .

Definition 10 Let a balanced functional equation .s = t be given and let equivalences .∼ and .≈ on .o be defined as in Definition 9. Then: . A .∼-class .oi is small if .|oi | ≤ 2; otherwise .oi is big. . A big .∼-class .oi is Abelian if .o0i = o1i = oi .

11.3.6 Generalized Balanced Functional Equations A. Sade was the first to try to generalize the results on generalized associativity and generalized bisymmetry. In the paper [37] he claimed the solution of all generalized balanced functional equations on quasigroups. Unfortunately, there is a mistake in his proof, as observed by V.D. Belousov in [7]. He showed that Sade’s result is true for balanced functional equations of the first kind only. Theorem 13 (V.D. Belousov [7] after A. Sade [37]) Let .s = t be a balanced functional equation of the first kind. Then: . All quasigroups from an isotopy class .oi (i = 1, . . . , r) are isotopic to a loop .◦i . . Loops .◦i (i = 1, . . . , r) satisfy the identity .s ' = t ' obtained from .s = t replacing all quasigroups from .oi by the loop .◦i . The problem was finally solved by B. Alimpi´c [5]: Theorem 14 Let .s = t be an arbitrary balanced functional equations on quasigroups. . All quasigroups from one class .o≈ i (i = 1, . . . , r) are isotopic to the same loop .◦i ; . The isotopy is of the form .σA A(u, v) = σA A1 u ◦i σA A2 v;

512

A. Krapež

. Loops .◦i satisfy the identity obtained from .s = t by replacing all operations from ≈ .o by the operation .◦i (i = 1, . . . , r). i Taking into account later results a variant of this theorem (for generalized .s = t) can be restated thus: Theorem 15 Let .s = t be a generalized balanced functional equation on quasigroups and let Tree(s), Tree(t) be trees of terms .s, t respectively. . Each tree which is a homomorphic image of both Tree(s), Tree(t) provides a solution to .s = t. The largest such tree determines a general solution (It is assumed that A the same variable from .s, t have the common image). . .s = t ⇔ ri=1 si = ti , where .si = ti is a .Vi -consequence of .s = t for appropriate .Vi ⊆ V . Equation .si = ti contains exactly the operations from .oi . . All operations from .oi (i = 1, . . . , r) are diisotopic to the same loop .◦i . . All loops .◦i (i = 1, . . . , r) have the common unit e. . If the class .oi (i = 1, . . . , r) is big then .◦i is a group. . If the class .oi (i = 1, . . . , r) is big but not Abelian then both .si and .ti reduce to .x1 ◦i . . . ◦i xri , the difference being just in bracketing. Therefore, .si = ti is some consequence of associativity. . If the class .oi (i = 1, . . . , r) is Abelian then the group .◦i is also commutative. . If the class .oi (i = 1, . . . , r) is Abelian then .si reduces to .x1 ◦i . . . ◦i xri and .ti reduces to .y1 ◦i . . . ◦i yri where .y1 , . . . , yri is some permutation of .x1 , . . . , xri and they have different bracketing. Therefore, in this case, .si = ti is some consequence of both associativity and commutativity.

11.4 Generalized Balanced Equations on n-Ary Quasigroups When B. Alimpi´c found the way to solve all (generalized) balanced functional equations for binary quasigroups [5], the main problem was to find solution of arbitrary balanced functional equation on n-ary quasigroups. V.D. Belousov was working on it and he published several papers by himself [7] or jointly with E.S. Livšic [11, 12] (see also [29–31]). There were many others working on it. Finally, I solved the problem in 1978 and published results in three papers [19]. The solution was similar to results of B. Alimpi´c (see [5]) with some technical difficulties. So, instead of cumbersome general theory, I am choosing several examples showing both main ideas and a quick look at tiresome details.

11.4.1 n-Ary Quasigroups n is a sequence .a , a Further, we assume that .am m m+1 , . . . , an if .m < n and is just .am n

if .m = n. If a non-indexed element a repeats n times we write .a. We have two definitions of n-ary quasigroups.

11 Generalized Quadratic Quasigroup Functional Equations

513

Definition 11 Algebra .(S; A) where .A : S n → S is an n-ary groupoid. This n-ary groupoid is an n-ary quasigroup if for all sequences .a1n,b with .n + 1 elements from n S and all .i = 1, . . . , n equation .A(i−1 1 , x, ai+1 ) = b has a unique solution for x. In particular, a unary or 1-groupoid is a 1-quasigroup iff .A : S → S is a bijection. n ) uniquely defines For a given n-quasigroup .(S; A), the sequence .(a1i−1 , b, ai+1 (i n+1) : S n → S which is also solution x. This defines an i-inverse operation .A n-quasigroup.

(S; A, A(1 n+1) , . . . , A(n n+1) )

Definition 12 Algebra quasigroup iff:

.

is

an

equational

n-

n A(a1i−1 , A(i n+1) (a1n ), ai+1 ) = ai

.

n A(i n+1) (a1i−1 , A(a1n ), ai+1 ) = ai

.

for all .i = 1, . . . , n. Definition 13 Let .(S; A) be an n-quasigroup. Define: Aσ (xσ (1) , . . . , xσ (n) ) = xσ (n+1) iff A(x1n ) = xn+1

.

for any permutation .σ of .{1, . . . , n + 1}. This is a parastrophe of the operation A. Lemma 1 A parastrophe of an n-quasigroup is also an n-quasigroup. Definition 14 Let .(S; A) and .(T ; B) be two n-quasigroups. We say that they are isotopic if there is a sequence of bijections .π1 , . . . , πn+1 : S → T such that n .πn+1 (A(x )) = B(π1 x1 , . . . , πn xn ). 1 Two n-quasigroups are isostrophic if one is isotopic to a parastrophe of the other. Definition 15 We say that element e from S is an i-unit of n-quasigroup .(S; A) if i−1

n−i

i ≤ n and .A( e , x, e ) = x for all .s ∈ S. If e is an i-unit for all .1 ≤ i ≤ n, then e is a unit of .(S; A). n-quasigroup with unit is a n-loop. For .n > 2 a unit in .(S; A) need not be unique.

.

Definition 16 For n-ary quasigroup operation A we can define retracts. These are m-operations (.m ≤ n) obtained from A by fixing some (exactly .n − m of them) of the arguments of A. All retracts are quasigroups of corresponding arrity, except for the 0-ary retract which is an element of the base set S. For more on n-ary quasigroups are [10].

11.4.2 Trivial Equations For ternary operations A and B, there are six such equations:

514

A. Krapež

A(x, y, z) = B(x, y, z) . A(x, y, z) = B(y, x, z) A(x, y, z) = B(z, x, y)

A(x, y, z) = B(x, z, y) A(x, y, z) = B(y, z, x) A(x, y, z) = B(z, y, x)

(11.42)

The similarity with the binary case is easy to see and we shall not go into details. We just mention rather obvious conclusion that there are .6 = 3! trivial equations with ternary operations and analogously .n! such equations with n-ary operations. In all cases A and B are parastrophes of each other while no inverse operations are used.

11.4.3 Reducibility Much more interesting is the following type of equation: A(x, y, z) = B(x, C(y, z)).

(11.43)

.

Let us pretend that we have just a ‘normal’ equation with binary operation symbols and apply the familiar plan to solve it: Choose .b1 , c1 , c2 ∈ S and define: c = C(c1 , c2 ), b2 = c, b = B(b1 , b2 ), .a1 = b1 , a2 = c1 , a3 = c2 , a = A(a1 , a2 , a3 ).

.

It follows that .a = b. We can define translations .C1 , C2 , B1 , B2 , but also: A1 x = A(x, a2 , a3 ),

A2 y = A(a1 , y, a3 ),

.

A3 z = A(a1 , a2 , z).

There are 1-consequences: .

A1 = B1

A2 = B2 C1

A3 = B2 C2

and 2-consequences: .

A12 (x, y) = B(x, C1 y)

A13 (x, z) = B(x, C2 z)

A23 (y, z) = B2 C(y, z).

Therefore: A(x, y, z) = B(x, C(y, z)) = A12 (x, C1−1 C(y, z)) = −1 −1 −1 .= A12 (x, C 1 B2 A23 (y, z)) = A12 (x, A2 A23 (y, z)).

.

We say that operation A is reducible, as we can express it using operations of lesser arities. The equation (11.43) becomes: A12 (x, A−1 2 A23 (y, z)) = B(x, C(y, z))

.

11 Generalized Quadratic Quasigroup Functional Equations

515

which is similar to Eq. (11.17). By Theorem 8, it follows that: A12 (x, y) = π L(π1 x, π2 y) B(x, y) = βL(β1 x, β2 y), .A23 (x, y) = ω(ω1 x · ω2 y) .C(x, y) = γ (γ1 x · γ2 y) . .

for appropriate .π, π1 , π2 , β, β1 , β2 , ω, ω1 , ω2 , γ , γ1 , γ2 . This yields: A(x, y, z) = αL(α1 x, α2 y · α3 z) for appropriate .α1 , α2 , α3 , α

.

(i.e. .α = β = Id, β2 γ = Id, α1 = β1 , α2 = γ1 , α3 = γ1 ). We proved that the chosen solution is of the proper form for the description of the solution of equation (11.43). ∗

.

Here is a more complicated and more realistic case: A(x, B(y, C(z, z' ), u), D(v, w)) = E(F (z' , y), x, H (w, v), K(z, u)).

.

(11.44)

There are five binary operation symbols: .C, D, F, H, K, two ternary symbols .A, B and a quaternary symbol E. Because E is the only quaternary symbol, it follows that the respective operation E is reducible. We chose .{x, y, u, v}-consequence—one of the six possible consequences of (11.44) to do the job: E(F2 y, x, H2 v, K2 u) = A(x, B13 (y, u), D1 v).

.

(11.45)

Two relevant consequences of (11.45) (and (11.44)) are: A(x, B1 y, D1 v) = E123 (F2 y, x, H2 v)

.

A2 B13 (y, u) = E14 (F2 y, K2 u).

.

We use them to express operation E in terms of retracts .E123 and .E14 : E(y, x, v, u) = E123 (E1−1 E14 (y, u), x, v).

.

(11.46)

This results in a system of equations—(11.46) and: A(x, B(y, C(z, z' ), u), D(v, w)) = E123 (E1−1 E14 (F (z' , y), K(z, u)), x, H (w, v)) (11.47)

.

equivalent to (11.44), but without quaternary operations. We concentrate on the second equation (11.47). In particular it contains three ternary operations: .A, B and .E123 . A and .E123 are mutually isostrophic, but the subterm .B(. . . ) corresponds to a subterm of .E123 (. . . ) with only binary operations. Therefore B must be reducible as well. We have: A2 B(y, C1 z, u) = E14 (F2 y, K(z, u))

.

516

A. Krapež

which yields: B(y, z, u) = B13 (y, B3−1 B23 (z, u)).

.

(11.48)

Now, we have a new system with three equations—(11.46), (11.48) and: A(x, B13 (y, B3−1 B23 (C(z, z' ), u)), D(v, w)) = .

= E123 (E1−1 E14 (F (z' , y), K(z, u)), x, H (w, v)).

(11.49)

The equation (11.49) is irreducible, i.e. none of the remaining operations from (11.48) can be reduced further. In this situation we can apply the methods just like in previous cases—separation into diisotopy classes and formation of equations with only mutually diisotopic operations. The word diisotopy is emphasized because it is not the proper one to use here. For example, we have isostrophy between A and .E123 , but it is not isotopy and diisotopy was not defined for ternary operations—it is just similar to diisotopy. To be precise, we have the isotopy of .E123 with .A(12) , where .A(12) (x, y, z) = A(y, x, z). As a result of separation of classes, we end up with the following system of equations which are together equivalent to (11.49): A(x, B1 y, D1 v) = E123 (F2 y, x, H2 v)

.

A2 B13 (y, B3−1 B23 (C(z, z' ), u)) = E14 (F (y, z' ), K(z, u))

.

A3 D(v, w) = E3 H (w, v)

.

(11.50) (11.51) (11.52)

The new system consisting of Eqs. (11.46), (11.48), (11.50), (11.51) and (11.52) is equivalent to the starting equation (11.44). Moreover, Eqs. (11.50), (11.51) and (11.52) are irreducible. We introduce loops .L, · and .Λ: −1 −1 L(x, y, z) = A(A−1 1 x, A2 y, A3 z)

.

x · y = A2 B13 ((A2 B1 )−1 x, (A2 B2 )−1 y)

.

Λ(x, y) = A3 D((A3 D1 )−1 x, (A3 D2 )−1 y).

.

Therefore, we have: A(x, y, z) = L(A1 x, A2 y, A3 z)

.

B13 (x, y) = A−1 2 (A2 B1 x · A2 B3 y)

.

D(x, y) = A−1 3 Λ(A3 D1 x, A3 D2 y)

.

11 Generalized Quadratic Quasigroup Functional Equations

517

and consequently: E123 (x, y, z) = L(E2 y, E1 x, E3 z), E14 (F2 y, K2 u) = A2 B13 (y, u) which implies .E14 (x, y) = E1 x · E4 y, ' ' −1 (A B C y, .A2 B2 C(z, z ) = E14 (F1 z , K1 z) which implies .C(x, y) = (A2 B2 ) 2 2 2 A2 B2 C1 x), −1 ' ' .A2 B23 (C2 z , u) = E14 (F1 z , K2 u) which implies .B23 (x, y) = A 2 (A2 B2 x · A2 B3 u), −1 −1 ' ' .E1 F (z , y) = A2 B13 (y, B 3 B2 C2 z ) which implies .F (x, y) = E1 (E1 F2 y · E1 F1 x), −1 .E4 K(z, u) = A2 B23 (C1 z, u) which implies .K(x, y) = E 4 (E4 K1 x · E4 K2 y) and −1 .E3 H (w, v) = A3 D(v, w) which implies .H (x, y) = E 3 Λ(E3 H2 y, E3 H1 x). . .

