Non-Associative Algebras and Related Topics: NAART II, Coimbra, Portugal, July 18–22, 2022 (Springer Proceedings in Mathematics & Statistics, 427) 3031327063, 9783031327063

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Springer Proceedings in Mathematics & Statistics

Helena Albuquerque Jose Brox Consuelo Martínez Paulo Saraiva   Editors

Non-Associative Algebras and Related Topics NAART II, Coimbra, Portugal, July 18–22, 2022

Springer Proceedings in Mathematics & Statistics Volume 427

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Helena Albuquerque · Jose Brox · Consuelo Martínez · Paulo Saraiva Editors

Non-Associative Algebras and Related Topics NAART II, Coimbra, Portugal, July 18–22, 2022

Editors Helena Albuquerque Department of Mathematics Centre for Mathematics of the University of Coimbra Coimbra, Portugal Consuelo Martínez Department of Mathematics University of Oviedo Oviedo, Spain

Jose Brox Department of Mathematics Centre for Mathematics of the University of Coimbra Coimbra, Portugal Paulo Saraiva CMUC-Centre for Mathematics of the University of Coimbra, CeBER-Centre for Business and Economics Research Faculty of Economics, University of Coimbra Coimbra, Portugal

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-031-32706-3 ISBN 978-3-031-32707-0 (eBook) https://doi.org/10.1007/978-3-031-32707-0 Mathematics Subject Classification: 17-XX, 17AXX, 46-XX, 46HXX, 68P25, 68P30 © 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

Alberto Elduque at the University of Zaragoza

Organization

Non-Associative Algebras and Related Topics: NAART II, Coimbra, Portugal, July 18–22, 2022, was a joint organization of CMUC-Centre for Mathematics of the University of Coimbra, CMUP-Centre for Mathematics of the University of Porto and CMA-UBI—Centre of Mathematics and Applications of the University of Beira Interior.

Organizing Committee Helena Albuquerque, CMUC and Department of Mathematics, University of Coimbra, Portugal Elisabete Barreiro, CMUC and Department of Mathematics, University of Coimbra, Portugal Patrícia Beites, CMA-UBI and Department of Mathematics, University of Beira Interior, Portugal Jose Brox, CMUC, University of Coimbra, Portugal Isabel Cunha, CMA-UBI and Department of Mathematics, University of Beira Interior, Portugal Jesús Laliena, University of La Rioja, Spain Samuel Lopes, CMUP and Department of Mathematics, University of Porto, Portugal Carla Rizzo, CMUC, University of Coimbra, Portugal Paulo Saraiva, CMUC and Faculty of Economics, University of Coimbra, Portugal Amir Fernández Ouaridi, University of Cádiz, Spain and CMUC, University of Coimbra, Portugal

Scientific Committee Yuri Bahturin, Memorial University of Newfoundland, Canada vii

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Organization

Vyacheslav Futorny, University of São Paulo, Brazil Santos González, University of Oviedo, Spain Victor Kac, Massachusetts Institute of Technology, USA Mikhail Kotchetov, Memorial University of Newfoundland, Canada Consuelo Martínez, University of Oviedo, Spain Fernando Montaner, University of Zaragoza, Spain Ivan Shestakov, University of São Paulo, Brazil Efim Zelmanov, Southern University of Science and Technology, China

Sponsoring Institutions CMUC, Centre for Mathematics of the University of Coimbra, Portugal CMUP, Centre for Mathematics of the University of Porto, Portugal CMA-UBI, Centre of Mathematics and Applications of the University of Beira Interior, Portugal FCT, Fundação para a Ciência e a Tecnologia, Portugal

NAART II group photo, Department of Mathematics, University of Coimbra

Preface

On October 6, 2020, Alberto Elduque turned 60. Several months before, a big group of friends (and colleagues) had started the preparation of a conference, to be held in July 2020, to celebrate this special birthday, including the publication of some proceedings that would gather those results presented during the conference whose authors agreed to contribute. Everything was planned and agreed with Springer, when the COVID-19 pandemic completely changed the scenario… and our lives. The complicated sanitary situation and the difficulties to travel obliged us to postpone the conference (and consequently its proceedings) for two years. One of the (minor) consequences of COVID was that instead of a nice conference, listening to math talks and having the chance to hug Alberto after singing to him “Happy birthday to you, happy birthday dear Alberto…”, we had to organize, exactly on October 6, 2020, and with the “needed cooperation and complicity” of Alberto’s close relatives, a short online meeting with a small number of people. But at least we had the chance to offer an unexpected present to Alberto and congratulate him on the screen. And we even had a birthday cake, real for a few people and virtual for the rest! Fortunately, two years later (in July 2022), the NAART II Conference in honor of Alberto Elduque took place in Coimbra as it had been planned. And this time, we all enjoyed the birthday cake! It is true that we had some very dear people absent, but they were so present in our hearts that we feel that they have also congratulated Alberto in the way he deserves. From very early on in his academic career, Alberto always sought to maintain collaborations with mathematicians from different parts of the world, an attitude that reveals a willingness to learn and an openness to share knowledge, the engine of scientific advancement. The fruits of these collaborations were innumerable, and the influence that his works have on the development of certain themes within nonassociative algebra is remarkable. The NAART II Conference allowed many of us to witness both the importance of such results and their influence on the work of other algebraists (particularly, but not exclusively, on his Ph.D. students).

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Alberto is a brilliant mathematician, a wonderful professor and a great human being. It is not easy to combine these three qualities in the same person. During the conference, it was clear how much Alberto is loved, respected and valued. He has advised many Ph.D. students, creating a recognized Algebra group, and his contributions to the field of Non-associative Algebra are numerous and outstanding. He is always ready to talk about mathematics, to explain anything to the newer generations, to help anyone who needs it, to collaborate in any activity, even if it demands a lot of time and effort, and always with a friendly smile on his face! During the online meeting in 2020, Alberto’s wife, Pili, thanked him for “making their life together so easy-going”. We think this beautiful phrase expresses perfectly what Alberto is like. It is easy to learn with him, it is easy to teach or to work with him, it is easy to research or to collaborate with him, and it is easy to be with him. This book includes several papers that were presented in some of the conference talks and shows the affection of Alberto’s teachers, fellows, students and collaborators. We would like to express our sincere gratitude to all those who collaborated to make this book a reality: authors, referees and the editorial staff… The work of all of them was essential. Special thanks go to all the speakers present at the conference, in particular those who made an effort to participate online, since they could not be in Coimbra. And we extend our thanks to all the participants in the NAART II Conference who traveled to Coimbra, some of them from very distant places, to be with Alberto, to share some emotive moments with him and to remember many previous experiences, enjoying listening to math talks and discussing mathematics. This book is divided into four parts. The first one is dedicated to Lie algebras, superalgebras and groups. The second one is devoted to Leibniz algebras. In the third one, results about associative and Jordan algebras and other related structures are included. Finally, papers considering other non-associative structures appear in Part four. Alberto has contributed in the past to the four subjects that define the parts of this book, mainly the first and fourth. In this case, his contribution appears in Part 4. So, his contribution is the finishing touch to this book. The rest of the papers (Alberto’s being an exception) appear in the corresponding part following alphabetical order of the first author’s surname. Needless to say, this book is dedicated to Alberto; it is a book for Alberto, a very small gift compared to what Alberto deserves. But we want it to be also a gift for Pili and Eva, their daughter. We hope that, as mathematicians, they will both appreciate it. Their presence in Coimbra with Alberto was, apart from the best tribute to Alberto, a pleasure for all of us participants in the conference, and it allowed us to share some time with them, get to know them in some situations and verify that, very often, beside a great man there is a great family supporting his activity. Congratulations Alberto

Preface

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on your birthday, on your family and on your human and mathematical achievements and success! Coimbra, Portugal Coimbra, Portugal Oviedo, Spain Coimbra, Portugal December 2022

Helena Albuquerque Jose Brox Consuelo Martínez Paulo Saraiva

Contents

Lie Algebras, Superalgebras and Groups Local Derivations of Classical Simple Lie Algebras . . . . . . . . . . . . . . . . . . . Shavkat Ayupov and Karimbergen Kudaybergenov

3

Examples and Patterns on Quadratic Lie Algebras . . . . . . . . . . . . . . . . . . . Pilar Benito and Jorge Roldán-López

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Reductive Homogeneous Spaces of the Compact Lie Group G 2 . . . . . . . . Cristina Draper and Francisco J. Palomo

29

On Certain Algebraic Structures Associated with Lie (Super)Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noriaki Kamiya

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Schreier’s Type Formulae and Two Scales for Growth of Lie Algebras and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Victor Petrogradsky

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Leibniz Algebras Universal Central Extensions of Compatible Leibniz Algebras . . . . . . . . . José Manuel Casas Mirás and Manuel Ladra

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On Some Properties of Generalized Lie-Derivations of Leibniz Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 José Manuel Casas Mirás and Natália Pacheco Rego Biderivations of Low-Dimensional Leibniz Algebras . . . . . . . . . . . . . . . . . . 127 Manuel Mancini Poisson Structure on the Invariants of Pairs of Matrices . . . . . . . . . . . . . . . 137 Rustam Turdibaev

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Associative and Jordan Algebras and Related Structures Automorphisms, Derivations and Gradings of the Split Quartic Cayley Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Victor Blasco and Alberto Daza-García On a Theorem of Brauer-Cartan-Hua Type in Superalgebras . . . . . . . . . . 163 Jesús Laliena Growth Functions of Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Consuelo Martínez and Efim Zelmanov The Image of Polynomials in One Variable on the Algebra of 3 × 3 Upper Triangular Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Thiago Castilho de Mello and Daniela do Nascimento Rodrigues Other Non-associative Structures Simultaneous Orthogonalization of Inner Products Over Arbitrary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Yolanda Cabrera Casado, Cristóbal Gil Canto, Dolores Martín Barquero, and Cándido Martín González Invariant Theory of Free Bicommutative Algebras . . . . . . . . . . . . . . . . . . . . 231 Vesselin Drensky An Approach to the Classification of Finite Semifields by Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 J. M. Hernández Cáceres and I. F. Rúa On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Abdennour Kitouni, Stephen Mboya, Elvice Ongong’a, and Sergei Silvestrov Okubo Algebras with Isotropic Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Alberto Elduque Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Lie Algebras, Superalgebras and Groups

Local Derivations of Classical Simple Lie Algebras Shavkat Ayupov and Karimbergen Kudaybergenov

With deep respect, we dedicate the article to the 60th anniversary of Professor Alberto Elduque