Finally, from (11.46) and (11.48) we get: E(x, y, z, u) = E123 (E1−1 E14 (x, u), y, z)

.

and B(x, y, z) = B13 (x, B3−1 B23 (y, z)).

.

Operation L is a ternary loop. Since operations .C ∗ , D, F ∗ and .K ∗ all belong to the same .∼-class, operation .· must be a group. Let us define: α α2 β . β2 γ α3 δ δ1

= = = = =

Id Id Id Id χ2

= ε2 = ϕ2 = κ1 = ϕ1 = κ2

α1 β1 γ1 γ2 β3

ε ε1 ϕ ε3 χ ε4 κ δ2

= = = = =

Id Id Id Id χ1 .

(11.53)

Then, the solution of (11.44) may be presented thus: εL(ε2 y, ε1 x · ε4 u, ε3 z) ϕ(ϕ2 y, ϕ1 x) χ Λ(χ2 y, χ1 x) κ(κ1 x · κ2 y) (11.54) with addition of relationships among permutations which are yielded by 1consequences of (11.44) and renamed by (11.53). A(x, y, z) B(x, y, z) . C(x, y) D(x, y)

= αL(α1 x, α2 y, α3 z) = β(β1 x · β2 y · β3 z) = γ (γ2 y · γ1 x) = δΛ(δ1 x, δ2 y)

E(x, y, z, u) F (x, y) H (x, y) K(x, y)

= = = =

∗

.

We see that there are two sharply different cases—reducible and irreducible equations.

518

A. Krapež

The case of irreducible operations is almost the same as the case of binary operations. The main reason is the following: Lemma 2 Let Eq be an irreducible functional equation on quasigroups, including an n-ary operation F of arity .n > 2. Then, F belongs to a small .∼-class. For reducible equations (i.e. these with at least one reducible operation) we have to add the process of transformation of the original equation by the system(s) with smaller number of reducible operations of maximal arity. Namely, the reduction process might introduce several retracts (which are not necessarily irreducible) but they are all of smaller arities than the original operation. We can recognize a double induction here, but it works and the process of the elimination of reducible operations always terminates. For the proof, see Krapež [19]. The final result is that we end up with an irreducible equation, which we know how to solve and then we find the solution for the original quasigroups as expressions which are functions of their irreducible retracts. Just as it was shown in two preceding examples (Eqs. (11.43) and (11.44)).

11.4.4 Generalized n-Ary Mediality We already solved one version of mediality—the one involving only binary quasigroup operations (11.28). The most general mediality is the .m × n-mediality which involves m-ary and n-ary quasigroup operations. We shall solve just a modest version—.2 × 3-mediality: A(B(x, y, z), C(u, v, w)) = E(D(x, u), F (y, v), H (z, w))

.

(11.55)

The generalization to .m ( × n is quite straightforward. ) x y z u v w We make a choice . and define other constants according to b1 b2 b3 c1 c2 c3 plan. In particular we get the following .2- and .3-consequences:

.

A(B1 x, C1 u) = E1 D(x, u) A1 B(x, y, z) = E(D1 x, F1 y, H1 z) A(B2 y, C2 v) = E2 F (y, v) A(B3 z, C3 w) = E3 H (z, w) A2 C(u, v, w) = E(D2 u, F2 v, H2 w)

and the most important one: .E(D1 x, F1 y, H2 w) = A(B12 (x, y), C3 w), which proves that the operation E, and consequently B and C, are reducible. This yields E(u, v, z) = E13 (E1−1 E12 (u, v), z).

.

The isotopy of B and C to E implies:

(11.56)

11 Generalized Quadratic Quasigroup Functional Equations

519

B(x, y, z) = B13 (B1−1 B12 (x, y), z)

(11.57)

C(u, v, w) = C13 (C1−1 C12 (u, v), w).

(11.58)

.

and .

Replacing this into (11.55), we get: A(B13 (B1−1 B12 (x, y), z), C13 (C1−1 C12 (u, v), w)) = −1 .= E13 (E 1 E12 (D(x, u), F (y, v)), H (z, w)).

.

The previous equation is irreducible and together with (11.56), (11.57) and (11.58) it makes a system equivalent to (11.55). We already have .A ≈ D, A ≈ F and .A ≈ H . Also: A(B1 x, C2 v) = E12 (D1 x, F2 v) hence .A ≈ E12 , A(B1 x, C3 w) = E13 (D1 x, H2 w) hence .A ≈ E13 , .A1 B12 (x, y) = E12 (D1 x, F1 y) hence .B12 ≈ E12 , .A1 B13 (x, y) = E13 (D1 x, H1 z) hence .B12 ≈ E12 , .A2 C12 (u, v) = E12 (D2 u, F2 v) hence .C12 ≈ E12 , .A2 C13 (u, w) = E13 (D2 u, H2 w) hence .C13 ≈ E13 . . .

−1 Therefore, there is a group .+, defined by .x +y = A(A−1 1 x, A2 y) which is isotopic to all of them:

= A1 x + A2 y = A−1 1 (A1 B1 x + A1 B2 y) = A−1 1 (A1 B1 x + A1 B3 z) = A−1 2 (A2 C1 u + A2 C2 v) = A−1 2 (A2 C1 u + A2 C3 w)

A(x, y) B12 (x, y) . B13 (x, z) C12 (u, v) C13 (u, w)

E13 (u, v) = E12 (y, z) = D(x, u) = F (y, v) = H (z, w) =

E1 u + E3 v E1 y + E2 z E1−1 (E1 D1 x + E1 D2 u) E2−1 (E2 F1 y + E2 F2 v) E3−1 (E3 H1 z + E3 H2 w)

Replacing all binary operations in the irreducible consequence of the equation (11.55), we get: A1 B1 x + A1 B2 y + A1 B3 z + A2 C1 u + A2 C2 v + A2 C3 w =

.

.

= E1 D1 x + E1 D2 u + E2 F1 y + E2 F2 v + E3 H1 z + E3 H2 w

which implies commutativity of .+. Moreover: B(x, y, z) = A−1 1 (A1 B1 x + A1 B2 y + A1 B3 z)

.

C(u, v, w) = A−1 2 (A2 C1 u + A2 C2 v + A2 C3 w)

.

E(x, y, z) = E1 x + E2 y + E3 z.

.

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A. Krapež

It follows that a general solution can be written in the form: P (x, y) = π(π1 x + π2 y) for .P ∈ o2 = {A, D, F, H }, Q(x, y, z) = ω(ω1 x + ω2 y + ω3 z) for .Q ∈ o3 = {B, C, E}

. .

where: α = Id α1 β = α2 γ = . β1 = β2 = β3 =

=ε Id Id δ1 ϕ1 χ1

ε1 δ ε2 ϕ ε3 χ γ1 γ2 γ3

= Id = Id = Id = δ2 = ϕ2 = χ2 .

11.4.5 System of Generalized n-Ary Associativity Following J. Ušan [45–47], we are solving a system of generalized associativity for ternary operations. The process is quite similar for n-ary operations. It is a best way to understand the present result as the first step in the induction proof of generalized associativity for n-ary operations. Assume: A(x, y, B(z, u, v)) = C(x, D(y, z, u), v) = E(F (x, y, z), u, v).

.

(11.59)

All operations are ternary but they are reducible: A(x, y, B1 z)=C12 (x, D12 (y, z)) which implies: .A(x, y, u)=A12 (x, A−1 2 A23 (y, u)), −1 .A3 B(z, u, v)=C23 (D23 (z, u), v) which implies: .B(z, u, v)=B23 (B B 12 (z, u), v), 2 .

and similarly: C(x, u, v) = C12 (x, C2−1 C23 (u, v)) −1 .E(x, u, v) = E23 (E 2 E12 (x, u), v) .

D(y, z, u) = D12 (y, D2−1 D23 (z, u)) −1 .F (x, y, z) = F12 (x, F 2 F23 (y, z)).

.

After the replacement of all ternary operations with appropriate binary operations, we get the irreducible equation: .

−1 −1 23 A12 (x, A−1 2 A (y, B23 (B2 B12 (z, u), v))) = C1 y + C2 C2 (C2 D2 x + C2 D1 z) −1 = E2 E2 (E2 F1 x + E2 F2 y) + E2 z. (11.60)

It is easy to see that all operations from (11.60) are mutually isotopic. Therefore, there is a group .· such that all operations from (11.60) are isotopic to. The end result is the following:

11 Generalized Quadratic Quasigroup Functional Equations

521

Theorem 16 A general solution of the generalized 3-associativity equation (11.59) is given by: P (x, y, z) = π(π1 x · π2 y · π3 z)

.

P ∈ o = {A, B, C, D, E, F }

.

where: . .· is an arbitrary group (with unit e) . .π1 , π2 , π3 , π are arbitrary permutations of S, related to .P ∈ o and such that: α=γ α3 β . γ2 δ ε1 ϕ

= ε = Id = Id = Id = Id

α1 α2 β1 β2 β3

= γ1 = = δ1 = = δ2 = = δ3 = = γ3 =

ϕ1 ϕ2 ϕ3 ε2 ε3 .

11.5 Generalized Quadratic Equations on Binary Quasigroups Quadratic equations is the name given by S. Krsti´c to equations which were introduced in A. Krapež [22] under the name strictly quadratic equations. They are characterized by the property that every object variable appears in them exactly twice, but not necessarily just once on each side of the equation symbol .= (which is the property of balanced equations).

11.5.1 Quadratic Equation with One Variable This is the equation: F (x, x) = e

.

(11.61)

for which we say that entails the existence of the middle unit. There is a formal problem that it contains an extra constant e. But, not to include it among relevant equations does not solve the problem as we have the equation .F (x, x) = G(y, y) with two variables (and no constants) which reduces to (11.61) as soon as y gets a value in S. Another problem is that although there are quasigroups (with middle unit) on a set S with any number of elements, there are not always such loops and groups. For example, there are not loops/groups with middle unit on any set with three elements. However Eq. (11.61) still has models which are loops/groups, but with .2m elements.

522

A. Krapež

11.5.2 Quadratic Equations with Two Variables Just as in the case of balanced equations, the most important for quadratic equations are 2-consequences. For any generalized quadratic functional equation Eq, a 2consequence of Eq has one of the following 24 forms (translations are ignored): F (x, x) = G(y, y) F (x, y) = G(x, y) F (x, y) = G(y, x) F (x, G(x, y)) = y F (x, G(y, x)) = y F (x, G(y, y)) = x . F (G(x, x), y) = y F (G(x, y), x) = y F (G(x, y), y) = x E(x, F (x, G(y, y))) = e E(x, F (y, G(x, y))) = e E(x, F (y, G(y, x))) = e

(0) (2) (1) (1) (2) (0) (0) (2) (1) (0) (2) (1)

E(x, F (G(x, y), y)) = e E(x, F (G(y, x), y)) = e E(x, F (G(y, y), x)) = e E(F (x, x), G(y, y)) = e E(F (x, y), G(x, y)) = e E(F (x, y), G(y, x)) = e E(F (x, G(x, y)), y) = e E(F (x, G(x, y)), y) = e E(F (x, G(x, y)), y) = e E(F (G(x, x), y), y) = e E(F (G(x, y), x), y) = e E(F (G(x, y), y), x) = e

(1) (2) (0) (0) (2) (1) (1) (2) (0) (0) (2) (1)

We see that in some cases quasigroups F and G are parastrophes or, since translations were ignored, isostrophes of each other. Some of these isostrophies are even and some are odd. Specifically, F and G are mutually even (odd) isostrophic iff they satisfy any of the 2-consequences denoted by (2) ((1)). In other cases, denoted by (0), we cannot draw any conclusion about isostrophy of F and G. In no case we can infer isostrophy of E to either F or G. Analogously to the balanced case, the equivalence relation .∼ related to isostrophy of quasigroup operations in Eq is defined. Recall, from balanced equations we were getting diisotopic operations. Likewise, the relations .≈ and .↔ are defined, pertaining to even .(·, \\, //) and odd .(\, /, ∗) parastrophes of .·. Paper [22] (see also [42]) gives a general solution in case of parastrophically uncancellable equations i.e. when .∼ is full relation on .o. .F ∼ G iff F is isostrophic to G. .F/ ∼ = (F/ ≈) ∪ (F/ ↔). If .|F / ∼ | > 2 then F is isotopic to a group (.F/ ∼ is a big class). If .F/ ≈ = F/ ↔ then F is isotopic to an Abelian group (.F/ ∼ is an Abelian class). Every operation/symbol F (class .F/ ∼) from .s = t is a loop operation/symbol (class). A class .F/ ∼ is either small (.|F/ ∼ | ≤ 2) or big (.|F/ ∼ | > 2). F is a group operation iff .F/ ∼ is big. Group operation (big class) is Abelian iff .F/ ≈= F/ ↔.

11.5.3 Generalized Transitivity Equation This is the equation:

11 Generalized Quadratic Quasigroup Functional Equations

A(B(x, y), C(y, z)) = E(x, z)

.

523

(11.62)

which was solved in [3]. Let us do it again. Equation (11.62) is equivalent to .A(B(x, y), u) = E(x, C −2 (y, u)). This has the form of the generalized associativity equation which entails existence of a group .· isotopic to .A, B and E i.e.:

.

A(x, y) = A1 x · A2 y B(x, y) = A−1 1 (A1 B1 x · A1 B2 y) E(x, z) = E1 x · E2 z).

But then .A(B2 y, C(y, z)) = E2 z and .A1 B2 y · C(y, z) = A2 C2 z, therefore: C(y, z) = A−1 2 (IA1 B2 y · A2 C2 z).

.

If we define:

.

α = ε = Id α1 β = Id α2 γ = Id

β1 = ε1 β2 y · γ1 y = e γ2 = ε2

(11.63)

we get: Theorem 17 General solution of the generalized transitivity equation (11.62) on the set S is given by: P (x, y) = π(π1 x · π2 y)

.

for .P ∈ o = {A, B, C, D},

where .· is an arbitrary group on S, while .π, π1 , π2 related to P are arbitrary permutations of S satisfying (11.63). ∗

.

Just as an exercise for the reader, we give the following equation: A(B(x, y), C(y, z)) = E(D(x, u), F (u, z))

.

(11.64)

which we call double transitivity. The equation has a corresponding Krsti´c graph .G1 (see Fig. 2, page 493 ).

11.5.4 Krsti´c Graphs S. Krsti´c [35] introduced the connection between generalized quadratic functional equations and particular cubic (multi)graphs which later got the name Krsti´c graphs.

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A. Krapež

We use standard graph-theoretic notions and facts, which we review next to make the Chapter more self-contained. This section serves also to fix terminology because it is not standardized in the literature. Recall, a multigraph is a triple .G = (V , E; I ), where V and E are finite disjoint sets whose elements are called vertices and edges respectively, while .I ⊆ V × E is an incidence relation. We also assume that every edge is incident to one or two vertices. In particular, an edge incident to one vertex is also called a loop. Edges incident to the same pair of vertices are referred as multiple edges. The degree of a vertex v, denoted by .deg(v), is equal to the number of edges incident to it, but provided that each loop is counted twice. Here, we will use shorter term graph for multigraphs, and assume that all graphs under considerations are finite (i.e. V and E are both finite), and also nontrivial (i.e. V is non-empty). A graph .G is cubic if every vertex has degree equal to 3. A cubic graph .G is of order .|V | = 2n and size .|E| = 3n for some .n > 0. In the rest of the Chapter we will consider only (finite) connected and cubic graphs. Definition 17 Two vertices u and v of a graph G are connected if there is a path between them. G is connected if every pair of its vertices is connected. Definition 18 Two (distinct) vertices u and v of a graph .G are 3-edge-connected (or 3-connected for short) if we need to remove at least three edges from .G to disconnect them. The 3-connectivity relation defined as above, if extended by reflexivity, is an equivalence relation on V denoted by the symbol .≡. The vertices of .G are partitioned by .≡ into equivalence classes. Definition 19 A graph .G is 3-edge-connected (or 3-connected for short) if the removal of at least three edges from .G makes it disconnected. Connectivity .c(G) of .G is the smallest number of edges whose removal disconnects .G. Since we are only interested in the notion of edge-connectivity in a graph (but not vertex-connectivity), we call the 3-edge-connectivity property simply 3connectivity. By Menger Theorem, a graph .G is 3-connected if and only if for any two vertices u and v of .G there are at least three edge-disjoint paths in .G from u to v. Obviously, for a cubic graph .G, .c(G) ≤ 3 and .G is 3-connected if and only if .c(G) = 3 if and only if the relation .≡ is the full relation on the vertices of .G.

11.5.5 Generalized Quadratic Equations Definition 20 Let a generalized quadratic functional equation s = t be given. The Krsti´c graph K(s = t) of the equation s = t is defined by: . K(s = t) is the graph (V , E; I ); . The vertices of K(s = t) are operation symbols of s = t;

11 Generalized Quadratic Quasigroup Functional Equations

525

. The edges of K(s = t) are subterms of s and t which are considered to be a single edge. Likewise, any variable (which appears twice in s = t) is taken to be a single edge; . If F (p, q) is a subterm of s or t then the vertex F is incident to edges p, q, F (p, q) and no other. In [26], S. Krsti´c proved: Theorem 18 Generalized quadratic functional equations Eq and Eq' are parastrophically equivalent iff graphs K(Eq) and K(Eq' ) are isomorphic. Theorem 19 Generalized quadratic functional equations Eq is parastrophically uncancellable iff K(Eq) is 3-connected. Theorem 20 Let Eq be a generalized quadratic functional equation with operation symbols F and G. The following statements are equivalent: . In every solution of Eq, quasigroups (i.e. interpretations of operation symbols) F and G are isostrophic. . F ∼ G in Eq. . F ≡ G in K(Eq). . F and G are 3-connected in K(Eq). Theorem 21 Let s = t be a generalized quadratic functional equation and K(s = t) the corresponding Krsti´c graph. Then: . Every operation symbol F is a loop symbol. . F is a group symbol iff F /∼ is big iff tetrahedron K4 is homeomorphically embeddable in K(s = t) within F /≡ iff K4 is homeomorphically embeddable in K(si = ti ) for F ∈ oi . . F is an Abelian symbol iff F /∼ = F /≈ iff subgraph F /≡ is not planar iff the complete bipartite graph K3,3 is homeomorphically embeddable in K(s = t) within F /≡ iff K3,3 is homeomorphically embeddable in K(si = ti ) for F ∈ oi . See graphs K4 and K3,3 in Figs. 11.1 and 11.2. Graph K4 corresponds to both generalized associativity and generalized transitivity equations and graph G1 to the equation of double transitivity. Graph K3,3 corresponds to the equation of 2 × 2-mediality, related to planarity of graphs by the famous result of K. Kuratowski [27, 28].

526

A. Krapež

K4

Fig. 11.1 Uncancellable equation, n = 3

G1

K3,3

Fig. 11.2 Uncancellable equations, n = 4

11.6 Generalized Quadratic Equations on n-Ary Quasigroups We just mention two early attempts by F. Sokhatsky, H. Krainichuk, A. Tarasevych [41] and F. Sokhatsky, A. Tarasevych [42] at classification of such equations but without the requirement of being generalized. Another item is an unpublished manuscript by A. Krapež and B. Laskovi´c [25].

11.7 Beyond Quasigroups In the last section of this Chapter, we show how to solve some of the equations where operations are weaker than quasigroups. Many cases are left out as the number of results grows daily. For example, we skip all results on GD-groupoids which are 3-sorted division groupoids (see [32, 38, 43]). Many results about them were used to prove results on n-ary quasigroups. Another hot topic is a number of results about other structures weaker than quasigroups such as division algebras. We recommend the recent book [35] by Yu. Movsisyan and S. Davidov on this topic as it includes many results on various second order type of formulas which are treated like equations.

11 Generalized Quadratic Quasigroup Functional Equations

527

We also mention a paper Infinitary quasigroups [14] by V. D. Belousov and Z. Stojakovi´c, the first and one of only a few papers on this topic.

11.7.1 On Semilattices This is my favorite of all results presented. Simple, elegant and with a surprising conclusion. The authors are solving the generalized associativity equation for semilattices, i.e. groupoids which are associative, commutative and idempotent (.xx = x). Theorem 22 (S. Mili´c, A. Tepavˇcevi´c [33]) If semilattice operations .A, B, C, D defined on the same set .S = / ∅ satisfy generalized associativity equation: A(x, B(y, z)) = C(D(x, y), z)

.

(11.65)

then all four operations are mutually equal. Proof A(x, y) = A(y, x) = A(y, B(x, x)) = C(D(y, x), x) = C(D(x, y), x)

.

= A(x, B(y, x)) = A(x, B(x, y)) = C(D(x, x), y) = C(x, y). A(x, y) = A(x, B(x, y)) = A(x, B(B(x, y), B(x, y)))

.

= C(D(x, B(x, y)), B(x, y)) = C(D(B(x, y), x), B(x, y)) = A(B(x, y), B(x, B(x, y))) = A(B(x, y), B(x, y)) = B(x, y). Similarly, .C = D.

u n

Natural? Easy? Let’s try it for the similar generalized transitivity equation, i.e. assume that we have four semilattices .A, B, C, D satisfying A(B(x, y), C(y, z)) = D(x, z).

.

Then: .x = D(x, x) = A(B(x, y), C(y, x)) = A(B(y, x), C(x, y)) = D(y, y) = y. The conclusion is that there are no such operations except in the trivial case .|S| = 1. Does this change your perspective of Mili´c, Tepavˇcevi´c result? With me, it does.

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A. Krapež

11.7.2 On Almost Trivial Groupoids J.F. Berglund and M.W. Mislove in [14] defined semigroups with almost trivial multiplication as semigroups whose translations are either surjections or constant mappings. They proved that such semigroup belongs to one of five well known semigroup classes. Almost trivial groupoids (ATGs) are defined in A. Krapež [21] as groupoids with translations which are either permutations or constant mappings. Although similar, two groupoid classes are incomparable—one requires associativity, the other that translations are permutations, not just surjections. All results presented in this subsection are from [21]. Definition 21 Groupoid .(S; ·) is: . Left groupoid if .xy = ϕx (.ϕ-permutation) . Right groupoid if .xy = ϕy (.ϕ-permutation) . Quasigroup with quasizero .(p, q, r) if: – – – –

For all x: .px = r For all x: .xq = r All left translations: .λa x = ax (a /= p) are permutations. All right translations: .ρb x = xb (b /= q) are permutations.

As usual, the best way to understand the definition is to see a good example: Example 1 Quasigroup .(S; ·) with a quasizero .(p, q, r): · a b . p q r

a q a r p b

b a b r q p

p b p r a q

q r r r r r

r p q r b a

The following representation theorems are proved in [21]. Theorem 23 Any ATG is one of: . . . . .

Zero-semigroup (.xy = 0) Left groupoid (.xy = ϕx) Right groupoid (.xy = ϕy) Quasigroup Quasigroup with quasizero.

Theorem 24 Any ATG is a principal isotope of the one of: . . . .

Zero semigroup Left zero semigroup (.xy = x) Right zero semigroup (.xy = y) Loop

11 Generalized Quadratic Quasigroup Functional Equations

529

. Loop with (externally added) zero. Groupoids mentioned in the last Theorem are principal ATGs. The following is proved in [21]: Theorem 25 The general solution of the generalized associativity equation: A(x, B(y, z)) = C(D(x, y), z)

.

on ATGs is given by: P (x, y) = π1 x ·P π2 y

.

where .·P are principal ATGs satisfying: x·A (y·B z) = (x·D y)·C z.

.

Example 2 There are 16 different groupoids with just two elements. All of them are ATGs. Consequently, there are .164 = 65,536 quadruples of (binary) operations which are candidates for the satisfaction of generalized associativity on 2-element groupoids. In [21] it is proven that only 1344 quadruples satisfy it.

11.7.3 On Groupoids Here is the solution of the most general version of associativity equation—on groupoids, see A. Krape.ž [20]. We fix the terminology and notation first. Definition 22 For a given function .P : S −→ S ' we have .x(ker P )y iff P (x) = P (y). Bijection .fP : S/ ker P −→ P (S), defined by .fP (x ker P ) = P (x), is naturally associated with P . Definition 23 Translations .λ, μ, ρ of a ternary operation T on S are defined by: λxy (z) = μxz (y) = ρyz (x) = T (x, y, z).

.

Also, .TS = {f |f : S −→ S} and .T1 = {ρyz |y, z ∈ S}, T2 = {μxz |x, z ∈ S}, T3 = {λxy |x, y ∈ S}.

.

Definition 24 If .α is an equivalence on .S 2 , then: (a, b, c)α1 (p, q, r) iff a = p and (b, c)α(q, r),

.

(a, b, c)α2 (p, q, r) iff b = q and (a, c)α(p, r),

.

(a, b, c)α3 (p, q, r) iff (a, b)α(p, q) and c = r.

.

530

A. Krapež

Definition 25 Let .ker B, ker D and .ker T be the kernels of operations .B, D : S 2 −→ S and .T : S 3 −→ S respectively. We define: ΓA : (x, (y, z)ker B ) −→ (x, y, z)ker T ,

.

ΓC : ((x, y)ker D , z) −→ (x, y, z)ker T .

.

Functions .ΓA and .ΓC are not always well defined but we can easily formulate conditions under which they are. These conditions are satisfied in the following theorem: Theorem 26 The general solution (on a nonempty set S) of the generalized associativity equation: A(x, B(y, z)) = C(D(x, y), z)

.

(GA)

is given by: . B and D are on S, { arbitrary groupoids fT ΓA (x, fB−1 y), for (x, y) ∈ S × B(S, S) . .A(x, y) = arbitrary, otherwise { fT ΓC (fD−1 x, y), for (x, y) ∈ D(S, S) × S . .C(x, y) = arbitrary, otherwise where T is an arbitrary 3-groupoid such that .(ker B)1 ∨ (ker D)3 ⊆ ker T . There are several important consequences of this theorem: Corollary 1 Ternary operation T (on S) is reducible iff .|T1 | ≤ |S| or .|T2 | ≤ |S| or .|T3 | ≤ |S|. Corollary 2 Any ternary operation on an infinite set is reducible. Theorem 27 Let A be a binary operation on S and let .T (x, y, z) = A(x, A(y, z)). A is associative iff . .g0 = {(A(x, y), λxy )|x, y ∈ S} is a function from .A(S, S) to .T3 , and . .(g0 x)y = A(x, y) for all .x ∈ A(S, S). In the recent paper [39] by D. A. Shahnazaryan, above results were used to prove the following: Theorem 28 Let four operations .A, B, C, D be right quasigroups on S. If these operations satisfy generalized associativity equation, then: . there is a right loop .(S; ·) such that A and D are isotopic while B and C are homotopic to .(S; ·) and . there are mappings .A1 , A2 , B1 , B2 , C1 , C2 , D1 , D2 : S → S (with .A1 , B1 , C1 , D1 being bijections) and such that:

11 Generalized Quadratic Quasigroup Functional Equations

A(x, y) A2 B(x, y) . C(x, y) C1 D(x, y)

= = = =

531

A1 x · y C1 D 2 x · B 2 y C1 x · B 2 y A1 x · B1 y

and

.

A1 = C1 D1 A2 B1 = C1 D2 A2 B2 = C2 .