Abstract The paper is devoted to study of local derivations on classical simple Lie algebras. We prove that every local derivation on an arbitrary classical simple Lie algebra g over an algebraically closed field F of characteristic >5 is a (global) derivation, excluding the algebra g = psln (F) in the case when charF divides n. We also give a description of local derivations on certain simple Lie algebras over fields of charF = 2, 3 and show that they admit local derivations which are not derivations. Keywords Classical simple Lie algebra · Chevalley basis · Derivation · Local derivation

S. Ayupov (B) · K. Kudaybergenov V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, University street, 9, Olmazor district, Tashkent 100174, Uzbekistan e-mail: [email protected] K. Kudaybergenov e-mail: [email protected] S. Ayupov National University of Uzbekistan, University street, 4, Olmazor district, Tashkent 100174, Uzbekistan K. Kudaybergenov Department of Mathematics, Karakalpak State University, Ch. Abdirov, 1, Nukus 230112, Uzbekistan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_1

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S. Ayupov and K. Kudaybergenov

1 Introduction There exist various types of linear operators which are close to derivations (see e.g. [4–6, 13, 14]). In particular, R.V. Kadison [13] and, independently, D.R. Larson and A.R. Sourour [14] introduced and investigated so-called local derivations on Banach algebras. Let A be an algebra (not necessary associative). Recall that a linear mapping δ : A → A is said to be a derivation, if δ(x y) = δ(x)y + xδ(y) for all x, y ∈ A. A linear mapping Δ is said to be a local derivation, if for every x ∈ A there exists a derivation δx on A (depending on x) such that Δ(x) = δx (x). R.V. Kadison in [13] proves that every continuous local derivation from a von Neumann algebra M into a dual M-bimodule is a derivation. Later this result has been extended in [4] to a larger class of linear operators Δ from M into a normed M-bimodule E satisfying the identity Δ( p) = Δ( p) p + pΔ( p) for every idempotent p ∈ M. The above papers gave rise to a series of works devoted to the description of mappings which are close to automorphisms and derivations of C ∗ -algebras and operator algebras. In [14] D.R. Larson and A.R. Sourour proved that if A = B(X ), the algebra of all bounded linear operators on a Banach space X, then every invertible local automorphism of A is an automorphism. Thus automorphisms on B(X ) are completely determined by their local actions. Later, several papers have been devoted to similar notions and corresponding problems for derivations and automorphisms of Lie algebras. For a given Lie algebra g, the main problem concerning local derivations is to prove that they automatically become a (global) derivation or to give examples of local derivations of g, which are not derivations. In [3] we established that every local derivation on semi-simple Lie algebras over an algebraically closed field of characteristic zero is a derivation and gave examples of nilpotent finite-dimensional Lie algebras with local derivations which are not derivations. In [19] Yao considered the Witt algebra g = W1 over an infinite field of prime characteristic and proved that every local derivation on W1 is a derivation. He also showed that on the maximal subalgebra g0 of g there exist local derivations which are not derivations. In [2] the authors in collaboration with Alberto Elduque obtained a general form of the local derivations on a Cayley algebra C over an arbitrary field F with a norm n. Namely, showed that the space LocDer (C) of all local derivations of C is the Lie algebra {d ∈ so(C, n) | d(1) = 0}. Further, we have proved that the group of all local automorphisms of C coincides with the group {ϕ ∈ O(C, n) | ϕ(1) = 1}. Finally, we have applied these results to the description of local derivations and automorphisms of the seven-dimensional simple Malcev algebras over a field of characteristic =2. In [21] Zotov have studied local automorphisms of nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring with identity. In the present paper we shall consider local derivations on classical simple Lie algebras. We suggest a new technique and generalize the above mentioned results of [3] for classical simple Lie algebras over a field of positive characteristic. We prove that all local derivations on classical simple Lie algebras over an algebraically closed

Local Derivations of Classical Simple Lie Algebras

5

field of characteristic >5 are derivations, except the case of projective special linear algebra g = psln (F) such that charF divides n. Further, we show that the simple Lie algebra psl3 (F) over a field charF = 3 admits local derivations which are not derivations. With this aim, we give a description of local derivations on psl3 (F) over a field F of characteristic 3 in terms of Cayley algebra (C, n) with a norm n over this field. Namely, we consider C0 – its subspace of trace 0 elements. We show that the space of all local derivations of psl3 (F) coincides with the Lie algebra {d ∈ so(C, n) | d(1) = 0} which is isomorphic to the orthogonal Lie algebra so(C0 , n). Comparing this with the description of derivations on the algebra psl3 (F) from [9], we obtain that this algebra admits local derivations which are not derivations. Also we describe in terms of matrices local derivations on the 3-dimensional simple Lie algebra over the field charF = 2 and show the existence of local derivations which are not derivations.

2 Main Result The main result of this paper is the following theorem. Theorem 1 Let g be a finite-dimensional classical simple Lie algebra over an algebraically closed field F with charF > 5, excluding the algebra g = psln (F) in the case when charF divides n. Then every local derivation Δ on g is a derivation.

2.1 A Classical Simple Lie Algebra By a classical Lie algebra over a field of positive characteristic we mean a Lie algebra g obtained from a complex simple Lie algebra gC by the well-known procedure of Chevalley (see [7, 9, 10, 16]). The Cartan matrix of a finite-dimensional semi-simple Lie algebra gC over the field of complex numbers C is a matrix   (αi , α j ) A= 2 , (α j , α j ) i, j=1,...,l where α1 , . . . , αl is a system of simple roots of gC with respect to a fixed Cartan subalgebra hC and (·, ·) is the scalar product on the dual space of hC defined by the (α ,α ) Killing form on gC . The entries αi j = 2 (α ij ,α jj ) of a Cartan matrix have the following properties: αii = 2; αi j ≤ 0 and αi, j ∈ Z for i = j; if αi j = 0 ⇒ α ji = 0.

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Denote by R the root system of gC relative to C. It is known [10] that there exists a basis {eα }α∈R ∪ {h i }1≤i≤l in gC such that (1) (2) (3) (4)

[h i , h j ] = 0, for all i, j ∈ {1, . . . , l}; ; [h i , eα ] = α, αi eα for all 1 ≤ i ≤ l and α ∈ R, α, β = 2 (α,β) (β,β) [eα , e−α ] = h α is a Z-linear combination of h 1 , . . . , h l ; if α and β are independent roots, β − r α, . . . , α, . . . , β + qα the α-string through β, then [eα , eβ ] = 0 if q = 0, while [eα , eβ ] = ±(r + 1)eα+β if α + β ∈ R.

Such a basis {eα }α∈R ∪ {h i }1≤i≤l is called a Chevalley basis of gC [10]. Let F be an algebraically closed field of positive characteristic and let gZ be the Lie ring of Z-span of a Chevalley basis. Let hZ be the Z-span of {h 1 , . . . , h r }. Set gF := gZ ⊗ F and hF := hZ ⊗ F. Then gF is a Lie algebra over F, and hF is its abelian subalgebra. We shall keep writing h i and eα for the basis elements of gF . For α ∈ R we denote by α the corresponding linear map hF → F, given by α(h i ) = α, αi , where the right-hand side is regarded as an element of F. In general, the algebra gF is not simple and it depends on the charF. Assume that charF > 5. Then the quotient of gF by its center is a simple Lie algebra [17, Sect. 2.6]. Note that the center is Z (gF ) = {h ∈ hF : α(h) = 0 for all α ∈ R}. The simple Lie algebras gF constructed in this way from irreducible root systems are called classical Lie algebras. Below for convenience we shall write g instead of gF . Note that Π = {α1 , . . . , αl } is a system of simple roots such that any root α ∈ R is a linear combination of the {αi }1≤i≤l with integer coefficients. The height of the root α is the sum |α| of the coefficients in the expansion of α in the basis Π in R (see [10]). positive and non positive roots in R, respectively. Set Let  R+ and R− be the  Feα and n− = Feα . Then n+ = α∈R+

α∈R−

g = h ⊕ n+ ⊕ n− .

(1)

The classical simple Lie algebras over a field F of characteristic = 2, 3 of types Ar , Br , Cr and Dr have the following matrix realizations: pslr +1 (F) has type Ar (r ≥ 1), so2r +1 (F) has type Br (r ≥ 1, with B1 = A1 ), sp2r (F) has type Cr (r ≥ 1, with C1 = A1 and C2 = B2 ), and so2r (F) has type Dr (r ≥ 3, with D3 = A3 ). Further, the exceptional simple Lie algebras are g2 , f4 , e6 , e7 , e8 . ItisknownthateveryderivationofaclassicalsimpleLiealgebraoveranalgebraically closed field F of charF = 2, 3 is inner, except of type An with charF | n + 1 (see [12, Sect. 6.8, 6.9], [9, Corollaries 3.2, 3.4, Proposition 5.34], [16, Chap. V, Sect. 5]). Also in this case there is a basis {1 , . . . , l } of h such that [i , eα j ] = δi, j eα j , i, j = 1, . . . , l, where δi, j is the Kronecker delta.

Local Derivations of Classical Simple Lie Algebras

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2.2 Local Derivations of Algebras of Type A In this short subsection we shall present the proof of Theorem 1 for the algebras of type A. Let R be a unital commutative ring and let Mn (R) be the ring of all n × n matrices over R, where n ≥ 2. It is known that any local derivation on Mn (R) is a derivation [5, Corollary 3.8] (see also [6, Corollary 2.8]). Let A = Mn (F), where F is an algebraically closed field of characteristic = 2, 3. As usual, we write [A, A] for the linear span of all [x, y], x, y ∈ A. Then psln (F) = [A, A]/ (Z (A) ∩ [A, A]) , where Z (A) is the center of A. Since charF = 2, 3, by [9, Corollary 3.4] any derivation on g = psln (F) has the following form x ∈ g → [a, x] ∈ g,

(2)

where a ∈ A/Z (A). Further, if charF does not divide n, then sln (F) = psln (F) and every derivation of g is inner. If charF divides n, then the quotient space Der(g)/Inn(g) is 1-dimensional. Lemma 1 Assume that charF does not divide n. Then any local derivation Δ on sln (F) is a derivation. Proof Let Δ be a local derivation on sln (F). Since charF does not divide n, any element x ∈ Mn (F) can be represented in the form x = λ1 + y, where λ ∈ F, y ∈ (F1) = sln (F), 1 is the identity matrix in Mn (F). Let us extend Δ on A = Mn (F) by Δ 0. We have that (2)  = Δ(λ1  Δ(x) + y) = Δ(y) = [a y , y] = [a y , λ1 + y] = [a y , x].

 is a local (associative) derivation on Mn (F). By [5, Corollary 3.8] This means that Δ  is an associative derivation. Hence Δ is also a derivation on sln (F). we obtain that Δ The proof is complete. 

2.3 The Proof of the Main Result Below in Lemmas 2–5, g is a classical simple Lie algebra over an algebraically closed field F of characteristic >5 except of type A. In this case we have already mentioned in subsection 2.1 that all derivations of g are inner. We also note that the dimension of the Cartan subalgebra l = dim h ≥ 2.

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Let sl2 (F) be the 3-dimensional simple Lie algebra and let h, e, f be its canonical basis, that is, [h, e] = 2e, [h, f ] = −2 f, [e, f ] = h. Lemma 2 Let β ∈ R be an arbitrary root and let sl2 (β) = h β , eβ , e−β . If Δ is a local derivation that maps sl2 (β) into itself, then the restriction Δ|sl2 (β) is a local of sl2 (β). Moreover, if Δ(h β ) = 0, there is a number λβ such that   derivation Δ e±β = ±λβ e±β . Proof Let x = ξh β + ξβ eβ + c−β e β be an arbitrary non zero element in sl2 (β). cα eα such that Δ(x) = [ax , x], where h β , h ∈ Take an element ax = 2λh β + h + α∈R

h are elements such that β(h β ) = 1 and h ∈ ker β. We have that

 Δ(x) = 2λh β + h + cα eα , ξh β + ξβ eβ + c−β eβ α∈R

 = 2λh β + cβ eβ + c−β e−β , ξh β + ξβ eβ + c−β eβ

  + cα eα , ξh β + ξβ eβ + c−β eβ . α=±β

Since Δ(x) ∈ sl2 (β), it follows that the last product is zero. Hence,  Δ(x) = λh β + cβ eβ + c−β e−β , ξh β + ξβ eβ + c−β eβ ,   because 2h β , e±β = ±2e±β = h β , e±β . This means that Δ|sl2 (β) is a local derivation of sl2 (β). By Lemma 1 it is a derivation of sl2 (β), and hence Δ|sl2 (β) = ad(aβ )|sl2 (β) , where aβ ∈ sl2 (β), because charF does notdivide 2. Further, if Δ(h β ) = 0, then aβ = λβ h β . Thus Δ e±β = ±λβ e±β . The proof is complete.  Lemma 3 Let Δ be a local derivation of g and let α ∈ R. Then (a) there exists aΔ ∈ n+ ⊕ n− such that Δ|h = ad(aΔ )|h ; (b) if Δ(h) = 0 for all h ∈ ker α , then aΔ ∈ span {e±α } . Proof Let i ∈ {1, . . . , l}. By the definition of local derivation there is an element ai ∈ g such that Δ(i ) = [ai , i ]. Thus [ai , i ] ∈ n+ ⊕ n− , and therefore the element Δ(i ) can be represented as follows Δ(i ) =



cα(i) eα .

α∈R

Let us first show that

  cγ(i) γ  j = cγ( j) γ (i )

for all 1 ≤ i = j ≤ l and γ ∈ R.

(3)

Local Derivations of Classical Simple Lie Algebras

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Let γ ∈ R be a fixed root. Consider the element x = γ(i ) j − γ( j )i . We have that Δ(x) = γ(i )Δ( j ) − γ( j )Δ(i )     cα( j) eα − γ( j ) cα(i) eα . = cγ( j) γ(i ) − cγ(i) γ( j ) eγ + γ(i ) α=γ

Now take an element ax = h + Then Δ(x)

 α∈R

α=γ

μα eα , where h ∈ h, such that Δ(x) = [ax , x].

= [ax , x] = −μγ γ(i )γ( j )eγ + μγ γ( j )γ(i )eγ   −γ(i ) α=γ μα α( j )eα + γ( j ) α=γ μα α(i )eα   = −γ(i ) α=γ μα α( j )eα + γ( j ) α=γ μα α(i )eα .

Comparing the last two equalities we obtain (3).   For every α ∈ R there is an element i(α) ∈ {1 , . . . , l } such that α i(α) = 0 (see [17, Sect. 2.6]). Set  c(i)  α  eα . aΔ = − (α) α∈R α i Taking into account (3), for j = 1, . . . , l, we have that [aΔ ,  j ] =



c(i)  α  eα h j, α i(α) α∈R (3)

=



α∈R

=



c(i)  α  α( j )eα α i(α) α∈R

cα( j) eα = Δ( j ).

Hence, Δ|h = ad(aΔ )|h . Now suppose that Δ(h) = 0 for all h ∈ ker α , where α is a fixed root. Then 0 = Δ(h) = [aΔ , h] = −



cα α(h)eα .

α∈R

Thus cα α(h) = 0 for all α ∈ R and h ∈ ker α . Take α = ±α . Since α and α are not collinear, there is an element h ∈ ker α such that α(h) = 0. Hence cα = 0 for  all α = ±α . This means that a ∈ span {e±α } . The proof is complete. Let Δ be a local derivation of g and let Φ be an automorphism of g. Set Δ(Φ) (x) = Φ −1 (Δ(Φ(x))) , x ∈ g.

(4)

It is well known that if D is a derivation, then D (Φ) defined as above is also a derivation.

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Let x ∈ g. Then for the element y = Φ(x) we consider a derivation D y on g such that Δ(y) = D y (y). Then   Δ(Φ) (x) = Φ −1 (Δ(Φ(x))) = Φ −1 D y (Φ(x)) = D (Φ) y (x). So, Δ(Φ) is a local derivation on g. Let α, β ∈ R. There exist integers r and q such that all the elements of the sequence β − r α, . . . , β, . . . , β + qα are roots, and this finite sequence is said to be the α-string through β. The α-string through β contains at most four roots (see [10, 11]). For an element a = eα ∈ n± , where α ∈ R, we set Φa (x) = exp(ad(a))(x) =

3  k=0

1 ad(a)k (x). k!

Since charF > 5, it follows that exp(ad(a)) is well-defined, and it is an inner automorphism of g. In the following two Lemmas, using conjugated local derivations of the form (4), firstly we show that a local derivation vanishing on Cartan subalgebra h acts diagonally on roots eα , α ∈ R. Secondly, we show that a local derivation vanishing on Cartan subalgebra and on simple roots vanishes on the whole algebra. Lemma 4 Let Δ be a local derivation of g such that Δ|h = 0. Then (a) for each α ∈ R there exists a number λα such that Δ(e±α ) = ±λα e±α ; (b) there is an element h Δ ∈ h such that Δ|span{e±αi :i=1...,l} = ad(h Δ )|span{e±αi :i=1...,l} . Proof (a) Take an element a = −eα , where α is a fixed root. Let Φa be an inner automorphism of g generated by the element a. We have that Φa (h) =

3  1 ad(a)k (h) = h + α (h)eα k! k=0

for all h ∈ h. In particular, Φa (h) = h for all h ∈ ker α . Thus Δ(Φa ) (h) = Φa−1 (Δ(Φa (h))) = Φa−1 (Δ(h)) = Φa−1 (0) = 0

(5)

for all h ∈ ker α . By Lemma 3(a) there exists an element b ∈ n+ ⊕ n− such that Δ(Φa ) |h = ad(b)|h .

(6)

Local Derivations of Classical Simple Lie Algebras

11

Taking into account (5) and Lemma 3(b), we conclude that b ∈ span {e±α } . On the other hand Δ(Φa ) (h α ) = Φa−1 (Δ(Φa (h α ))) = Φa−1 (Δ(h α + 2eα )) = 2Φa−1 (Δ(eα )). Thus   (6)    2Δ(eα ) = Φa Δ(Φa ) (h α ) = Φa [b, h α ] = Φa (b), Φa (h α ) = [Φa (b), h α + 2eα ].

Since b ∈ span {e±α } , it follows that Φa (b) ∈ sl2 (α ). Thus Δ(eα ) ∈ sl2 (α ). This means that Δ maps sl2 (α ) into itself. By Lemma 2, Δ|sl2 (α ) is a derivation on sl2 (α ). Since Δ|h = 0, again by Lemma 2, there is a number λα such that Δ (e±α ) = ±λα e±α .   (b) For each 1 ≤ i ≤ l take a number λi such that Δ e±αi = ±λi e±αi . Setl  ting h Δ = λi i , we see that Δ − ad (h Δ ) vanishes on the direct sum h ⊕ i=1    span e±αi : i = 1, . . . , l . The proof is complete. Lemma 5 Let Δ be a local derivation of g such that Δ vanishes on  h ⊕ span e±αi : i = 1, . . . , l . Then Δ(eγ ) = 0 for all γ ∈ R. Proof The proof is by induction on the height |γ|. Let |γ| = 1. This means that γ = ±αi . So, for |γ| = 1 is true by the assumption of Lemma. Assume that the assertion is true for all |γ| ≤ m − 1, where m ≥ 2. Take γ ∈ R such that |γ| = m. Without loss of generality we may assume that γ is a positive root. For negative root the proof is similar. Then γ = α + β, where |α| = 1 and |β| = m − 1. Then, by the assumption, it follows that Δ(eα ) = Δ(eβ ) = 0.

(7)

Let Φta be an inner automorphism of g generated by the element ta, where a = eα , 0 = t ∈ F. Direct computations show that Φta (h) = h − α(h)teα for all h ∈ h. Hence, −1 −1 (Δ(Φa (h))) = Φta (Δ(h − α(h)teα )) Δ(Φta ) (h) = Φta (7)

−1 −1 (α(h)tΔ(eα )) = Φta (0) = 0 = −Φta

for all h ∈ h. Thus Δ(Φta ) |h = 0. Since −β − α, −β ∈ R and α-string through −β contains at most four roots, it follows that the only possible roots of the form −β + kα with k ≥ 1 may be

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only −β + α and −β + 2α ∈ R. Note that the height of −β + kα is | − β + kα| = |β| − k|α| = m − 1 − k < m, where k = 1, 2. By the inductive assumption Δ ad(ta)k (e−β ) = 0 for all k ≥ 1, because ad(ta)k (e−β ) ∈ span{e−β+α , e−β+2α }. (Φta ) Thus Δ(Φta ) (e−β ) = 0. Applying Lemma , we have  4 (a) to the local derivation Δ (Φta ) (eβ ) = 0, and therefore Δ Φta (eβ ) = 0. Hence, that Δ Δ

 3  tk k=0

and hence

3  tk k=0

k!

k!

 ad(eα ) (eβ ) = 0, k

  Δ ad(eα )k (eβ ) = 0,

that is,   t2   t3     Δ eβ + tΔ [eα , eβ ] + Δ [eα , [eα , eβ ]] + Δ [eα , [eα , [eα , eβ ]]] = 0. 2 6 Since t is arbitrary, it follows that   Δ ad(eα )k (eβ ) = 0 for all k = 0, 1, 2, 3. In particular, Δ(eγ ) = 0, because [eα , eβ ] = n α,β eγ , n α,β = 0. The proof is complete.  Proof of Theorem 1. Let Δ be a local derivation of g. By Lemma 3 there is an element aΔ ∈ n+ ⊕ n− such that Δ|h = ad(aΔ )|h . Further, applying Lemma 4 to the local derivation Δ − ad(aΔ ) we can find an element h Δ ∈ h such that (Δ − ad(aΔ )) − ad(h Δ ) vanishes on h ⊕ span e±αi : i = 1, . . . , l . By Lemma 5 we obtain that Δ − ad(aΔ ) − ad(h Δ ) ≡ 0. Thus Δ = ad(aΔ + h Δ ) is a derivation. The proof is complete. 