References 1. Aczél, J.: Functional Equations and Their Applications, Academic Press, New York, London (1966). 2. J. Aczél ed.: Functional Equations: History, Applications and Theory, D. Reidel Publishing Company, (1984). 3. Aczél, J. Belousov V.D., Hosszú M.: Generalized associativity and bisymmetry on quasigroups. Acta Math. Acad. Sci. Hungar. 11, 127–136 (1960). 4. Albert, A.A.: Quasigroups I, Trans. Amer. Math. Soc. 54, 507-519 (1943). 5. Alimpi´c, B.: Balanced laws on quasigroups (Serbian), Mat. Vesnik 9(24), 249–255 (1972). 6. Andjeli´c T.P. (ed.): Symposium en Quasigroupes et Equation Fonctionnelles (Belgrade–Novi Sad, 1974). Zbornik Rad. Mat. Inst. Beograd (N.S.) 1(9), Matematiˇcki institut, Beograd, (1976). 7. Belousov, V.D.: Balanced identities in quasigroups (Russian), Mat. Sb. (N.S.) 70 (112) 55-97 (1966). 8. Belousov, V.D.: Foundations of the Theory of Quasigroups and Loops (Russian), Nauka, Moscow (1967). 9. V.D. Belousov: Configurations in algebraic nets (Russian), Shtiinca, Kishinev (1979). 10. Belousov, V.D.. n-Ary Quasigroups (Russian), Stiintsa, Kishinev (1972). 11. Belousov, V. D., Livšic, EE.S.: Functional equation of generalized associativity on binary quasigroups (Russian). Mat. Issled. 4 (34), 1974. 12. V. D. Belousov, EE.S. Livšic. Balanced functional equations on quasigroups of arbitrary arity (Russian). Mat. Issled. 43 (1974). 13. V. D. Belousov, Z. Stojakovi´c. Infinitary quasigroups, in the book [6], 31–42 (1976). 14. Berglund, J.F., Mislove, M.W.: A class of semigroups having almost trivial multiplications, Semigroup Forum 4/2, (1972) 15. Dénes J., Keedwell A.D.: Latin squares and their applications, Académiai Kiadó, Budapest (1974). 16. Fisher, R.A,: The design of experiments, 8th edition, Oliver & Boyd, Edinburgh (1966). 17. Ibragimov, S. G.: On the prehistory of quasigroup theory. From the neglected work of E. Schroeder, 19th Century, In: Abstracts of lectures: First All–Union Conference of quasigroups, pp. 15-16, Suhumi (1967). ˇ ˇ 93/2, 18. J. Ježek, T. Kepka: Medial groupoids, Rozpravy Ceskoslovenske Academie VED Academia, Praha (1983). 19. A. Krapež: On solving a system of balanced functional equations on quasigroups I–III, Publ. Inst. Math. (Beograd) (N.S.) 23 (37), (1978), 25 (39), (1979), 26 (40), (1979). 20. Krapež, A.: Generalized associativity on groupoids, Publ. Inst. Math. (Beograd) (N.S.) 28 (42) 105-112 (1980).

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21. Krapež, A.: Almost trivial groupoids, Publ. Inst. Math. (N.S.) (Belgrade)28 (42) 113-124 (1980). 22. Krapež, A.: Strictly quadratic functional equations on quasigroups I. Publ. Inst. Math (Belgrade) (N.S.) 29 (43) 125–138 (1981). 23. Krapež, A. (ed.): A Tribute to S. B. Preši´c: Papers Celebrating his 65th Birthday, Matematiˇcki institut SANU, (2001). 24. Krapež A.: Generalized quadratic quasigroup equations with three variables, Quasigroups and Related Systems 17, 253–270 (2009). 25. Krapež, A., Laskovi´c, B.: 3–diagonal equation and planarity of graphs, manuscript, (2022). 26. S. Krsti´c: Quadratic quasigroup identities (Serbian), PhD thesis, University of Belgrade, (1985). 27. K. Kuratowski: Sur le problème des courbes gauches en Topologie, Fund. Math. 15 (1930), 271–283. 28. K. Kuratowski: On the problem of skew curves in topology (English translation of [27], in Borowiecki et al, Editors: Graph Theory (Lagow 1981), Lecture Notes in Math. 1018, Springer, Berlin (1983), 1–13. 29. EE.S. Livšic. Functional equations of the 2nd kind on binary quasigroups (Russian). Mat. Issled. 36 (1975). 30. EE.S. Livšic. Balanced functional equations of the 1st kind on quasigroups of arbitrary arity (Russian). Mat. Issled. 39 (1976). 31. EE.S. Livšic. Balanced functional equations on quasigroups of arbitrary arity (Russian). Mat. Issled. 43, (1976). 32. S. Mili´c: On GD–groupoids with application to n–ary quasigroups, Publ. Int. Math. (Beograd) (N.S.) 13 (27) 65–76, (1972). 33. Mili´c, S., Tepavˇcevi´c, A.: On P-fyzzy correspondences and generalized associativity, Fuzzy sets and systems 96, 223-229 (1998). 34. R. Moufang: Zur Struktur von Alternativkörpern, Math. Ann. 110, 416–430, (1935). 35. Movsisyan, Yu.M., Davidov, S.S.: Algebras That are Nearly Quasigroup (Russian), URSS, Moscow, 2018. 36. H.O. Pflugfelder. Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990. 37. Sade, A.: Entropie demosienne de multigroupoides et de quasigroupes, Ann. Soc. Sci. Bruxelles 73, 302-309 (1959). 38. Satyabhama, V.: Generalized Distributivity Equation on GD–groupoids, Mat. Vesnik 29/1, 137146 (1972). 39. Shahnazaryan, D.A.: On Functional Equations of associativity with right quasigroup operations, Proceedings of the Yerevan State University, Physical and Mathematical Sciences 53/3, 150-155, (2019). 40. V. Shcherbacov, Elements of Quasigroup Theory and Applications, CRC Press, Boca Raton, 2017. 41. F. Sokhatsky, H. Krainichuk, A. Tarasevych: A classification of generalized functional equations on ternary quasigroups, Visnyk DonNU, A: Natural Sciences no. 1-2, (2017). 42. F. M. Sokhatsky, A. Tarasevych: Classification of generalized ternary quadratic quasigroup functional equations of the length three, Carpathian Math. Publ. 11(1) 179-192 (2019). http:// www.journals.pu.if.ua/index.php/cmp 43. Stojakovi´c, Z.: Generalized Entropy on GD–groupoids with Applications to Quasigroups of various Arities, Mat. Vesnik 24/1, 159-166 (1972). 44. A. Ungar: Beyond the Einstein Addition Law and its Gyroskopic Thomas Precession – The Theory of Gyrogroups and Gyrovector Spaces, Kluwer Academic Publishers, Dordrecht, Boston, London (2001). 45. Ušan, J.: A generalization of a theorem of V. D. Belousov on four quasigroups to the ternary case (in Russian). Bull. Soc. Math. Phys. Macédoine 20, 13–17, (1969). 46. Ušan, J.: An n–ary analogue of the Belousov’s theorem on four quasigroups and some corollaries of it (in Russian). Bull. Soc. Math. Phys. Macédoine 21, 5–17 (1970). 47. Ušan, J.: n–groups in the light of the neutral operations. Math. Moravica, Special vol., ˇ cak, (2003). University of Kragujevac, Technical Faculty of Caˇ

Chapter 12

A Primer on Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism José F. Cariñena, Héctor Figueroa, and Partha Guha

Mathematics Subject Classification (2000) 17B70, 53D17 PACS Numbers 11.10.Nx, 02.40.Yy, 45.20.Jj

12.1 Introduction The study of exotic particle models with non-commutative position coordinates was started about 20 years ago [1, 2]. There are several physical phenomena appearing in condensed matter physics [3, 4], namely semiclassical Bloch electron phenomena, fractional quantum Hall effect [5, 6], double special relativity models, etc., that exhibit such feature. All these models share the somewhat unusual feature that the Poisson brackets of the planar coordinates do not vanish. This class of dynamical structures has appeared in geometric mechanics and geometric control theory [7– 10]. In her thesis, Sánchez de Álvarez [11] indicates a characterization of the Poisson structure in terms of Poisson brackets of particular functions on the tangent bundle T P of a Poisson manifold P , and discusses its functorial properties. A very noble derivation of a pair of Maxwell equations was originally proposed by Feynman, but the exact details of his argument came to the scientific community

J. F. Cariñena Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza, Spain e-mail: [email protected] H. Figueroa Departamento de Matemáticas, Universidad de Costa Rica, San Pedro, Costa Rica e-mail: [email protected] P. Guha (O) Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi, UAE SNBNCBS, Kolkata, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1_12

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from the work of Dyson [12]. According to Dyson, Feynman showed him the construction and examples of the Lorentz force law and the homogeneous Maxwell equations in 1948. A derivation of a pair of Maxwell equations and the Lorentz force is based on the commutation relations between positions and velocities for a single non-relativistic particle. In general the locality property, that is, different coordinates commute, is assumed. Due to increasing interest in non-commutative field theories, it is worthwhile to consider the non-commutative analogue of Feynman approach. This deletes the axiom of locality, which, according to Dyson, was the original aim of Feynman. Tanimura [13] gave both a special relativistic and a general relativistic versions of Feynman’s derivation. Land et al. [14] examined Tanimura’s derivation in the framework of the proper time method in relativistic mechanics and showed that Tanimura’s result then corresponds to the five-dimensional electromagnetic theory previously derived from a Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. An extension of Tanimura’s method has been achieved [15] by using the Hodge duality to derive the two groups of Maxwell’s equations with a magnetic monopole in flat and in curved spaces. A rigorous mathematical description of Feynman’s derivation connected to the inverse problem for Poisson geometry has been formulated in [16] (see also [17]). Hughes [18] considered Feynman’s derivation in the framework of the Helmholtz inverse problem for the calculus of variations (see also [19] and [20]). In fact, it was pointed out by Jackiw that Heisenberg suggested in a letter to Peierls that spatial coordinates may not commute, Peierls communicated the same idea to Pauli, who informed it to Openheimer; eventually the idea arrived to Snyder [21, 22] who wrote the first paper on the subject. Nowadays the physics in non-commutative planes is relevant not only in string theory but also in condensed matters physics [3, 4]. In the context of the Feynman’s derivation of electrodynamics, it has been shown that non-commutativity allows other particle dynamics than the standard formalism of electrodynamics [23]. Noncommutative quantum mechanics is recently the subject of a wide range of works from particle physics to condensed matter physics. This has also been studied from the point of view of Feynman’s formalism in [24]. The examples of exotic mechanics started to appear around 1995. Physicists obtained various models such that the Poisson brackets of the planar coordinates do not vanish. Souriau’s orbit method [1, 5, 25, 26] was used to construct a classical mechanics associated with Lévy-Leblond’s exotic Galilean symmetry. In terms of the Souriau 2-form a wide set of Hamiltonian dynamical systems have been described in [2, 27–31]. Lévy-Leblond [32] has realized that due to the commutativity of the rotation group .O(2, R), the Lie algebra of the Galilei group in the plane admits a second exotic extension defined by [K1 , K2 ] = iκ I,

.

where .κ is the new extension parameter. For a free particle the usual equations of motions are unchanged and .κ only contributes to the conserved quantities. It yields the non-commutativity of the position coordinates. The Carroll group was originally

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introduced by Lévy-Leblond by considering the contraction of the Poincaré group as .c → 0, and it ass been shown in [33, 34] that Carroll group share with Galilei group a kind of duality of times. Feynman procedure to obtain Maxwell’s equation in electrodynamics has been reviewed under different kind of settings, and several nontrivial and interesting generalizations are possible, see for instance [35–42]. Recently, Duval and Horváthy [35] successfully applied the techniques of Souriau’s orbit method [43–45] to various models. Incidentally, one of these models can be viewed as the nonrelativistic counterpart of the relativistic anyon considered before by Jackiw and Nair [38, 46]. Mathematically, the ‘exotic’ model arises due to the particular properties of the plane. A wide set of dynamical systems can be derived from the Lagrange-Souriau 2-form approach in three dimensions and the generalizations to higher number of degrees of freedom have been outlined in [28]. Wong’s equations describe the interaction between the Yang-Mills field and an isotopic-spin carrying particle in the classical limit [47]. Feynman-Dyson’s proof offers a way to check the consistency of these equations [48]. See also [47] for a more recent paper. In a slightly different context Kauffman [49] introduced discrete physics based on a non-commutative calculus of finite differences. This gives a context for the Feynman–Dyson derivation of non-commutative electromagnetism. More recently, Kauffman [50] found an interesting way to describe mechanics in a curved background interacting with gauge fields in such a way that the physical equations of motion emerge automatically from underlying algebraic relations in a non-commutative geometry and this construction depends largely on the FeynmanDyson construction. In an interesting paper, Cortese and García [51] studied a variational principle for noncommutative dynamical systems in the configuration space. In particular they showed that the non-commutative consistency conditions (NCCC), that come from the analysis of the dynamical compatibility, are not the Helmholtz conditions of the generalized inverse problem of the calculus of variations. It has been shown that the .θ -deformed Helmholtz conditions are connected to a third-order time derivative system of differential equations. Noncommutative phase spaces have been introduced by minimal couplings in [52] and then some of them are realized as coadjoint orbits of the anisotropic Newton-Hooke groups in two- and three-dimensional spaces. This has been further generalized to realize noncommutative phase spaces as coadjoint orbit extensions of the Aristotle group in a two dimensional space [53]. In this article we apply Souriau’s orbit method to study exotic mechanics on the tangent bundle or velocity phase space. Souriau first unified both the symplectic structure and the Hamiltonian into a single two-form. It has an exotic symplectic form and a free Hamiltonian and yields a generalized Hamiltonian mechanics. Duval and Horváthy used Souriau’s orbit method to construct a classical planar system associated with Lévy-Leblond’s twofold extended Galilean symmetry. The four dimensional phase space is endowed with the following exotic form O = dpi ∧ dqi +

.

θ eij dpi ∧ dpj , 2

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where summation on repeated indices is understood. The exotic term in the symplectic form only exists in the plane. Following [54, 55] we also explore a volume-preserving flow on a symplectic manifold from the Souriau form associated with the velocity phase space. Many authors [35, 36, 56] have generalized this modification of the symplectic form by introducing the so-called dual magnetic field such that 1 1 O = dpi ∧ dqi + gij dpi ∧ dpj + fij dqi ∧ dqj . 2 2

.