2.4 Local Derivations on psl3 (F) Over a Field of Characteristic 3 In order to describe local derivations on the algebra psl3 (F) over a field of charF = 3 we turn to description of this algebra in terms of Cayley algebras (for details see [9, Sect. 4.3]). Let C be a Cayley algebra over a field F of characteristic = 3 with a norm n. Define a bracket [·, ·] by [x, y] = x y − yx, x, y ∈ C.

Local Derivations of Classical Simple Lie Algebras

13

Then the space of trace 0 elements: C0 = {x ∈ C : n(x, 1) = 0}, with the bracket − [·, ·], is a simple Lie algebra which is denoted as C− 0 . Note that C0 is a central simple Lie algebra of type A2 (i.e., a twisted form of psl3 (F)) and, conversely, any central simple Lie algebra of type A2 is obtained in this way (see [1] or [9, Theorem 4.26]). Using [8, Chap. IV, Lemma 6.1] we see that ∼ Der (psl3 (F)) ∼ = Der (C− 0 ) = Der (C), in particular, we obtain that dim Der (psl3 (F)) = 14. On the other hand, by [2, Theorem 3.17], the space of all local derivations of psl3 (F) ∼ = C− 0 is the Lie algebra so(C0 , n) of all linear mappings on C0 skew-symmetric with respect to the norm n. Since dim LocDer (psl3 (F)) = dim so(C0 , n) = 21, it follows that the algebra psl3 (F) admits local derivations which are not derivations.

2.5 Local Derivations on 3-Dimensional Simple Lie Algebra Over a Field of Characteristic 2 Let F2 = {0, 1} be the two point field of characteristic 2. It is known [18] that there is a unique simple Lie algebra g3 of dimension 3 over F2 . It has a basis e1 , e2 , e3 with the following table of multiplications [e1 , e2 ] = e3 , [e2 , e3 ] = e1 , [e3 , e1 ] = e2 . Direct computations show that each derivation D on g3 has the following symmetric matrix form ⎛ ⎞ μ1 λ3 λ2 ⎝ λ3 μ2 λ1 ⎠ , (8) λ2 λ1 μ1 + μ2 where μ1 , μ2 , λ1 , λ2 , λ3 ∈ F2 . From (8) we see that the sum of all elements of the matrix of derivation is zero, and it is easy to show that the same is true for matrices of local derivations. Let us show that the latter property of matrices is also a sufficient condition to be a local derivation, i.e. any local derivation on g3 has the following matrix form ⎛

⎞ a11 a12 a13 ⎝ a21 a22 a23 ⎠ , a31 a32 a33 where ai j ∈ F2 , 1 ≤ i, j ≤ 3 such that

3  i, j=1

ai j = 0.

(9)

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Consider a linear mapping Δ1 on g3 defined on basis elements as follows Δ1 (e1 ) = 0, Δ1 (e2 ) = Δ1 (e3 ) = e1 . We shall show that it is a local derivation which is not derivation. Since Δ1 has the following matrix form ⎛

⎞ 011 ⎝0 0 0⎠, 000 comparing with (8) we see that it is not a derivation. Let us check that Δ1 is a local derivation. For any x =

3 

ξi ei ∈ g3 we need to

i=1

find a derivation Dx on g3 such that Δ1 (x) = Dx (x). We shall consider the possible two cases. Case 1. Let ξ2 = ξ3 . Since charF2 = 2, it follows that Δ1 (x) = ξ2 (e1 + e1 ) = 0, and therefore we can take Dx ≡ 0. Case 2. Let ξ2 = ξ3 . By a symmetry we may assume that ξ2 = 1 and ξ3 = 0. If ξ1 = 1, taking in (8), μ1 = 1 and all other components zero, we get a derivation Dx such that Δ1 (x) = Dx (x). Further, if ξ1 = 0, taking in (8), λ3 = 1 and all other components zero, we obtain a required derivation. Similarly, the linear mappings Δ2 and Δ3 with matrices ⎞ ⎛ ⎞ 000 101 ⎝ 1 0 1 ⎠ and ⎝ 0 0 0 ⎠ , 000 000 ⎛

respectively, are also local derivations which are not derivations. Since a non zero linear combination of Δ1 , Δ2 and Δ3 has non-symmetric matrix, it follows that the above three matrices generate a 3-dimensional space of local derivations which are not derivations, and together with the 5-dimensional space Der (g3 ) they form an 8-dimensional linear subspace LocDer (g3 ) in M3 (F2 ) which necessarily coincides with the 8-dimensional space of matrices of the form (9). Therefore, as far as we know, the algebras psl3 (F) and g3 are the first examples of simple Lie algebras which admits pure local derivations, i.e. local derivations which are not derivations. It should be noted that for simple Lie algebras over an algebraically closed field of characteristic zero all local derivations are global (see [3, Theorem 3.1]).

Local Derivations of Classical Simple Lie Algebras

15

2.6 Local Derivations on Borel Subalgebras Let g be a finite-dimensional classical simple Lie algebra over an algebraically closed field F with charF > 5, and let g = h ⊕ n+ ⊕ n− be the decomposition of g as in (1). A subalgebra b = h ⊕ n+ is called a Borel subalgebra of g. It is known that a Borel subalgebra of a simple Lie algebra over a field of characteristic 0 is complete, i.e. it is centerless and all derivations of g are inner (see [15, Corollary 4.3]). Note that the proofs and conclusions from Sect. 2.3 with slight modifications are applicable to b. Let b2 be the 2-dimensional solvable Lie algebra with a basis {h, e} and with the multiplication rule [h, e] = e. Direct computations show that b2 is complete and all local derivations of b2 are derivations. Applying this observation instead of Lemma 1, one can obtain versions of Lemmas 2–5 for complete Borel subalgebras. Note that the proof for Borel subalgebras will be even easier than for the case of simple Lie algebras, since we may operate only with positive roots. We omit the details and directly state the following result which is a generalization of [20, Theorem 2.1]. Theorem 2 Let g be a finite-dimensional classical simple Lie algebra over an algebraically closed field F with charF > 5. Assume that the Borel subalgebra b = h ⊕ n+ is complete. Then all local derivations on b are derivations. Acknowledgements We would like to thank professor Alberto Elduque for useful comments on derivations of exceptional Lie algebras in prime characteristic and for drawing our attention to the paper [12]. We are indebted to the Referee for very valuable remarks and comments.

References 1. Alberca-Bjerregaard, P., Elduque, A., Martín-González, C., Navarro-Márquez, F.: On the Cartan-Jacobson theorem. J. Algebr. 250(2), 397–407 (2002) 2. Ayupov, Sh.A., Elduque, A., Kudaybergenov, K.K.: Local derivations and automorphisms of Cayley algebras. J. Pure Appl. Algebr. 227(5), Paper No. 107277 (2023) 3. Ayupov, Sh.A., Kudaybergenov, K.K.: Local derivations on finite-dimensional Lie algebras. Linear Algebr. Appl. 493(6), 381–398 (2016) 4. Brešar, M.: Characterizations of derivations on some normed algebras with involution. J. Algebr. 152(2), 454–462 (1992) 5. Brešar, M.: Characterizing homomorphisms, derivations and multipliers in rings with idempotents. Proc. R. Soc. Edinb. 137A, 9–21 (2007) 6. Brešar, M.: Automorphisms and derivations of finite-dimensional algebras. J. Algebr. 599(1), 104–121 (2022) 7. Carter, R.W.: Simple Groups of Lie Type. Wiley and Sons, New York (1972) 8. Elduque, A., Myung, H.Ch.: Mutations of alternative algebras. Mathematics and Its Applications, vol. 278, pp. xiv+226. Kluwer Academic Publishers Group, Dordrecht (1994) 9. Elduque, A. Kochetov, M.: Gradings on simple Lie algebras. Mathematical Surveys and Monographs, vol. 189. American Mathematical Society, Providence, RI; Atlantic Association for Research in the Mathematical Sciences (AARMS), Halifax, NS (2013)

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10. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1972) 11. Jacobson, N.: Lie Algebras. Dover Publications, New York (1979) 12. Jantzen, J.C.: First cohomology groups for classical Lie algebras. Prog. Math. 95, 289–315 (1991) 13. Kadison, R.V.: Local derivations. J. Algebr. 130(2), 494–509 (1990) 14. Larson, D.R., Sourour, A.R.: Local derivations and local automorphisms of B(X ). Proc. Symp. Pure Math. 51, 187–194 (1990) 15. Leger, G., Luks, E.: Cohomology theorems for Borel-like solvable Lie algebras in arbitrary characteristic. Can. J. Math. 24(6), 1019–1026 (1972) 16. Seligman, G.B.: Modular Lie Algebras. Springer, New York (1967) 17. Steinberg, R.: Automorphisms of classical Lie algebras. Pac. J. Math. 11, 1119–1129 (1961) 18. Vaughan–Lee, M.: Simple Lie algebras of low dimension over G F(2). LMS J. Comput. Math. 9, 174–192 (2006) 19. Yao, Y.-F.: Local derivations on the Witt algebra in prime characteristic. Linear Multilinear Algebr. 70(15), 2919–2933 (2022) 20. Yu, Y., Chen, Zh.: Local derivations on Borel subalgebras of finite-dimensional simple Lie algebras. Commun. Algebr. 48(1), 1–10 (2020) 21. Zotov, I.N.: Local automorphisms of nil-triangular subalgebras of classical lie type Chevalley algebras. J. Sib. Fed. Univ. Math. Phys. 12(5), 598–605 (2019)