The coefficients .gij and .fij are responsible of the noncommutativity of momenta and positions, respectively. The classical dynamics in noncommutative space leads to noncommutative Newton’s second law [57, 58]. This generalization can be studied in various types of noncommutative space-times; for instance Harikumar and Kapur studied in [59] the modification to Newton’s second law due to the kappadeformation. Recently the modification of integrable models in the kappa-deformed scheme is analyzed [60] and kappa-Minkowski space-time through exotic oscillator is studied in [61]. The kappa-space-time is an example of a non-commutative spacetime with coordinates satisfying a Lie-algebraic-type relation [62]. The Kepler problem has singularities corresponding to collision orbits. In a very recent paper [63] Guha et al. have extended the regularization methods due to Moser and to Ligon and Schaaf to the kappa-deformed space-time and demonstrate the regularization of the kappa-deformed Kepler problem. Zhang et al. [64, 65] studied the 3D mechanics with non-commutativity, where the potential may also depend on the momentum. They obtained the conserved quantities by using van Holten’s covariant framework. It is known that the Snyder model has the remarkable property of leaving the Lorentz invariance intact. Recently, motivated from loop quantum gravity an idea has been proposed to extend the Snyder model [21] to space-times of constant curvature, by introducing a new fundamental constant whose inverse is proportional to the inverse of the cosmological constant. More recently, classical dynamics on Snyder space-times has drawn a lot of attention to physicists [66–75]. Moreover, Snyder dynamics in curved space-time has been extended by Mignemi et al. in [76, 77]. See also [78]. The noncommutative quantum mechanics (NCQM) [79–85] has also been considered as being a non-relativistic approximation of noncommutative quantum field theory (NCQFT) [86] where the underlying fields are considered as functions of a noncommutative space-time. Quantum mechanics in phase space has been studied in [87] in this noncommutative framework. The most widely advocated physical applications of NCQM is provided by quantum mechanical systems coupled to a constant background magnetic field [88]. The Landau levels have also been studied in this noncommutative quantum mechanics framework [89–94]. The difference of the exotic NCQM suggested by Duval and Horvathy from the conventional planar NCQM lies in the coupling of an external magnetic field. Instead of a naive, or algebraic approach, used in conventional NCQM, the minimal, or symplectic, coupling is used there, in the spirit of Souriau. The enlarged exotic Galilei symmetry

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in the noncommutative plane was used in [95] to study the electric Chern-Simons term. In a very recent paper [96], Chowdhury demonstrated that there exists a 2parameter family of vector potentials associated with a 2-dimensional quantum system in a constant external magnetic field and showed how these choices of gauge potentials are related with the unitarily equivalent irreducible representations of .GN C . It is worth noting that noncommutativity causes fundamental problems and one of them is the violation of weak equivalence principle. In an interesting paper [97] this has been recovered in two-dimensional noncommutative phase space and the conditions on the parameters of noncommutativity to recover this equivalence principle have been obtained, hence the motion of the center-of-mass of the composite system and the relative motion are independent, the additivity property of kinetic energy of composite system is preserved. The main theme of our paper is to show that non-commutativity between coordinates allows us to construct various other generalized classes of dynamical systems. Tools of non-commutative geometry often appear in quantum gravity. Using a differential geometric theory on non-commutative space-time Aschieri et al. [98] defined .θ -deformed Einstein-Hilbert action, and by means of the technique of the deformation of the algebra of diffeomorphisms one can derive .*-deformed integrable systems [99] and Newtonian mechanics [100]. Today we find noncommutativity in various fields of modern physics such as, graphene, Hydrogen atom spectrum, etc. [101–103]. Construction of the noncommutative principal bundles by deforming the principal bundles with a Drinfeld twist (2-cocycle) was given in [104]. It is shown that if the twist is associated with the structure group then a deformation of the fibers is obtained, whereas if the twist is associated with the automorphism group of the principal bundle, then noncommutative deformations of the base space is obtained. This paper is organized as follows: in order to the paper be self-contained, Sect. 12.2 is devoted to a review of multivector fields, Poisson bivector [105], Schouten–Nijenhuis bracket [106, 107] and various other geometrical tools. We give a brief geometrical description of Poisson manifolds in Sect. 12.3. Section 12.4 is devoted to Souriau’s formalism of generalized symplectic forms. We illustrate Souriau’s construction through examples. Section 12.5 is focused on the construction of Feynman-Dyson’s scheme and its connection to Souriau’s method. Section 12.6 relates volume preserving mechanical systems and Souriau’s form. We finish our paper with an outlook in Sect. 12.7.

12.2 Geometrical Background Let .F(M) be the associative and commutative algebra of .C ∞ -class functions (the algebra of classical observables) on a manifold M (the classical state space). We denote by .Op (M) the space of .C ∞ -class differentiable p-forms, and by .Ap (M)

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the space of .C ∞ -class skew-symmetric contravariant tensor fields of order p, often called p-vectors. By convention we set .A0 (M) = O0 (M) = F(M) and .Ap (M) = Op (M) = 0, when .p < 0. Then, O

O(M) =

.

Op (M)

and

A(M) =

p∈Z

O

Ap (M),

p∈Z

are .Z-graded algebras under their exterior products; moreover both are anticommutative, so, for instance, if .P ∈ Ap (M) and .Q ∈ Aq (M) P ∧ Q = −(−1)pq Q ∧ P .

.

When .α is a 1-form and X is a vector field the .C ∞ (M)-class function . given by ( ) (x) := := α(x) X(x) ,

.

∀x ∈ M,

defines a pairing between .O1 (M) and .X(M) ≡ A1 (M). More generally when .η in .Oq (M) and P in .Ap (M) are decomposable, so .η = α1 ∧ · · · ∧ αq and .P = X1 ∧ · · · ∧ Xp , for .αi in .O1 (M) and .Xj in .A1 (M), we set { := =

.

if p /= q, ( ) det if p = q.

0

Since the value of . at a point only depends on the value of .η and P at this point, we can extend by bilinearity, in a unique way, this pairing to arbitrary elements .η in .O(M) and P in .A(M). Furthermore, it is easy to check that if .η is in p .O (M), then = η(X1 , . . . , Xp ).

.

If X is a vector field on M, the inner product .i(X) is a derivation of degree .−1 on the graded algebra .O(M) and since the exterior derivative d is a derivation on .O(M) of degree 1, it follows that the Lie derivative with respect to X, given by Cartan’s formula: LX := [i(X), d] = i(X) ◦ d + d ◦ i(X),

.

where .[·, ·] means graded commutator, is a graded derivation of degree 0 on .O(M). LX can also be defined on .A(M) as the unique derivation of degree 0 such that, for f in .A0 (M) and Y in .A1 (M),

.

LX f = X(f )

.

and

LX Y = [X, Y ],

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where .[X, Y ] is the usual Lie bracket on vector fields. Furthermore the Schouten– Nijenhuis bracket is defined as a natural extension of the Lie derivative with respect to a vector field on .A(M). More specifically, it is defined as the unique bilinear map .[·, ·]SN : A(M) × A(M) → A(M) such that, for f and g in .A0 (M) = F(M), 1 p q r .X ∈ A (M), .P ∈ A (M), .Q ∈ A (M) and .R ∈ A (M), (a) (b) (c) (d)

[f, g]SN = 0 [X, Q]SN = LX Q (p−1)(q−1) [Q, P ] .[P , Q]SN = −(−1) SN (p−1)q Q ∧ [P , R] .[P , Q ∧ R]SN = [P , Q]SN ∧ R + (−1) SN . .

From these properties it readily follows that .[P , Q]SN belongs to .Ap+q−1 (M), therefore the last property means that the endomorphism .dP : A(M) → A(M) given by dP Q := [P , Q]SN ,

(12.1)

.

is a derivation of .A(M) of degree .p − 1. A somewhat long, but otherwise easy, induction, based on the defining properties, gives [ ] [ ] (−1)(p−1)(r−1) P , [Q, R]SN SN + (−1)(q−1)(p−1) Q, [R, P ]SN SN [ ] + (−1)(r−1)(q−1) R, [P , Q]SN SN = 0, (12.2)

.

which is called graded Jacobi identity. This, together with bilinearity, (c), and the fact that .[P , Q]SN belongs to p+q−1 (M) means that .A(M), equipped with the Schouten–Nijenhuis bracket, is a .A graded Lie algebra when the degree of P in .Ap (M) is declared to be .p − 1, not p. So, for instance, vector fields would be the homogeneous elements of degree 0 under this new grading of .A(M). To perform computations with the Schouten–Nijenhuis bracket it is convenient to extend the definition of the interior product. If .η is in .O(M), f is a function and .X1 , . . . , Xp are vector fields we set i(f )η := f η

.

i(X1 ∧ · · · ∧ Xp )η := i(X1 ) ◦ · · · ◦ i(Xp )η.

and

( ) In general, ( .i(P ))is defined in such a way that the map .i(·) : A(M) → E O(M) , where .E O(M) is the space of endomorphism of .O(M), is .F(M)-linear. In particular, .i(P ∧ Q)η = i(P )(i(Q)η), for all P and Q in .A(M). Furthermore, when .η is a p-form i(X1 ∧· · ·∧Xp )η=i(X1 )◦· · ·◦i(Xp )η=η(Xp , . . . , X1 )=(−1)

.

(p−1)p 2

η(X1 , . . . , Xp ),

therefore for any .P ∈ Ap (M) i(P )η = (−1)

.

(p−1)p 2

.

(12.3)

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Unfortunately .i(P ), in general, is not a derivation, which complicates computations. Nevertheless, another simple induction gives ) [ ] ( i [P , Q]SN = [i(P ), d], i(Q) ,

(12.4)

.

where the brackets on the right are the usual brackets on the algebra of endomorphisms of .A(M). Notice that when .P = X is a vector field this reduces to the well-known relation among interior products and Lie derivatives: .i(LX Q) = [LX , i(Q)]. If P is a p-vector and .η is a .(p − 1)-form, then .i(P )η = 0 and i(P ) ◦ i(f )η = i(P )(f η) = 0.

.

These, together with (12.3) and (12.4) entail ] (p−2)(p−1) (p−2)(p−1) [ 2 2 i([P , f ]SN )η = (−1) = (−1) [i(P ), d], i(Q) η ( (p−2)(p−1) 2 i(P ) ◦ d ◦ i(f )η − (−1)p d ◦ i(P ) ◦ i(f )η = (−1) ) − i(f ) ◦ i(P ) ◦ dη − (−1)p i(f ) ◦ d ◦ i(P )η ( ) (p−2)(p−1) 2 i(P ) ◦ d ◦ i(f )η − i(f ) ◦ i(P ) ◦ dη = (−1)

.

= (−1)

(p−2)(p−1) 2

i(P )(df ∧ η)

2

= (−1)(p−1) = (−1)(p−1)(p−2) = . Repited use of this gives = = · · · ] [ ] [ . = · · · [P , fp ]SN , fp−1 SN , · · · , f1

.

SN

(12.5)

12.3 Poisson Manifolds A Poisson structure on M is a skew-symmetric .R-bilinear map .{·, ·} : F(M) × F(M) → F(M) satisfying the Jacobi identity: {f, {g, h}} + {h, {f, g}} + {g, {h, f }} = 0,

.

∀f, g, h ∈ F(M) ,

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and such that the map .Xf = {·, f } is a derivation of the associative and commutative algebra .F(M), for each .f ∈ F(M), or in other words, .Xf is a vector field, usually called a Hamiltonian vector field, and f is said to be the Hamiltonian of .Xf . Observe that .(F(M), {·, ·}) is a real Lie algebra. This property characterizing derivations of the associative and commutative algebra .F(M), .{g1 g2 , f } = g1 {g2 , f } + g2 {g1 , f }, called Leibniz’ rule, is very important and gives a compatibility condition of the associative and commutative algebra structure in .F(M) with the Lie algebra given in .F(M) by the Poisson bracket. To construct Poisson structures let .A be an element of .A2 (M), if .f ∈ A0 (M) and .g ∈ A0 (M) are two functions, using (12.5), we define a third function by [ ] {f, g} := A(df, dg) = −A(dg, df ) = − = − [A, f ]SN , g SN . (12.6)

.

As .A is skewsymmetric, we have that .{f, g} = −{g, f }. By construction .Xf := [A, f ]SN is a vector field, and the defining property (b) entails Xf (g) = LXf g = [Xf , g]SN = −{f, g} = {g, f }.

(12.7)

.

In particular [ [ ] ] {g, {h, f }}= [A, g]SN , [A, h]SN , f SN

.

SN

] [ = Xg , [Xh , f ]SN SN =LXg ◦ LXh f.

By the same token .{h, {f, g}} = −{h, {g, f }} = −LXh ◦ LXg f . On the other hand, the graded Jacobi identity (12.2), the defining property (b), and (12.7) entail [[ ] ] {f, {g, h}} = −{{g, h}, f } = {[Xg , h]SN , f } = − A, [Xg , h]SN SN , f SN [[ ] [[ ] ] ] = − Xg , [h, A]SN SN , f − h, [A, Xg ]SN SN , f SN SN [[ ] ] [ ] = − [Xg , Xh ]SN , f SN − h, [A, Xg ]SN SN , f SN [[ ] ] = −L[Xg ,Xh ] f − h, [A, Xg ]SN SN , f .

.

SN

Altogether gives ) ( {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = LXg ◦ LXh − LXh ◦ LXg − L[Xg ,Xh ] f [[ ] ] − h, [A, Xg ]SN SN , f SN [[ ] ] = − h, [A, Xg ]SN SN , f .

.

SN

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Furthermore, from the graded Jacobi identity ] ] ] [ [ [ 0 = A, [A, g]SN SN + A, [A, g]SN SN + g, [A, A]SN SN ] ] [ [ = 2 A, [A, g]SN SN + g, [A, A]SN SN [ ] = 2[A, Xg ]SN + g, [A, A]SN SN ,

.

(12.8)

therefore, by (12.5) [ ] ] ] 1 [ [ h, g, [A, A]SN SN ,f .{f, {g, h}} + {g, {h, f }} + {h, {f, g}} = SN 2 SN [[ [[ ] ] ] ] [ ] ] 1 [ 1 h, [A, A]SN , g SN [A, A]SN , g SN , h ,f =− = ,f SN SN 2 2 SN SN 1 = − . 2

(12.9)

It follows that the bracket defined via a bivector field .A is a Poisson structure exactly when .[A, A]SN = 0, and when this happens we say that .A is a Poisson tensor. This elementary, but clever, computation was first performed by Lichnerowicz in [108], who also realized that, when .A is a Poisson tensor and P is in .Ap (M), the graded Jacobi identity implies [ [ ] ] [ ] 0 = (−1)p−1 A, [A, P ]SN SN − A, [P , A]SN SN + (−1)p−1 P , [A, A]SN SN [ ] = 2(−1)p−1 A, [A, P ]SN SN .

.

In other words, for Poisson tensors, the derivation .dA , defined as in (12.1) by .dA P = [A, P ]SN , satisfies the cocycle condition dA ◦ dA = 0.

.

On the other hand, from (12.8) we see that, for Poisson tensors, .[Xf , A]SN = 0, this together with the graded Jacobi identity and (12.7) give [ [ [ ] ] ] [Xf , Xg ] = Xf , [A, g]SN SN = − A, [g, Xf ]SN SN + g, [Xf , A]SN SN ] [ = A, [Xf , g]SN SN = −[A, {f, g}]

.

= X−{f,g} .

(12.10)

The converse is also true: a Poisson structure determines a Poisson tensor. To see this let .ξa denote a set of local coordinates on M, then, using the summation index convention,

12 A Primer on Noncommutative Classical Dynamics on Velocity Phase. . .

Xf = Xf (ξa )

.

543

∂ ∂ = {ξa , f } , ∂ξa ∂ξa

hence {f, g} = Xg (f ) = {ξa , g}

.

∂f . ∂ξa

Thus, {ξa , g} = −{g, ξa } = −{ξb , ξa }

.

∂g ∂g = {ξa , ξb } , ∂ξb ∂ξb

(12.11)

and the local coordinate expression of the Poisson Bracket becomes {f, g} = {ξa , ξb }

.

∂g ∂f . ∂ξb ∂ξa

(12.12)

Therefore to compute the Poisson bracket of any pair of functions is enough to know the fundamental Poisson brackets Aab = {ξa , ξb }.

.

Moreover, the value of .{f, g} at a point .m ∈ M does not depend neither on f nor on g but only on df and dg, as explicitly shown in (12.12), hence from the Poisson structure we get a twice contravariant skew-symmetric tensor A(df, dg) := {f, g}.

.

Indeed, the local coordinate expression of .A is A = Aab

.

∂ ∂ ∧ , ∂ξa ∂ξb

and if .ξ¯ = φ(ξ ) is another set of local coordinates on M, then, ¯ ab = {ξ¯a , ξ¯b } = {φa , φb } = {ξc , ξd } A

.

∂φa ∂φb ∂φa ∂φb = Acd , ∂ξc ∂ξd ∂ξc ∂ξd

so the components of .A do change like the local coordinates of a skew-symmetric twice contravariant tensor which, by (12.9), it is a Poisson tensor. We are using the convention that in the local expression of the wedge product only summands whose subindex on the left hand side term is smaller than the subindex on the right hand side term do appear. For any function .h ∈ F(M) the integral curves of the dynamical vector field .Xh are precisely determined by the solutions of the system of differential equations

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.

dξa = {ξa , h} , dt

(12.13)

and the dynamical evolution of a function f on M is given by .

df = {f, h}, dt

or in local coordinates .

∂f ∂h df = Aab . ∂ξa ∂ξb dt

Interesting examples of Poisson manifolds are symplectic manifolds. A symplectic form .ω on M determines a bundle map .ωb : T M → T ∗ M over the identity, which gives rise to the corresponding linear map between their spaces of sections, defined by ( .

) ωb (X) (Y ) := = ω(X, Y ).

Since .ω is non-degenerate .ωb is actually an isomorphism; we denote .ω# the inverse map and define a bivector .A by ( ) A(α, β) := ω ω# (α), ω# (β) ,

.

if .α and .β are 1-forms. When .α = df is exact the corresponding vector field is denoted by .Xf := ω# (df ), and we say .Xf is the vector field associated to f with respect to .ω. It is actually defined by the equation .i(Xf )ω = df . Furthermore, let .{·, ·} be the bracket associated to .A via (12.6). Then the vector field associated to f with respect to .ω is also the Hamiltonian vector field given in (12.7), explaining why we use the same notation. If .ξa , .a = 1, . . . , m, denote local coordinates and 1 .∂/∂ξ1 , . . . , ∂/∂ξm , and .dξ1 , . . . , dξm are, respectively, the local basis of .A (M) and 1 b .O (M) associated to .ξa , let .B = (Bab ) be the matrix of the linear map .ω relative to these bases, and .ω = ωab dξa ∧ dξb the local expression of the symplectic form, then ) ( ( )) ( ) ( ( )( ∂ ) ∂ ∂ ∂ ∂ = ωb = Bac dξc = Bab . .ωab = ω , ∂ξa ∂ξb ∂ξb ∂ξa ∂ξb Thus, .B = (ωab ), and the matrix of .ω# associated to these bases is the inverse of B. On the other hand, ( ( ( ) ) ) dω(Xf , Xg , Xh ) = Xf ω(Xg , Xh ) + Xg ω(Xh , Xf ) + Xh ω(Xf , Xg )

.

− ω([Xf , Xg ], Xh ) − ω([Xg , Xh ], Xf ) − ω([Xh , Xf ], Xg ).

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Nevertheless, .ω(Xg , Xh ) = A(dg, dh) = {g, h}, so (12.7) entails ( ) Xf ω(Xg , Xh ) = Xf ({g, h}) = −{f, {g, h}}.

.

Also, by (12.10), .[Xg , Xh ] = [Xg , Xh ]SN = X−{g,h} , hence ω([Xg , Xh ], Xf ) = ω(X−{g,h} , Xf ) = −{{g, h}, f } = {f, {g, h}}.

.

It follows that 0 = dω(Xf , Xg , Xh ) = −2({f, {g, h}} + {g, {h, f }} + {h, {f, g}}),

.

(12.14)

so .A is a Poisson tensor. Reciprocally, from a Poisson tensor .A we get a bundle map .A# : T ∗ M → T M, defined by := A(α, β).

.

In general .A# is not a bundle isomorphism. We say the Poisson structure is regular when that is the case, and then we denote the inverse map by .Ab . Thus we have an identification of .Tx M and .Tx∗ M, for each point x of M and, therefore, an identification of higher order contravariant and covariant tensors. In particular, we define a 2-form .ω by ( ) ω(X, Y ) := A Ab (X), Ab (Y ) .

.

Notice that, from this point of view, .Xf = A# (df ). Indeed, from (12.7) = Xf (g) = {g, f } = A(dg, df ) = .

.

Therefore, ( ( ( ) )) ω(Xf , Xg ) = A Ab A# (df ) , Ab A# (dg) = A(df, dg).

.

In particular the brackets associated to .ω and .A coincide. Since the Poisson bracket satisfies the Jacobi identity, (12.14) entails, .dω(Xf , Xg , Xh ) = 0. Given that locally one can consider a basis consisting of Hamiltonian vector fields, we conclude that .ω is a closed 2-form, moreover by definition it is non-degenerate, hence .ω, so defined, is indeed a symplectic form. Using local coordinates as above, we see that the matrix of .A# with respect to the standard bases is .(Aab ), where ∂ ∂ .A = Aab ∂ξa ∧ ∂ξb is the local expression of .A.

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12.4 Souriau’s Prescription As far as we know, Souriau [43–45] was the first to realize that since a Hamiltonian dynamical system has two pieces, the symplectic form and the Hamiltonian, the equations of motion can be described by different data, modifying one or the other component. Thus, a perturbed dynamical system can be described starting from the free case by modifying the Hamiltonian, as was classically done, or simply by changing the symplectic form (see also [109]). This idea of adding an extra term to the symplectic form was successfully exploited by Souriau in his study of the orbit method, and it is what is behind the exotic mechanics, and several other models where non-commutativity of the variables is employed. But before we tackle that, let us consider a more down to earth example. The classical method to derive the Lorentz equations in a relativistically invariant way is to use the so called minimal coupling, which consists in making the substitution .p |→ p−eA inside the free Hamiltonian, where A is the vector potential of the electromagnetic field and e is the electric charge. Thus, the starting point is the cotangent bundle .T ∗ M, of a manifold M, endowed with its canonical symplectic form .ω0 = −dθ0 , where .θ0 is the canonical 1-form given, in local cotangent bundle coordinates .(qi , pi ), induced from local coordinates .(qi ) on M, by .θ0 = pi dqi , −1 together with a Hamiltonian .H : T ∗ M → R, which one replaces by .HA = H ◦ φA , ∗ ∗ where .φA : T M → T M is the bundle map over the identity given by ( ) φA (q, p) = q, p + eA(q) ,

.

and .A = Ai (q) dqi is a basic 1-form on .T ∗ M. The Hamiltonian vector field .XHA −1 ∗ associated to this new Hamiltonian .HA = (φA ) H , that leads to the equation of motion, is given by .

−1 ∗ − i(XHA )dθ0 = dHA = (φA ) (dH ).

On the other hand, and with an abuse of notation, we denote .σ both the 1-form on M defined by .σ = e Ai (q) dqi , as well as the basic 1-form on .T ∗ M obtained by pulling back .σ by the canonical projection .π : T ∗ M → M. Then, ( ) ∗ ∗ ∗ ∗ φA (dθ0 ) = φA (dpi ∧dqi ) = dφA (pi )∧dφA (qi ) = d pi +eAi (q) ∧ dqi = dθ0 +dσ,

.

and since .φ is a diffeomorphism, .

( ) ( ) ∗ − i φA∗ (XHA ) (dθ0 + dσ ) = −i φA∗ (XHA ) (φA dθ0 ) ( ∗ −1 ) ∗ ∗ = φA (φA ) (dH ) = dH. (i(XHA )dθ0 ) = φA

In other words, by adding the extra term .dσ to the symplectic form, which, by the way, it is a basic 2-form (i.e. locally it is of the form .gij (q) dqi ∧ dqj , so it does not

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involve the p’s), we see that the vector field .φA∗ (XHA ) is the Hamiltonian vector field associated to the original Hamiltonian H with respect to this new symplectic form, and we obtain the same equations of motion. If .ω = ω0 + 12 gij dqi ∧ dqj , where .gij (q, p) is a skew-symmetric matrix, and the Hamiltonian vector field .XH is .XH = Vi ∂qi + Wi ∂pi , then 1 i(XH )ω = (i(XH )dqi ) ∧ dpi − dqi ∧ (i(XH )dpi ) + gij (i(XH )dqi ) ∧ dqj 2 1 − gij dqi ∧ (i(XH )dqj ) 2

.

= −Wi dqi + Vi dpi + gij Vj dqi The equation .i(XH )ω = dH , entails Vi =

.

∂H ∂pi

and

Wi = −

∂H ∂H + gki , ∂qi ∂pk

therefore by (12.7) the Poisson bracket associated to .ω is given by {F, H } =

.

∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F − + gij = {F, H }0 + gij ≡ XH (F ), ∂qi ∂pi ∂pi ∂pj ∂pi ∂pj ∂pi ∂qi

where .{·, ·}0 stands for the Poisson bracket corresponding to .ω0 . It follows that the generalized (Hamiltonian) vector field is XH = XH + gij

.

∂H ∂ . ∂pi ∂pi

The equations of motion are given by .

dqk ∂H , = ∂pk dt

∂H dpk ∂H + gik . =− dt ∂qk ∂pi

Our construction can be extended easily to a more general Souriau form ω = ω0 + 12 gij dqi ∧ dqj + 21 fij dpi ∧ dpj ,

.

and the equations of motion are then given by .

∂H ∂H dqk = + fki , ∂pk ∂qi dt

∂H ∂H dpk =− + gik . dt ∂qk ∂pi

Then if we assume the Hamiltonian is of the form

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H (q, p) =

.

δ ij pi pj + V(q), 2m

with the potential energy .V depending only on the configuration coordinates .qi , the equations of motion are .

∂V dqk pk , + fki = m ∂qi dt

pi dpk ∂V + gik . =− dt ∂qk m

These are equivalent to the modified Newton’s second law [36, 52, 57, 58] d pi d 2 qk ∂V + gik + m .m =− 2 ∂qk m dt dt

) ( ∂V . fki ∂qi

(12.15)

The second term of Eq. (12.15) is a correction due to the noncommutativity of momenta and the third term is that of noncommutativity of coordinates. It follows that even for the case .V = 0 the particle accelerates because of the noncommutativity of momenta. The second procedure has the advantage that it works even when only the 2form is globally defined, with no reference to the 1-form .θ0 made. In [110] this idea was generalized to the case of a classical particle in the presence of a Yang-Mills field. When the Poisson manifold M is the tangent bundle T Q of a n-dimensional manifold, so M is known as the velocity phase space, Souriau also proposed to describe the dynamics not on phase space but in what he called evolution space, with coordinates .(xi , x˙j , t). His idea was to join the symplectic form .ω on phase space with the Hamiltonian by considering the two-form .ω − dH ∧ dt on the evolution space, and then perform the minimal coupling recipe. This allows to recover the Euler-Lagrange equations, and it is equivalent to Faddeev-Jackiw construction [35, 111]. Recently, Bolsinov and Jovanovi´c [112] considered G-invariant magnetic geodesic flows on coadjoint orbits of a compact Lie group G, where .σ is the Kirillov-Kostant two-form.