Examples and Patterns on Quadratic Lie Algebras Pilar Benito and Jorge Roldán-López

Abstract A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. In characteristic 0, semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadratic algebras. The class of quadratic algebras is large, but at first sight it is not clear whether an algebra is quadratic. Some necessary structural conditions appear due to the existence of an invariant form forcing elementary patterns. Through the paper we overview classical features and constructions on this topic and focus on the existence and constructions of local quadratic Lie algebras. Keywords Quadratic · Bilinear form · Lattice · Double extension · T ∗ -extension

1 Introduction A bilinear form ϕ on a Lie algebra g with product [x, y] that satisfies, ϕ([x, y], z) = ϕ(x, [y, z]) ∀x, y, z ∈ g

(1)

is said to be invariant. In addition, if ϕ is symmetric and non-degenerate, the pair (g, ϕ) is named quadratic Lie algebra (some authors also use the terminology quasiclassical, self-dual, or even orthogonal). Finite-dimensional real quadratic algebras appear as metric or metrizable in the literature. They were introduced in 1957 by S. T. Tsou and A. G. Walker [15] who provided basic well known facts on existence In the memory of our colleague and friend Georgia Benkart. P. Benito · J. Roldán-López (B) Departamento de Matemáticas y Computación, Complejo Científico Tecnológico, Universidad de La Rioja, Calle Madre de Dios 53, 26006 Logroño, La Rioja, Spain e-mail: [email protected] P. Benito e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_2

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and structure for this class of algebras. Tsou and Walker were looking for nice properties to distinguish metrizable from non-metrizable Lie groups. The structure results they published are a compendium of Tsou’s PhD Thesis (Liverpool, 1955), completed under the supervision of Walker. In the mid-1980s and along the 1990s, two different methods of construction of quadratic Lie algebra appeared: the double extension and the T ∗ -extension processes (applicable to general algebras in characteristic different from 2). The first method is a multi-step process that was introduced independently by several authors: Kac, Keith, Hoffmann, Favre, Santharoubane, Medina and Revoy. According to [12], any non-simple, non-abelian, indecomposable quadratic Lie algebra is a double extension either by a one-dimensional or by a simple Lie algebra. The T ∗ -extension is a one-step method that produces quadratic Lie algebras as central extensions of an arbitrary Lie algebra g by using a cyclic 2-cocycle ω ∈ Z 2 (g, g∗ ) (see [4] for definitions). The most basic example is the null extension T0∗ (g) = g ⊕ g∗ , with ω = 0, with the hyperbolic form q(a + β, a  + β  ) = β  (a) + β(a  ) and product [a + β, a  + β  ] = [a, a  ]g + β ◦ ad a  − β  ◦ ad a.

(2)

For arbitrary ω, the term ω(a, a  ) must be added to the expression (2). The double extension procedure is a bit more complex. We start with a quadratic Lie algebra (a, ϕa ) and another Lie algebra g such that there exists a Lie homomorphism ρ : g → Der ϕa a where Der ϕa a is the subalgebra of ϕa -skew derivations (see definition in Sect. 2). From ρ we get the 2-cocycle ω : a × a → g∗ given by ω(a, b)(x) = ϕa (ρ(x)(a), b) that allows us to define the central extension a ⊕ω g∗ , under the multiplication [a + β, a  + β  ] = [a, a  ]a + ω(a, a  ). Now, from the coadjoint representation ad∗ of g (ad∗ (x)(β) = −β ◦ ad x) and ρ, we reach the homomorphism of Lie algebras ψ : g → Der(a ⊕ω g∗ ), ψ(x)(a + β) = ρ(x)(a) − β ◦ ad x. The semidirect product a(g) = g ⊕ψ (a ⊕ω g∗ ) is a Lie algebra with bracket [b + a + β, b + a  + β  ] := [b, b ]g + ρ(b)(a  ) − ρ(b )(a) + [a, a  ]a + w(a, a  ) + β  ◦ ad b − β ◦ ad b , (3)

and the bilinear form ϕ which is non-degenerate and symmetric on a(g): ϕ(b + a + β, b + a  + β  ) := β(b ) + β  (b) + ϕa (a, a  ).

(4)

So, the pair (a(g), ϕ) is a quadratic Lie algebra, over a field of characteristic zero, and is called the double extension of (a, ϕa ) by (g, ρ). Along the 2000s, many efforts were made on small dimension classifications of quadratic Lie algebras or in deepening the structure of certain families related to symmetric spaces (Kath, Olbrich, Duong, Benayadi, Hilgert, Neeb). From then until

Examples and Patterns on Quadratic Lie Algebras

19

today (see [13] and references therein), we find papers on classification by using non-classical procedures (Benito, Laliena, de-la-Concepción) or relating quadratic algebras, geometric structures and applications (Rodriguez-Vallarte, Salgado, Cornulier, del Barco, Conti, Rossi). Throughout this paper we will summarise features and results on quadratic Lie algebras in three sections, apart from the Introduction. Section 2 includes basic terminology, existence results and duality. Local quadratic algebras, their structure and constructions, are treated in Sect. 3. The final section is devoted to explaining the 2014 Elduque and Benayadi contribution [3] on mixed quadratic Lie algebras as a tribute to Alberto on his 60th birthday. We restrict to finite-dimensional algebras over generic field K of characteristic zero. These results, collected from different articles, have been revisited, expanded and exemplified along this paper.

2 Patterns and Duality Along this section (g, ϕ) is a quadratic Lie K-algebra with product [x, y]. In general, for U and V subspaces of g, U ⊥ = {x ∈ g : ϕ(x, u) = 0 ∀u ∈ U } and [U, V ] = span [u, v] : u ∈ U, v ∈ V . The lower (respectively upper) central series of g is inductively defined as g1 = g and gt+1 = [g, gt ] (Z 0 (g) = {0}, Z 1 = Z (g), the centre, and Z t+1 = {x ∈ g : [x, g] ⊂ Z t }). The terms in the derived series are g(1) = g and g(t+1) = [g(t) , g(t) ]. A Lie algebra g is called nilpotent (respectively solvable) if gt = 0 for some t ≥ 1 (g(t) = 0 for some t ≥ 1). The left multiplication by x ∈ g, ad x = [x, · ], is a derivation known as an inner derivation. In general, a derivation d of g such that ϕ(d(x), y) + ϕ(x, d(y)) = 0 will be called ϕ-skew symmetric and Der ϕ g will denote the set of skew-symmetric derivations with respect to ϕ, while Der g and Inner g will denote the whole set of derivations and inner derivations, respectively. We note that condition (1) is equivalent to saying that Inner g is a subalgebra of Der ϕ g (in fact, it is an ideal). The algebra g is reduced if its centre is contained in the derived algebra g2 . And, g is called decomposable if it contains a proper and non-degenerate ideal I (so I ∩ I ⊥ = 0). Otherwise, we say that g is indecomposable. Semisimple Lie algebras by means of their Killing form are quadratic. Abelian quadratic are just orthogonal vector spaces (v, ϕ), [u, v] = 0 for all u, v ∈ v. Orthogonal direct sums of quadratic algebras (as ideals) produce new quadratic algebras. Therefore, any reductive Lie algebra g = s ⊕ Z (g), s semisimple, is quadratic. But the class of quadratic algebras is very large and assorted. There are a wide variety that includes non-abelian nilpotent, solvable (non-nilpotent) and mixed (non-solvable and non-semisimple). In fact, denoting by (r, s) = (dim g2 , dim Z (g)), in [14, Theorem 5.1] we find the following result: Theorem 1 (Tsou, 1962) There exist reduced quadratic Lie algebras of arbitrary type (r, s) except for (5, 0), (7, 0) and (4, s), 0 ≤ s ≤ 4.

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The proof of this theorem, based on multilinear arguments and tools, does not make it clear how to build them. Since ϕ is non-degenerate and invariant, the centre and the derived algebra of g are orthogonal: (g2 )⊥ = {x ∈ g : ϕ([x y], z) = 0 = −ϕ(y, [x z]) ∀x, y ∈ g} = Z (g),

(5)

and therefore dim g = dim g2 + dim Z (g).

(6)

Equalities (5) and (6) are two of the most famous patterns on quadratic algebras. The following proposition covers other well-known features. Proposition 1 Let (g, ϕ) a quadratic Lie algebra, U any subspace and I an ideal. Then: (a) dim g = dim U + dim U ⊥ and, if U is non-degenerate, g = U ⊕ U ⊥ . (b) U is an ideal of g if and only if [U ⊥ , U ] = 0. (c) Any minimal and non-degenerate ideal of g is simple or one-dimensional. Minimal degenerated ideals are isotropic and abelian. (d) The algebra g decomposes as the orthogonal direct sum, as ideals, of a reduced quadratic Lie algebra and an abelian Lie algebra. (e) If g is indecomposable and I is proper, then I ∩ I ⊥ = 0 and Z (g) ⊆ g2 . In particular, g is a reduced algebra and has no simple ideals. (f) The centre of a nonzero solvable quadratic algebra is nonzero. (g) If g is indecomposable, then either g is one-dimensional or simple or isometrically isomorphic to a double extension of an algebra h which is one-dimensional or simple. Proof A detailed proof of items (b) to (c) and (g) can be found in [6]. Item (d) is just [15, Proposition 6.2] and (e) follows easily from previous items. For item (f), apply  (5) and g2 = g because of g is nonzero and solvable. We introduce examples on quadratic and non quadratic algebras regarding previous Proposition 1 and classical constructions. Example 1 The smallest solvable non-abelian quadratic algebra is the oscillator (d4 , ψ) with basis x1 , x2 , x3 , z, nonzero products [x1 , x2 ] = x3 , [x1 , x3 ] = −x2 , [x2 , x3 ] = z and form ψ(x1 , z) = 1 and ψ(x2 , x2 ) = ψ(x3 , x3 ) = 1. The derived algebra d24 = [d4 , d4 ] = h1 , the Heisenberg Lie algebra (see Example 3 below). An easy computation yields Der ψ d4 = Inner d4 . This algebra arises over the reals in the quantum mechanical description of a harmonic oscillator and ψ is a Lorentzian form.  Example 2 Following [9, Proposition A], the tensor product (g ⊗ a, ϕ ⊗ q) where (g, ϕ) is quadratic Lie with bracket [x, y] and (a, q) is an associative and commutative algebra with invariant symmetric and nondegenerate form q and product ab produces the quadratic algebra [x ⊗ a, y ⊗ b] = [x, y] ⊗ ab under the form