12.4.1 Souriau’s Formalism and Exotic Mechanics For concreteness let us ponder the ‘exotic’ plane studied by Horváthy in [25]. Thus, we consider the dynamical system .(T ∗ R2 , ωϑ , H0 ), where .ϑ ∈ R, ωϑ = dq1 ∧ dp1 + dq2 ∧ dp2 − ϑ dp1 ∧ dp2

.

and

H0 =

p12 + p22 , 2m

i.e. .ω = ω0 − ϑ dp1 ∧ dp2 . The 2-form .ωϑ is not only closed but exact, and as the associated map b 1 ∗ 2 1 ∗ 2 .ω : A (T R ) → O (T R ) is given by the matrix ϑ

12 A Primer on Noncommutative Classical Dynamics on Velocity Phase. . .

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0 ⎜ 0 b .ω = ⎜ ϑ ⎝1 0

549

⎞ 0 −1 0 0 0 −1 ⎟ ⎟, 0 0 ϑ⎠ 1 −ϑ 0

which is regular for any value of .ϑ, the 2-form .ωϑ is symplectic. The inverse matrix is ⎛

0 ϑ ⎜−ϑ 0 # .ω = ⎜ ϑ ⎝ −1 0 0 −1

⎞ 10 0 1⎟ ⎟, 0 0⎠ 00

which corresponds to the Poisson structure associated to the bi-vector Aϑ = ϑ

.

∂ ∂ ∂ ∂ ∂ ∂ ∧ + ∧ + ∧ , ∂q1 ∂q2 ∂q1 ∂p1 ∂q2 ∂p2

so the fundamental Poisson commutators are {q1 , q2 } = ϑ , {q1 , p1 } = {q2 , p2 } = 1 , {q1 , p2 } = {q2 , p1 } = 0 , {p1 , p2 } = 0 ,

.

and the dynamical vector field is given by .

q˙1 ={q1 , H0 } =

p1 p2 , q˙2 ={q2 , H0 }= , p˙ 1 ={p1 , H0 } = 0, p˙ 2 = {p2 , H0 } = 0 . m m

In this way we obtain a 1-parameter family of symplectic structures for the free particle. These symplectic structures are the sum of two symplectic structures on .T ∗ R2 that are invariant under rotations in the plane, one of them being the canonical symplectic structure .ω0 on .T ∗ R2 . The generating function of such 1-parameter group will be the function f satisfying i(X)ωϑ = df,

.

where X is the vector field that is the cotangent lift of the rotation generator in configuration space, X = q1

.

∂ ∂ ∂ ∂ − q2 + p1 − p2 . ∂q2 ∂q1 ∂p2 ∂p1

Since i(X)ωϑ = p2 dq1 − p1 dq2 − q2 dp1 + q1 dp2 + ϑ(p1 dp1 + p2 dp2 ) ,

.

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we find that the generating function is given by f (q1 , q2 , p1 , p2 ) = q1 p2 − q2 p1 +

.

) ϑ( 2 p1 + p22 . 2

We now apply Souriau’s minimal coupling procedure, so we introduce a basic 2form .σ = B(q1 , q2 ) dq1 ∧ dq2 , which is closed and can be interpreted as a magnetic field, and consider the closed 2-form ωϑ,σ := ωϑ − π ∗ σ .

.

The corresponding linear map .ωϑ,σ : A1 (T ∗ R2 ) → O1 (T ∗ R2 ) is represented by the matrix b

⎛

b

ωϑ,σ

.

0 ⎜−B =⎜ ⎝ 1 0

⎞ B −1 0 0 0 −1 ⎟ ⎟, 0 0 ϑ⎠ 1 −ϑ 0

whose determinant is .(1−ϑ B)2 , therefore .ωϑ,σ is regular in the points where .B ϑ /= 1; the inverse matrix being given by ⎛

#

ωϑ,σ

.

0 ϑ 1 ⎜−ϑ 0 0 1 ⎜ = 1 − ϑ B ⎝ −1 0 0 0 −1 −B

⎞ 0 1⎟ ⎟, B⎠ 0

which corresponds to the Poisson structure defined by the bi-vector Aϑ,σ

.

1 = 1−ϑB

) ( ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ . ϑ ∧ + ∧ + ∧ +B ∧ ∂q1 ∂q2 ∂q1 ∂p1 ∂q2 ∂p2 ∂p1 ∂p2

The corresponding fundamental Poisson brackets are ϑ , 1−ϑB B {p1 , p2 } = , 1−ϑB {q1 , q2 } =

.

{q1 , p1 } = {q2 , p2 } =

{q1 , p2 } = {q2 , p1 } = 0 .

When the Hamiltonian is H =

.

1 , 1−ϑB

p2 + V (q1 , q2 ) , 2m

(12.16)

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using (12.11), (12.13) and (12.16), we see that the time evolution is given by q˙1 =

.

∂V p1 ϑ + , m(1 − ϑ B) 1 − ϑ B ∂q2

q˙2 = −

p˙ 1 = −

∂V 1 Bp2 + , 1 − ϑ B ∂q1 m(1 − ϑ B)

∂V ∂V ϑ p2 1 Bp1 , p˙ 2 = − . + − 1 − ϑ B ∂q1 m(1 − ϑ B) 1 − ϑ B ∂q2 m(1 − ϑ B)

The system is still invariant under rotations if .B(q1 , q2 ) is a rotationally invariant function, i.e. B is a function of .q12 + q22 , .B(q1 , q2 ) = b(q12 + q22 ), and the generating function for the infinitesimal generator of rotations is f (q1 , q2 , p1 , p2 ) = q1 p2 − q2 p1 +

.

ϑ 2 1 (p1 + p22 ) + B(q1 , q2 ). 2 2 b

On the other hand, when B is constant, and .B ϑ = 1, the determinant of .ωϑ,σ is b

zero and the rank of .ωϑ,σ is two, the kernel of the 2-form .ωϑ,σ being generated by the vector fields X1 = ϑ

.

∂ ∂ , + ∂p1 ∂q2

and

X2 = −ϑ

∂ ∂ + . ∂q1 ∂p2

The solutions of .X1 F = X2 F = 0 are to be found from the method of characteristics and turn out to be the functions which depend on .ξ1 = q1 + ϑ p2 and .ξ2 = q2 − ϑ p1 . This suggests the change of variables .(q1 , q2 , p1 , p2 ) |→ (ξ1 , ξ2 , p1 , p2 ), i.e. .q1 = ξ1 − ϑ p2 , .q2 = ξ2 + ϑ p1 . In such coordinates, .X1 = ∂/∂p1 , .X2 = ∂/∂p2 and .ωϑ,σ becomes ωϑ,σ = −B dξ1 ∧ dξ2 .

.

This means that the quotient manifold .T ∗ R2 / ker ωϑ,σ is parametrized by .ξ1 and .ξ2 which, moreover, are Darboux coordinates for such 2-dimensional symplectic manifold. Using the same idea with commutators Nair and Polychronakos [38] described quantum mechanics for both the non-commutative plane and the non-commutative sphere, and proved that the Landau problem for the non-commutative plane can be recovered as the limit of large radius of the Landau problem for the noncommutative sphere.

12.4.2 Nonrelativistic Anyon Model in Souriau Formalism Let .(q1 , q2 ) be orthogonal Cartesian coordinates in .Q = R2 , and consider the Lagrangian .L0 in .F(T Q) of the free particle

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L0 (q1 , q2 , v1 , v2 ) =

.

) 1 ( 2 m v1 + v22 . 2

Let M be the graph of the corresponding Legendre transformation. This is the submanifold of the Pontryagin bundle .T Q ⊕ T ∗ Q given by the constraint functions λi (q1 , q2 , v1 , v2 , p1 , p2 ) = pi − m vi .

.

Let .κ ∈ R be a constant and consider T Q endowed with the exact 2-form .ω1 defined by ω1 (q1 , q2 , v1 , v2 ) := κ dv1 ∧ dv2 .

.

In the spirit of Souriau’s idea, we consider the closed 2-form on .T Q ⊕ T ∗ Q ω := −pr∗1 ω1 + pr∗2 ω0 ,

.

where .pr1 and .pr2 are the natural projections .pr1 : T Q ⊕ T ∗ Q → T Q and .pr2 : T Q ⊕ T ∗ Q → T ∗ Q, .ω0 is the canonical symplectic structure in .T ∗ Q and .ω1 ∈ O2 (T Q) is as before. The corresponding map .ωb : X(T Q ⊕ T ∗ Q) → O1 (T Q ⊕ T ∗ Q) is represented by the matrix ⎞ 0 0 0 0 −1 0 ⎜0 0 0 0 0 −1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜0 0 0 κ 0 0⎟ b .ω = ⎜ ⎟, ⎜0 0 −κ 0 0 0⎟ ⎟ ⎜ ⎝1 0 0 0 0 0⎠ 01 00 0 0 ⎛

which is regular for .κ /= 0. In this case, the 2-form .ω is symplectic, and since the inverse matrix is ⎞ ⎛ 0 0 0 0 10 ⎜ 0 0 0 0 0 1⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 −1/κ 0 0⎟ # .ω = ⎜ ⎟, ⎜ 0 0 1/κ 0 0 0⎟ ⎟ ⎜ ⎝ −1 0 0 0 0 0⎠ 0 −1 0 0 00 we obtain the bi-vector field on .T Q ⊕ T ∗ Q A=

.

∂ ∂ ∂ ∂ 1 ∂ ∂ ∧ + ∧ − ∧ , ∂q1 ∂p1 ∂q2 ∂p2 κ ∂v1 ∂v2

12 A Primer on Noncommutative Classical Dynamics on Velocity Phase. . .

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and the corresponding Poisson structure is defined by the following fundamental relations: {q1 , p1 } = {q2 , p2 } = 1,

{q1 , q2 } = {p1 , p2 } = {q1 , p2 } = {p1 , q2 } = 0 ,

1 {v1 , v2 } = − , κ

{v1 , p2 } = {v2 , p1 } = 0 .

.

The two constraint functions .λ1 and .λ2 are second class constraints, because {λ1 , λ2 } = {p1 − mv1 , p2 − mv2 } = m2 {v1 , v2 } = −

.

m2 . κ

They define a four dimensional symplectic manifold.

12.5 Feynman–Dyson’s Method and Non-commutativity In this section we review the Feynman’s derivation of Maxwell’s equations [12], in the framework of a tangent bundle, so the Poisson manifold M is the tangent bundle T Q of a configuration space Q. In terms of local tangent bundle coordinates in T Q induced from local coordinates in Q, denoted .xi and .x˙i , a general Poisson bracket on T Q is locally given by {f, g} = {xa , xb }

.

∂g ∂f ∂g ∂f ∂g ∂f ∂g ∂f +{xa , x˙b } +{x˙a , xb } +{x˙a , x˙b } . ∂ x˙b ∂xa ∂xb ∂ x˙a ∂ x˙b ∂ x˙a ∂xb ∂xa (12.17)

The assumptions in [12] are Newton’s equations of motion mx¨j = Fj (x, x), ˙

.

i.e. the dynamics is given by the second order differential equation vector field r = x˙i

.

∂ ∂ , + Fi (x, x) ˙ ∂ x˙i ∂xi

together with the fundamental brackets {xi , xj } = 0

.

and

m{xi , x˙j } = δij .

(12.18)

The goal is to determine the other fundamental Poisson brackets, and as .{x˙i , x˙j } must be skew symmetric it can be written as {x˙i , x˙j } =

.

1 εij k Bk (x, x), ˙ m2

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where .εij k denotes the fully skew-symmetric Levi–Civita tensor and .B is defined as the magnetic field. Now (12.18) implies that {xi , Fj } =

.

1 ∂Fj , m ∂ x˙i

and using the derivation property for the time derivative of the second equation in (12.18), i.e. assuming that the vector field .r is a derivation of the Poisson structure, we get 1 1 {xi , Fj } = 2 εij k Bk (x, x) ˙ , m m

{x˙i , x˙j } = −

.

(12.19)

and the Jacobi identity for the functions .xi , x˙j , x˙k , {xi , {x˙j , x˙k }} + {x˙k , {xi , x˙j }} + {x˙j , {x˙k , xi }} = 0,

.

entails 0 = {xr , Bs } =

.

1 ∂Bs . m ∂ x˙r

In particular .B does not depend on the dotted variables. Moreover from the Jacobi identity with three different velocities we obtain .

div B = 0,

which reveals that the flux of the field .B through a closed surface is zero, and that magnetic monopoles do not exist! On the other hand, since .B does not depend on .x˙i , Eqs. (12.17) and (12.19) entail that .F is at most linear in such variables, therefore we can define another field .E, called the electric field, by .Ej = Fj − εj kl x˙k Bl . Using repeatedly all the equations above, one arrive to Maxwell’s equation corresponding to Faraday’s law of electrodynamics, in the setting suggested at the beginning of this section, namely .rot E = 0 in the autonomous case, or in general .

∂B + rot E = 0, ∂t

a magnetic field that is changing in time produces a non-conservative electric field. We refer the reader to [39, 40] for details. To obtain a dynamic different from the standard formalism of electrodynamics, one needs to modify the fundamental brackets in (12.18). The first idea is to use Souriau’s technique, namely to replace the first fundamental bracket by {xi , xj } = gij (x),

.

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where .gij is an arbitrary skew-symmetric matrix of functions, fulfilling the constraints that the Poisson bracket properties impose, and keep the other assumptions. In particular, the Jacobi identity {xi , {xj , x˙k }} + {x˙k , {xi , xj }} + {xj , {x˙k , xi }} = 0,

.

entails 0 = {x˙k , gij } = {x˙k , xl }

.

∂gij ∂gij 1 ∂gij + {x˙k , x˙l } =− . ∂xl ∂ x˙l m ∂xk

(12.20)

Then the matrix .gij is a constant skew-symmetric .3 × 3 matrix, which is an interesting, but somewhat restrictive, case. We then modify Souriau’s idea and settle for {xi , xj } = gij (x, x). ˙

.

Accordingly, (12.20) becomes 0 = {x˙k , gij } = −

.

∂gij 1 ∂gij + {x˙k , x˙l } . ∂ x˙l m ∂xk

This equation clearly relates the part of the Poisson structure on the base (the positions) with the part of the Poisson structure on the fibre (the velocities). Hence if the Poisson structure on the base is known one can compute the fundamental brackets on the fibre. On the other hand, from the Jacobi identity among .(xi , xj , xk ) we obtain {xi , gj k } + {xk , gij } + {xj , gki } = 0,

.

and this leads to another constraint: 0 = gil

.