Examples and Patterns on Quadratic Lie Algebras

21

(ϕ ⊗ q)(x ⊗ a, y ⊗ b) = ϕ(x, y)q(a, b). As a remarkable example we point out the , K ⊗ q) for any n ≥ 1, s semisimple, K (x, y) = tr(ad x ◦ ad y) the algebra (s ⊗ K[t]

t n  Killing form and q(x i , x j ) = 1 (q(x i , x j ) = 0) if and only if i + j + 1 = n (other wise), where x is the class of t modulo t n . Example 3 The generalized Heisenberg algebra series is determined through the property g2 = Z (g) = span z. They are 2-nilpotent (g3 = 0) of odd dimension. For any n ≥ 1 there exists a unique n th Heisenberg algebra of dimension 2n + 1 that we denote from now on as hn . The algebra hn has a standard basis e1 , . . . , en , en+1 , . . . e2n , z with nonzero brackets [ei , en+i ] = z. Note that Eq. (6) fails, so hn are not quadratic. Doubling by the dual space (hn )∗ , we get the trivial quadratic T ∗ -extension T0 (hn ) as it is defined in (2). In [4, Example 4.1], examples of nilpotent quadratic algebras with arbitrary nilindex follow from the non-quadratic nilpotent filiform  algebra series ln , for n ≥ 2. Example 4 According to [5], the 5-dimensional algebra n2,3 and the 6-dimensional n3,2 are the unique quadratic free nilpotent Lie algebras. Any algebra in the free nilpotent series nd,t (d ≥ 2 and t ≥ 2) satisfies: the nilindex is t + 1, d = codim n2d,t  and the centre, Z (nd,t ) = (nd,t )t , has dimension 1t a|t μ(a)d t/a , where μ the is Möbius function (see [7]). Then, the assertion is a corollary of item (a) in Propo5 and sition 1. The quadratic structure of n2,3 is given by the (Hall) basis {ai }i=1 nonzero products [a2 , a1 ] = a3 , [a3 , a1 ] = a4 , [a3 , a2 ] = a5 and the symmetric form ϕ(ai , a j ) = (−1)i−1 for i ≤ j and i + j = 6 and ϕ(ai , a j ) = 0 otherwise. For 6 and nonzero products [a2 , a1 ] = a4 , [a3 , a1 ] = a5 , n3,2 , take as (Hall) basis {ai }i=1 [a3 , a2 ] = a6 and symmetric form φ(ai , a j ) = (−1)i−1 for i ≤ j and i + j = 7 and  φ(ai , a j ) = 0 otherwise. The existence of an invariant non-degenerated bilinear form on g is equivalent to saying that the adjoint ρ = ad and coadjoint ρ∗ representations of g are isomorphic. This pattern has a significant influence on the lattice of ideals of quadratic algebras (see [9, 10]). Proposition 2 (Keith, Hofmann, 1984) For a quadratic Lie algebra (g, ϕ), the map Ω : Set of ideals of g → Set of ideals of g, Ω(I ) = I ⊥ is an involutive antiautomorphism of the lattice of ideals of g. In particular, lattices of ideals of quadratic Lie algebras are self-dual and, (a) I is a minimal ideal if and only if I ⊥ is maximal. So, the sum of the set of minimal ideals, the socle ideal, and the intersection of the set of maximal ideals, the Jacobson radical, are orthogonal. (b) The terms gt+1 and Z t (g) of the lower and upper central series are orthogonal and dim g = dim gt+1 + dim Z t (g). The lattice of ideals of the oscillator d4 is a 4-chain in case x 2 + 1 is irreducible over K. Otherwise, there are eigenvectors of ad x1 , [x1 , u] = λu and [x1 , v] = −λv and Z (d4 ) = [u, v] and the proper ideals of d4 are d24 , u, [u, v], v, [u, v] and the centre. Both lattices are self-dual. The next figure shows other lattices of quadratic Lie algebras (Fig. 1).

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P. Benito and J. Roldán-López

n2,3

···

n3,2

s⊗

K[t] t3

L1 and L2

··· ···

··· ···

···

Fig. 1 L 1 = s ⊕ s ⊕ s and L 2 = s ⊕ s ⊕ K, with s simple

Remark 1 Self-duality of lattices of ideals is a necessary condition for being quadratic, but it is not equivalent to it. However, it can be a very useful tool to rule out the possibility of being quadratic. The lattices of ideals of the non-quadratic series hn and ln are not self-dual. Remark 2 Let us define the split extension algebra of the 3-dimensional Heisenberg, a = h1 ⊕ span d, where {x, y, z} is the standard basis of h1 , so [x, y] = z, and [d, x] = x, [d, y] = y and [d, z] = 2z. This algebra is solvable and centreless. So, from item (f) in Proposition 1, it is not quadratic. But, a2 = h1 is the unique maximal ideal and Kz is the unique minimal. The rest of the ideals are of the form span z, λx + μy. So, the lattice of ideals of a is self-dual.

3 Local Quadratic Algebras A local algebra is an algebra with only one maximal ideal. Along this section, s, r and n will denote the Levi subalgebra of g, its solvable radical and its nilradical. We have the Levi decomposition g = s ⊕ r. In case s = 0 = r, we say that g is mixed. The Jacobson radical J(g) of g is the solvable radical of g2 . This ideal is nilpotent and, J(g) = [g, g] ∩ r = [g, r] ⊆ n. When g is solvable, J(g) = g2 . It is known that J(g) is the intersection of all maximal ideals. So, in any local algebra, this radical is the only maximal ideal. Lemma 1 For a Lie algebra g of dimension greater than one and not simple, the following is equivalent: (a) g is local, (b) J(g) = n, (c) g = s ⊕ n with s one dimensional or simple and

Examples and Patterns on Quadratic Lie Algebras

23

n2 = {x ∈ n : [y, x] ∈ n2 ∀y ∈ s}. Remark 3 Assertion n2 = {x ∈ n : [y, x] ∈ n2 ∀y ∈ s} in Lemma 1, is equivalent to saying that ad x is faithful on n/n2 for all x ∈ n\n2 if s is one-dimensional and, in the simple case, the ads -module n/n2 has not trivial modules. We start by revisiting Theorem 3.1 in [2]. Proposition 3 (Bajo, Benayadi, 2007) Up to isometric isomorphisms, any local and quadratic Lie algebra (g, ϕ) is of one of the following types: (a) g = span x and ϕ(x, x) = 1. (b) g is simple and there is λ = 0 such that λϕ(x, y) = tr(ad x ◦ ad y) is the Killing form if the base field is algebraically closed. (c) g = s ⊕ s∗ the central split extension of a simple Lie algebra s and its dual module s∗ and ϕ(x + α, y + β) = α(y) + β(x). (d) g is a solvable Lie algebra double extension of a nonzero and nilpotent quadratic Lie algebra (a, ψ) by a ψ-skew derivation δ which is invertible on the centre Z (a). (e) g = g2 is a mixed Lie algebra double extension of a nonzero and nilpotent quadratic Lie algebra (a, ψ) by a simple subalgebra s through a representation ψ : s → Der ϕ (a) such that Z (a) has non trivial s-modules. The Jacobson radical J(g) = n is non-abelian for any g in items (d) or (e), J(g) = s∗ in item (c) and null for simple and one-dimensional algebras. Proof If g = s ⊕ n is reductive, the only possibilities are (a) or (b). Assume then g non-reductive. From [2, Theorem 3.1, item (i)] we arrive at item (d) and from [2, Theorem 3.1, item (ii)] n⊥ = Z (n) we get either (c) if n = n⊥ or (e) otherwise. Note that if g solvable, 0 = Z (g) ⊆ n. Then g2 = n = [x, n] ⊕ n2 and n2 = 0 give  us ad x|n bijective, which is not possible. From the previous proposition we have the following conclusions: – Simple and one-dimensional algebras are the only local quadratic and reductive Lie algebras (types (a) and (b)). – Trivial T ∗ -extensions of simple Lie algebras are local and quadratic (type (c)). Recall that the T ∗ -extension is a more general construction. The trivial extension is the easier case. Quadratic local algebras (g, ϕ) of type (d) or (e) have some common structural patterns and remarkable differences. Using Proposition 3 and [4, Theorem 2.2], we can easily deduce the following: – Common patterns: r = n is the unique maximal ideal, n⊥ is the unique minimal ideal and n = n⊥ . In addition, 0 = n⊥ ⊆ n2 and g can be built as a double extension of the quadratic nilpotent algebra,  n  n . , ϕ| n⊥ n⊥

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– Extra patterns for g of type (d): g is solvable (s = 0) and n = g2 is of codimension 1, n⊥ = Z (g) = span z ⊆ n2 ⊂ n and the double extension is induced through the ϕ-skew ad x for any x ∈ g such that ϕ(x, z) = 0 and ad x| Z ( n⊥ ) is invertible. n – Extra patterns for g of type (e): g = s ⊕ n is a perfect mixed algebra and s is simple, n⊥ = Z (n) ⊆ n2 ⊂ n = [g, n], Z (n) ∼ = s∗ as ad s-modules and the second term of the lower central series Z 2 (n) has no trivial ads -modules. Here the double extension is induced by the adjoint representation of the Levi subalgebra, ad n⊥ : s → Der ϕ ( nn⊥ ). n

In both cases, by imposing n2 = n⊥ , the algebra g is a double extension of the abelian algebra nn⊥ and yields to the constructions given in Examples 5 (g solvable) and 6 (g perfect). Otherwise, n2 = n⊥ and local quadratic Lie algebras follow from nilpotent and nonabelian quadratic algebras. In our Examples 7 and 8 we will present some algebras of low nilindex. Example 5 (Generalized oscillator algebras) Abelian quadratic Lie algebras are nothing else but a pair (vn , ϕ) where vn is an n-dimensional vector space endowed with a symmetric and non-degenerate symmetric form ϕ. Consider now any ϕ-skew linear automorphism δ of vn . This is only possible in the case n = 2m. Then, the double extension d(2m) = F · δ ⊕ v2m ⊕ F · δ ∗ is local and quadratic. Note that h = d(2m)2 = v2m ⊕ F · δ ∗ and its square is just [d(2m)2 , d(2m)2 ] = F · δ ∗ . So h = hm is a generalized Heisenberg algebra and add(2m) δ ∈ Inner d(2m) extends δ by declaring  [δ, δ ∗ ] = 0. Note that adh δ ∈ Der h is an outer derivation of h. Remark 4 Over the reals and starting with a positive definite form ϕ, since δ is ϕ-skew, there is a ϕ-orthonormal basis {e1 , . . . e2m } of the real space v2m such that [e2i−1 , e2i ]d(2m) = λi e2m+1 , e2m+1 = δ ∗ for some nonzero real numbers 0 < λ1 ≤ λ2 · · · ≤ λm and δ(e2i−1 ) = λi e2i and δ(e2i ) = −λi e2i−1 and δ(e2m+1 ) = 0. According to [11, Theorem 4.1, Sect. 4], the algebras g(λ) = span e0 (= δ), e1 , . . . e2m , e2m+1 (= δ ∗ ), λ = (λ1 , . . . , λm ) are named oscillator algebras. This family determines the set of connected and simply connected non simple Lie groups, endowed with a Lorentz-invariant metric that makes them indecomposable. The series of oscillator algebras also appeared in the characterization of Lorentzian cones given by Hilgert and Hofmann [8]. Example 6 Let s be a simple Lie algebra and (v, ϕ) an s-module without trivial submodules. Denote by ρ the representation of s on v. Assume that ϕ is a symmetric, nondegenerate and s-invariant bilinear form. The existence of ϕ only is possible if the 2-symmetric tensor power, S 2 v, has a trivial submodule. Then, the representation ρ : s → gl(v) satisfies ϕ(ρ(s)(u), v)) + ϕ(u, ρ(s)(v)) = 0 ∀s ∈ s. Equivalently, ρ : s → Der ϕ (v, ϕ). Let v(s) = s ⊕ψ (v ⊕ω s∗ ), the double extension of (v, ϕ) by ρ. The bracket and the bilinear form are given by Eqs. (3) and (4) where

Examples and Patterns on Quadratic Lie Algebras

25

g = s, a = v is an abelian algebra and ω(u, v)(s) = ϕ(ρ(s)(u), v), for all u, v ∈ v and s ∈ s. Taking s = sl2 (K) and v = V (n), the unique possibility is n = 2m and the algebra a(sl2 (K)) is a (2m + 7)-dimensional local algebra with 4 ideals in chain.  In [5], the authors compute Der ϕ n2,3 and Der φ n3,2 . Skew-derivations of quadratic are a natural source of examples of local quadratic algebras by double extension. Example 7 Let (n2,3 , ϕ) be the algebra described in Example 4 with (Hall) basis 5 . Then, Der ϕ n2,3 is 6-dimensional and it is given as the direct sum of the {ai }i=1 (nipotent) ideal Inner n2,3 , and its Levi subalgebra s ∼ = sl2 (K). Thus, any derivation decomposes as the sum (for short, D = D(m i ) ∈ s and d = d(vi ) ∈ Inner n2,3 ): m1 m2 m 3 −m 1 0 D+d = 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 + m1 m2 m 3 −m 1

0 0 v2 v3 0

0 0 v1 0 v3

0 0 0 v1 −v2

0 0 0 0 0

0 0 0 0 0

Note that n2,3 has two copies of the 2-dimensional natural s-module V (1) ( a1 , a2  and a4 , a5 ) and one of the trivial module V (0) = a3 . By double extension formulae (3) and (4) we get the 11-dimensional local (apply Lemma 1 or Proposition 3) perfect quadratic Lie algebra n2,3 (s) = s ⊕ n2,3 ⊕ s∗ . The lattice of ideals of n2,3 (s) is a 6-chain. Taking any derivation δ = D(m 1 , m 2 , m 3 ) such that m 3 m 2 − m 21 = 0 we get a 7-dimensional solvable local quadratic via the double extension by δ.  Example 8 Consider now the quadratic (n3,2 , φ) in Example 4 with (Hall) basis 6 . In this case, Der ϕ n3,2 is 10-dimensional and it is the direct sum of {ai }i=1 Inner n3,2 and a simple Lie subalgebra s ∼ = sl3 (K) (D = D(m 1 , . . . , m 5 ) ∈ s and d = d(v1 , v2 , v3 ) ∈ Inner n3,2 ):

D+d =

m1 m4 m7 0 0 0

m2 m5 m8 0 0 0

m3 m6 −m 1 − m 5 0 0 0

0 0 0 m1 + m5 m8 −m 7

0 0 0 m6 −m 5 m4

0 0 0 −m 3 m2 −m 1

+

0 0 0 v1 v3 0

0 0 0 v2 0 v3

0 0 0 0 v2 −v1

···

···

Here n3,2 decomposes as two copies of the 3-dimensional natural s-module. By double extension, we get the 22-dimensional local (just applying Lemma 1) perfect quadratic Lie algebra n3,2 (s) = s ⊕ n3,2 ⊕ s∗ and its lattice of ideals is a 5-chain.  Example 9 According to [6, Proposition 5.1], double extensions via inner derivations provide decomposable quadratic algebras. So, any double extension of (d4 , ψ)  in Example 1 is decomposable because Der ψ d4 = Inner d4 .

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4 Final Comments In 2014, A. Elduque and S. Benayadi classified real and complex indecomposable mixed quadratic Lie algebras of dimension ≤13 (see [3, Theorems 3.16 and 4.11]). The Levi subalgebra of this type of algebras in the complex case is the 3-dimensional split sl2 (C) and over the reals it is either sl2 (R) or su2 (R). Combining irreducible modules for these algebras, smart reasoning and elementary linear and multilinear algebra tools and previous knowledge on basic structure of quadratic Lie algebras, the authors achieve a very clean classification. In the complex case, apart from the double extension process, constructions of quadratic Lie algebras as tensor products (Example 2) also appeared. The authors arrived there by using Jordan algebras of dimension 2, 3, 4 and following the ideas given in 1976 by B.N. Allison [1, Theorem 1, Sect. 5]. Lie algebras in the complex classification are local, so indecomposable. Most part of this classification could be recovered from previous examples seen in Sect. 3. We also point out that their lattices of ideals are mainly n-chains with n ≤ 6.

Funding The authors have been supported by research grant MTM2017-83506-C2-1-P of ‘Ministerio de Economía, Industria y Competitividad, Gobierno de España’ (Spain) until 2022 and by grant PID2021-123461NB-C21, funded by MCIN/AEI/10.13039/5 01100011033 and by “ERDF A way of making Europe”. J. Roldán-López has been also supported by a predoctoral research grant FPI-2018 of ‘Universidad de La Rioja’.

References 1. Allison, B.: A construction of Lie algebras from j-ternary algebras. Am. J. Math. 98(2), 285–294 (1976) 2. Bajo, I., Benayadi, S.: Lie algebras with quadratic dimension equal to 2. J. Pure Appl. Algebr. 209(3), 725–737 (2007) 3. Benayadi, S., Elduque, A.: Classification of quadratic Lie algebras of low dimension. J. Math. Phys. 55(8), 081703 (2014) 4. Bordemann, M.: Nondegenerate invariant bilinear forms on nonassociative algebras. Acta Math. Univ. Comen. 66(2), 151–201 (1997) 5. Del Barco, V.J., Ovando, G.P.: Free nilpotent Lie algebras admitting ad-invariant metrics. J. Algebr. 366, 205–216 (2012) 6. Figueroa-O’Farrill, J.M., Stanciu, S.: On the structure of symmetric self-dual Lie algebras. J. Math. Phys. 37(8), 4121–4134 (1996) 7. Gauger, M.A.: On the classification of metabelian Lie algebras. Trans. Am. Math. Soc. 179, 293–293 (1973) 8. Hilgert, J., Hofmann, K.H.: Lorentzian cones in real Lie algebras. Mon. Für Math. 100(3), 183–210 (1985) 9. Hofmann, K.H., Keith, V.S.: Invariant quadratic forms on finite dimensional Lie algebras. Bull. Aust. Math. Soc. 33(1), 21–36 (1986)

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10. Keith, V.S.: On invariant bilinear forms on finite-dimensional Lie algebras. Tulane University (1984) 11. Medina, A.: Groupes de Lie munis de métriques bi-invariantes. Tohoku Math. J., Second Ser. 37(4), 405–421 (1985) 12. Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant. Ann. Sci. l’École Norm. Supérieure 18(3), 553–561 (1985) 13. Ovando, G.: Lie algebras with ad-invariant metrics: a survey-guide. Rend. Semin. Mat. Univ. Politec. Torino 74(1), 243–268 (2016) 14. Tsou, S.T.: Xi. On the construction of metrisable Lie algebras. Proc. R. Soc. Edinb. Sect. A: Math. 66(2), 116–127 (1962) 15. Tsou, S.T., Walker, A.G.: Xix. Metrisable Lie groups and algebras. Proc. R. Soc. Edinb. Sect. A. Math. Phys. Sci. 64(3), 290–304 (1957)

Reductive Homogeneous Spaces of the Compact Lie Group G 2 Cristina Draper and Francisco J. Palomo

Abstract The first author defended her doctoral thesis “Espacios homogéneos reductivos y álgebras no asociativas” in 2001, supervised by P. Benito and A. Elduque. This thesis contained the classification of the Lie-Yamaguti algebras with standard enveloping algebra g2 over fields of characteristic zero, which in particular gives the classification of the homogeneous reductive spaces of the compact Lie group G 2 . In this work we revisit this classification from a more geometrical approach. We provide too geometric models of the corresponding homogeneous spaces and make explicit some relations among them. Keywords Octonions · Exceptional algebra g2 · Exceptional group G 2 Homogeneous manifold

·

1 Introduction This paper extends the talk In the footsteps of Alberto Elduque given by the first author in the conference “Non-Associative Algebras and Related Topics-II” (Coimbra, July 2022), dedicated to honor Alberto Elduque on the occasion of his 60th birthday. We thought that seeing his student following his steps and developing his ideas along the years would be a nice way of thanking him for so many years of friendship, for Partially supported by Junta de Andalucía projects FQM-336, UMA18-FEDERJA119 and P20_01391, and by the Spanish MICINN projects PID2019-104236GBI00/AEI/10.13039/501100011033 and PID2020-118452GB-I00, all of them with FEDER funds. Partially supported by Spanish MICINN project PID2020-118452GB-I00 and Andalusian and ERDF projects FQM-494 and P20− 01391. C. Draper (B) · F. J. Palomo Universidad de Málaga, Málaga, Spain e-mail: [email protected] F. J. Palomo e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_3