∂gij ∂gj k ∂gij ∂gki 1 ( ∂gj k ∂gki ) . + gj l + + + + gkl ∂xl ∂xl m ∂ x˙i ∂ x˙k ∂ x˙j ∂xl

(12.21)

Note that if .dx denotes the exterior derivative on the vector space .Tx Q, for x in ˜ Q, then the term inside the parenthesis in (12.21) are the local coordinates of .dx ω, where .ω˜ x is the 2-form in .O2 (Tx Q) defined by .ω˜ x = gij (x, x)d ˙ x˙i ∧ d x˙j . On the ˜ is the bivector in .A2 (T Q) given by .A ˜ = gij (x, x) ˙ ∂x∂ i ∧ ∂x∂ j the terms other hand, if .A ˜ A] ˜ SN . Therefore outside the parenthesis in (12.21) are the local coordinates of .[A, Eq. (12.21) is fulfilled when dx ω˜ = 0

.

and

˜ = [A, ˜ A] ˜ SN = 0, dA˜ A

˜ is a Poisson tensor. that is, when .ω˜ x is closed and .A

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Furthermore, similar ideas as in the commutative case, using the other Jacobi identities, lead, see [23], to the modified Gauss law .

div B = −

1 B · ∇˙ × B, m

where .∇˙ = ( ∂∂x˙1 , ∂∂x˙2 , ∂∂x˙3 ), and also to (rot E)k +

.

) ( 1 ˙ k + B · ∂E − (∇˙ · E) Bk = 0 , (E · ∇)B ∂ x˙ k m

which is what replaces the Maxwell equation corresponding to Faraday’s law.

12.5.1

Generalized Lorentz Force Equations

Consider the Hamiltonian dynamical system on .T R3 , where the Hamiltonian and the symplectic form are given respectively by H =

.

1 δij x˙i x˙j + φ(x), 2m

and the 2-form on .T R3 , ω=

.

1 1 dxi ∧ d x˙i + B1 dx2 ∧ dx3 + B2 dx3 ∧ dx1 + B3 dx1 ∧ dx2 + gij d x˙i ∧ d x˙j . m 2

Assume that, in local coordinates, the Hamiltonian vector field is written as XH = Si ∂xi + Ri ∂x˙i .

.

The equation .i(XH )O = dH becomes .

1 1 x˙1 = S1 + g21 R2 + g31 R3 , m m

∂φ 1 = − R1 + B2 S3 − B3 S2 , ∂x1 m

1 1 x˙2 = S2 + g12 R1 + g32 R3 , m m

∂φ 1 = − R2 + B3 S1 − B1 S3 , ∂x2 m

1 1 x˙3 = S3 + g13 R1 + g23 R2 , m m

∂φ 1 = − R3 + B1 S2 − B2 S1 . ∂x3 m

On the other hand, from (12.13) and (12.7) we obtain .

dxi = Si dt

and

d x˙i = Ri . dt

(12.22)

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Therefore if we assume that .XH is a second order differential equation, namely that d 2 xi .Si = x ˙i , then . 2 = Ri , and the right column of (12.22) entails dt .

1 d 2 x1 ∂φ =− + x˙3 B2 − x˙2 B3 , m dt 2 ∂x1 1 d 2 x2 ∂φ =− + x˙1 B3 − x˙3 B1 , ∂x2 m dt 2

(12.23)

1 d 2 x3 ∂φ =− + x˙2 B1 − x˙1 B2 , 2 ∂x3 m dt whereas the left column provides the constraints 0 = mg21

.

0 = mg12 0 = mg13

( ∂φ ∂x2 ( ∂φ ∂x1 ( ∂φ ∂x1

) ) ( ∂φ − x˙2 B1 + x˙1 B2 , − x˙1 B3 + x˙2 B1 + mg31 ∂x3 ) ) ( ∂φ − x˙2 B1 + x˙1 B2 , − x˙3 B2 + x˙2 B3 + mg32 ∂x3 ) ) ( ∂φ − x˙1 B3 + x˙3 B1 . − x˙3 B2 + x˙2 B3 + mg23 ∂x2

In particular when .∇φ = −eE, (12.23) is a generalized Lorentz force: a force experienced by a charged particle moving in an electromagnetic field, subject to a system of constraints. Another interesting class of systems can be studied via the “generalized Souriau form” ω˜ 0 = dxi ∧ d x˙i + gij d x˙i ⊗ d x˙j .

.

It is a mixture of a symplectic and a gradient structure, known as a metriplectic system. The symmetric bracket associated to the metric tensor incorporates the dissipative structure of the system. The Leibniz vector field .Xh associated to a function .h ∈ C ∞ (M) satisfies .Xh = ∇h, i.e. .Xh generates a gradient dynamical system. In local coordinates the vector field .Xh is given by Xh = gij

.

∂h ∂ , ∂xj ∂xi

and the corresponding bracket in this context is called a Leibniz bracket.

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12.6 Volume Preserving Mechanical System Related to Souriau Form Another interesting class of dynamical systems that generalize the Hamiltonian systems, where noncommutativity is also possible, was introduced in [54, 55]. Let .(M, ω0 ) be a 2n-dimensional symplectic manifold, a vector field X is said to be symplectic, or locally Hamiltonian vector field, if .LX ω0 = 0, from the Cartan identity, this is equivalent to .i(X)ω0 being closed, in particular every Hamiltonian vector field is symplectic. On the other hand, we say that a vector field X preserves the volume if .LX ω0∧n = 0; here and in what follows powers are meant with respect to the wedge product, i.e. by .ω0n we understand .ω0∧n . Since .

| | d d ∗ | | ot+s ω0n | = o∗t o∗s ω0n | = o∗t LX ω0n = 0, s=0 s=0 ds ds

where .ot is the flow of X, it follows that .ot does preserves the volume form .ω0n . Furthermore, a simple induction gives LX ω0k = kLX ω0 ∧ ω0k−1 .

.

In particular, we see that every symplectic vector field preserves the volume, but the converse is not true in general. The divergence of a vector field Xwith respect to the volume form .ω0n is defined as the unique function .div X in .C ∞ (M) such that LX ω0n = div X ω0n .

.

Therefore X preserve the volume if, and only if, it is divergence free. Let (x , . . . , x2n ) be Darboux coordinates, then .ω0 = dxi ∧ dxn+i , and if .X = E12n i=1 Xi ∂xi it is easy to check that

.

.

div X =

2n E ∂Xi i=1

∂xi

.

For more details on this concept and that of the Jacobi multiplier and its applications see e.g. the recent papers [113, 114]. We now describe a procedure that produces dynamical systems that preserves the volume. First consider the map .F : A1 (M) → O2n−1 (M) given by .F (X) := i(X)ω0n . Using Darboux coordinates, simple combinatorial arguments entail ωk = (−1)

.

k(k−1) 2

k!

E

(

i ∧ · · · ∧ dx < dxn+i ∧ dx1 ∧ · · · ∧ dx n+i ∧ · · · ∧ dx2n ∂xk

∂Bki >i ∧ · · · ∧ dx < dxi ∧ dx1 ∧ · · · ∧ dx n+i ∧ · · · ∧ dx2n ∂xn+k ( ) (n−1)(n−2) ∂gki ∂Bki 2 = (n − 2)!(−1)n+ dxi ∧ dxn+i − ∂xn+k ∂xk

− (n − 2)!(−1)n

>i ∧ dx < dx1 ∧ dxn+1 ∧ · · · ∧ dx n+i ∧ · · · ∧ dxn ∧ dx2n = d(ω ∧ ωon−2 ). In other words, .Xω = X, so we can associate a volume preserving flow with the Souriau’s form, and the equations of motion are given by .

(−1)n ∂gki dxi = , n − 1 ∂xk dt

dxn+i (−1)n ∂Bki =− . n − 1 ∂xn+k dt A Nambu-Poisson system is a volume preserving flow, on a Nambu-Poisson manifold of order 2n, determined by .(2n − 1) Hamiltonian functions .H1 , . . . , H2n−1 ∈ C ∞ (M) as follows: .

dxi = XH1 ,...,H2n−1 (xi ) = {H1 , . . . , H2n−1 , xi }. dt

In general, .LXH1 ,...,H2n−1 f = {H1 , . . . , H2n−1 , f } if .XH1 ,...,H2n−1 is a NambuHamiltonian vector field. If .η is a Nambu-Poisson tensor, then Takhtajan [115] proved that .LXH1 ,...,H2n−1 η = 0.

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12.7 Conclusion and Outlook In this paper we have studied the classical non-commutative mechanical systems using Souriau’s method of generalized symplectic form. In particular we have explored a large class of non-commutative flows which includes the noncommutative magnetic geodesic flows, non-relativistic anyon model [116–118], generalized Lorentz force equation, etc. Souriau’s formalism allows us to study geometrically all these non-commutative dynamical systems in an unified manner. The dynamics of these systems boil down to generalized Hamiltonian dynamics where the Poisson structure can be complicated functions of phase space coordinates and momenta. However, some questions should be addressed in the future. On the quantum formulation the deformation is the fundamental commutation relations leads to a modified uncertainty principle and to a minimal length formalism in quantum gravity and string theory [119–124]. At first we must consider the quantization of these classical non-commutative system. There is an interesting paper [125] which addresses the connection between non-commutative quantum mechanics and Feynman–Dyson’s method. Actually, the generalization of Feynman–Dyson’s idea to the quantum world would be an interesting subject to be studied. This would lead to unveil the close relation existing between the non-commutative geometry and the geometric phases. Quantization of these models could give rise to new physics at some very high energy scale [93, 126– 136]. This method can be extended to other directions also. We can simply generalize this construction to supersymmetric non-commutative systems [122, 137, 138]. In other words, we can try to generalize the Feynman–Dyson’s scheme to supersymmetric framework. This would certainly yield a generalization of supersymmetric generalized Hamiltonian dynamics. There is a recent upsurge of interest in .(2 + 1)-dimensional model [139, 140] with a kind of nonstandard noncommutativity, where both coordinates and momenta get deformed commutators. So far most of the papers concern time-independent systems, it would be rather challenging to extend this framework to time-dependent systems. In a recent note Liang and Jiang [141] studied the time-dependent harmonic oscillator in a background of time-dependent electric and magnetic fields. Recently noncommutative quantum mechanics [125, 142–147] is becoming an exciting topic to study, it would be interesting for us to investigate this subject using geometrical methods of quantizing noncommutative phase space mechanics. Possibly one can study the Helmholtz condition analogous to standard classical dynamics. Then the corresponding variational formulation can be used to construct Noether symmetries and conserved densities. It is known [51] that the Helmholtz condition connected to .θ -deformed Poisson system is a third-order time derivative equation. One should analyse carefully all these new aspects. We can also study the field theoretic Poisson brackets on jet space. This will yield an interesting class of partial differential equations. One must try to explore its

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connection to other branches of mechanics and geometry, namely, non-holonomic systems, control theory, Finslerian mechanics, Lie algebroid theory, etc. Acknowledgments We thank Peter Horváthy, Parameswaran Nair, Ali Chamseddine, Frederik Scholtz and Bozidar Jovanovic for useful conversations over a span of one decade. JFC acknowledges financial support from research projects MTM–2012–33575 (MINECO, Madrid) and DGA-GRUPOS CONSOLIDADOS E/21. HF acknowledges support from the Vicerrectoría de Investigación of the Universidad de Costa Rica. HF and PG thank the Departamento de Física Teórica de la Universidad de Zaragoza for its warm hospitality. A part of the work was done was done while PG was visiting IHES. He would like to express his gratitude to the members of IHES for their warm hospitality. PG thanks Khalifa University of Science and Technology for its support towards this research, this work is partially supported by the grant FSU-2021-014. JFC acknowledge financial support from Spanish Ministerio de Ciencia, Innovación y Universidades project PGC2018-098265-B-C1 and Spanish Ministerio de Ciencia, Innovación y Universidades project PID2021-125515NB-C22.

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Index

A Adjoint representation, 59 Algebra graded, 193 Anti-dendriform algebra, 51 Anti-pre-Lie algebra, 51 Anti-Zinbiel algebra, 51, 58 Associative color algebra, 194

H Hom-Lie algebra, 339 Homogeneous element, 192

B Bicharacter skew-symmetric, 193 Binomial expansion, 385 Bol-Moufang, 107

L Lagrange’s theorem, 105 Left-symmetric algebra, 333 Leibniz-Poisson color algebra, 203 Linear space graded, 192 Loop, 97, 103

C CC-quasigroup, 134 C-loop, 124 Color bimodule, 221 Color Jordan triple, 231 D Dendriform algebra, 50 Doro’s conjecture, 142 G G-graded linear space, 192 G-loop, 135 Groupoid, 134

K Kinyon’s Conjecture, 134

M Morphism graded algebras, 193 Moufang loop, 137

N n-ary bracket, 367 Nucleus of algebra, 298

O Osborn loop, 137

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. N. Hounkonnou et al. (eds.), Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-031-39334-1

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570 P PAA algebra, 68 PAL algebra, 66 PAP algebra, 67 PCA algebra, 62 PCP algebra, 61 Poisson algebra, 52 Pre-Lie algebra, 49 PZA algebra, 65 PZL algebra, 63 PZP algebra, 64 Q QPS graphical presentation of, 424 Quasigroup, 97, 493 fractal, 424 non-fractal, 424 T TAAD algebra, 93 TAAO algebra, 80

Index TALD algebra, 90 TALO algebra, 78 TAPD algebra, 92 TAPO algebra, 79 TCAD algebra, 85 TCAO algebra, 73 TCPD algebra, 84 TCPO algebra, 72 Transposed Poisson algebra, 52 TZAD algebra, 89 TZAO algebra, 76 TZLD algebra, 86 TZLO algebra, 74 TZPD algebra, 88 TZPO algebra, 75

V VD-loop, 134

Z Zinbiel algebra, 50