29

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sharing ideas and adventures. And, of course, for sharing our love of exceptional Lie algebras and exceptional objects. The geometries associated to exceptional Lie groups often present interesting properties which permit both to understand these exceptional algebraic objects and to shed some light on the geometric features of homogeneous manifolds for such groups. The smallest exceptional Lie groups have real dimension 14, those of type G 2 , namely, the automorphism groups of the octonion division algebra O and of the split division algebra Os , and the double covering of the last one. We will focus on the connected and simply-connected compact Lie group G 2 = Aut(O). For the first author, the Lie group G 2 has been a traveling companion throughout the years. The first of these encounters occurred in her doctoral dissertation [19], where Lie-Yamaguti algebras with standard enveloping algebra of type G 2 over fields of characteristic zero were classified. Lie-Yamaguti algebras are binaryternary algebras defined to codify reductive homogeneous spaces to an algebraic structure, similarly to the way Lie triple systems are defined to codify algebraically symmetric spaces. In particular, the above mentioned classification essentially gives the list of the homogeneous reductive spaces of the Lie groups of type G 2 . Motivated by these facts and by the very interesting geometrical structures associated with G 2 , our main purpose will be to give geometric descriptions of the homogeneous reductive spaces of the compact Lie group G 2 . When we began to look for descriptions of these homogeneous manifolds, we found that some of them were well-known. However, not only these results were scattered in the literature, but also we found insufficient information flow between differential geometers and researchers in Algebra. For instance, until now the cites to [6] were from researchers working in topics related to nonassociative structures. Also the first reference we have found about the quotient G 2 /SO(3) is [26], only three years ago. We have tried to make accessible the results of [6], while giving a unified perspective. For instance, the homogeneous spaces G 2 /Sp(1)+ and G 2 /Sp(1)− are considered in the book Sasakian Geometry [9, Example 13.6.8], which offers a very nice panoramic of the diagram submersions, but we add not only a self-contained description, but also the relations with the remaining G 2 -homogeneous spaces. The exceptional Lie group G 2 occurs in different situations and in various guises in Differential Geometry (see [1] for an historical perspective). With no claim to be exhaustive, we would like to recall several such contexts. G 2 may be the holonomy group of certain non locally symmetric 7-dimensional Riemannian manifolds, according to the Berger list of irreducible holonomies (see for instance [7, Chap. 10]). Also note that G 2 is a subgroup of SO(7) and, then, one can consider G 2 -structures on 7-dimensional Riemannian manifolds. Essentially, a G 2 -structure consists of a 3-form which permits to construct a Riemannian metric, a volume form and a vector cross product [28, 31]. Another issue is related to generic distributions in dimension 5. Let us recall that a rank two distribution D on a 5-dimensional manifold M is said to be generic if and only if it is bracket generating with grow vector (2, 3, 5). That is, the Lie brackets of vector fields in D span a rank 3-distribution of T M, and triple Lie brackets of vector fields in D span all T M. The study of such distributions

Reductive Homogeneous Spaces of the Compact Lie Group G 2

31

has a long history starting with the “five variables paper” by E. Cartan in 1910 [12]. Rank two distributions arise in the mechanical system of a surface rolling without slipping and twisting on another surface. In this case, the configuration space has a rank two distribution which encodes the no slipping and twisting condition. When both surfaces are round spheres with ratio of their radii 1 : 3, the universal double covering of the configuration space is a homogeneous space for the real split form of the exceptional Lie group of type G 2 . As mentioned above, our main aim will be to describe the family of reductive G 2 homogeneous spaces in geometrical terms, as well as relationships between them. We have tried to do it in an accessible way, while keeping a unified perspective. The paper is organized as follows. Section 2 is devoted to generalities on the octonion division algebra O, and also to fix some terminology and notations. For instance, we describe G 2 and its related Lie algebra g2 , not only in terms of automorphisms and derivations of the octonion algebra, but also in terms of a convenient (generic) 3-form. The key result for this paper is Theorem 1, which introduces the complete list of non-abelian reductive Lie subalgebras of g2 , up to conjugation. We have labeled every such subalgebra as hi with i ∈ {1, . . . , 8}. In order to construct the corresponding G 2 -homogeneous manifold, we consider for each of our subalgebras hi of g2 the unique closed connected Lie subgroup Hi with Lie algebra hi by means of Theorem 1 and [42, Theorem 3.19]. All of these algebras hi admit natural descriptions in terms of quaternions, complex numbers and derivations, with the exception of h8 , a principal three-dimensional subalgebra of g2 . The description of h8 has to wait until Proposition 2, after devoting an effort in Sect. 2.6 to understand the main properties and the existence of the principal subalgebras of compact real forms. In Theorem 1, the corresponding 3-dimensional simple subalgebra of g2 is described by its properties, namely, the fact that O0 is an absolutely irreducible module for it. Its uniqueness is clarified in Proposition 3. The central part of this paper is Sect. 3, where we develop one by one every reductive G 2 -homogeneous space G 2 /H such that H is connected and its Lie algebra is a reductive subalgebra of g2 . We have included old and new geometric descriptions of such spaces and several features and uses in Differential Geometry. Section 3 begins with some generalities on homogeneous manifolds. Then, we realize G 2 as a hypersurface of the Stiefel manifold V7,3 of all orthonormal 3-frames in R7 . The long string of reductive G 2 -homogeneous manifolds starts with arguably the best known examples in the literature: the 8-dimensional quaternion-Kähler symmetric space G 2 /SO(4) and the nearly Kähler six dimensional sphere S6 ∼ = G 2 /SU(3). The directed tree in Fig. 1 describes the relationships between the manifolds in Sect. 3. According to the dimensions, the Lie group G 2 is in the top (the root), while the symmetric space G 2 /SO(4) and the nearly Kähler sphere S6 ∼ = G 2 /SU(3) are in the bottom. Our directed tree has three leaves: G 2 /SO(4), S6 and the irreducible space G 2 /SO(3)irr . Moreover, every arrow denotes a fiber bundle projection G/Hi → G/H j with standard fiber H j /Hi whenever Hi is a closed subgroup of H j . There are two branches with end on S6 . The first one has two nodes: the unit tangent bundle US6 and the complex quadric Q 5 ⊂ CP 6 (excluding the root, G 2 ). Alternative descriptions and properties of the manifolds in this branch can be found in Sects. 3.5,

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G2

14

11

G2 SU(2)l

10

G2 U(2)l

6&8

∼ = US6

G2 SO(3)

∼ = Q5

S6

G2 SU(2)r

G2 SO(3)irr

G2 U(2)r

G2 SO(4)

Fig. 1 Reductive homogeneous spaces of G 2

3.4 and 3.3. The other branch with end on S6 has only the node G 2 /SO(3). This G 2 homogeneous manifold G 2 /SO(3) is almost unknown, or at least one would think so from its virtually non-existent appearances in the literature. Section 3.6 is devoted to this case. Three branches end on the 8-dimensional quaternion-Kähler symmetric space G 2 /SO(4) (compare again with [9, Example 13.6.8]). Two of such branches agree with those ones ending on S6 . The third one has two nodes: G 2 /SU(2)r and G 2 /U(2)r . Sections 3.7 and 3.8 provide very concrete descriptions of such manifolds of twistor spaces over G 2 /SO(4). Observe that the first and third branches with end on G 2 /SO(4) contain topologically different manifolds. In fact, the two copies of U(2) and SU(2) in G 2 produce quotients with different homotopy types [9, Example 13.6.8] (see also [37], which computes their third homotopy groups making use of the notion of Dynkin index). Finally, the manifold G 2 /SO(3)irr is a leaf in our tree and its branch reduces only to one node, because the corresponding algebra is at the same time a maximal subalgebra and minimal among the non-abelian reductive ones. The material on this manifold, an isotropy irreducible space, is presented in Sect. 3.9, where we give the first concrete description (as far as we know) of this homogeneous space. Although Wolf gives in [44] a structure theory and classification for non-symmetric, isotropy irreducible homogeneous spaces, the truth is that the mere apparition of G 2 /SO(3)irr in a list leaves the reader with more questions than answers.

Reductive Homogeneous Spaces of the Compact Lie Group G 2

33

2 Background on Octonion Algebras and g2 In this paper we will focus on the real number field, due to the applications to Differential Geometry, although this section can be widely generalized. The main facts on Cayley algebras and its derivation algebras can be consulted in [41, 45].

2.1 Octonion Division Algebra The well-known octonion division algebra is denoted here by O and a basis is given by {1, ei : 1 ≤ i ≤ 7}, where the multiplication of such elements follows the rules ei ei+1 = ei+3 ,

ei2 = −1,

for the indices modulo 7. Moreover, if ei e j = ek , then we also have e j ek = ei and ek ei = e j , and ei e j = −e j ei for i = j. Alternatively, we can take O = H ⊕ Hl, where H = 1, i, j, k denotes the usual quaternion algebra with product, for any q1 , q2 ∈ H, q1 (q2 l) = (q2 q1 )l,

(q2 l)q1 = (q2 q1 )l

and

(q1 l)(q2 l) = −q2 q1 .

Recall that O is a quadratic algebra, that is, every x = a0 1 + isfies the following second degree polynomial equation

7 i=1

(1)

ai ei ∈ O sat-

x 2 − t (x)x + n(x)1 = 0, 7 ai ei , the trace is given by t (x) = x + x¯ = 2a0 and the norm where x¯ = a0 1 − i=1 7 by n(x) = x x¯ = i=0 ai 2 . Moreover the norm is multiplicative, n(x y) = n(x)n(y) for all x, y ∈ O, so that O is a composition algebra. We will also denote by n : O ×  O → R the polar form n(x, y) = 21 n(x + y) − n(x) − n(y) related to the positive ¯ definite quadratic form n. Every x ∈ O \ {0} has an inverse given by x −1 = x/n(x). The octonion algebra is an important example of a nonassociative algebra: if we denote the associator by (x, y, z) := (x y)z − x(yz), note that, for instance, (i, j, l) = 2kl = 0. Remark 1 A remarkable property of the octonion algebra is that every subalgebra Q of O of dimension 4 is isomorphic to H. Moreover, if we take v ∈ Q⊥ with n(v) = 1, then the isomorphism f : Q → H is extended to an automorphism O = Q ⊕ Qv → O = H ⊕ Hl by means of q1 + q2 v → f (q1 ) + f (q2 )l. The proof is a consequence of the fact that v and Q satisfy the relations in Eq. (1) (see for instance [45, Chap. 2, Lemma 6]).

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2.2 Cross Products and 3-Forms The projection of the product of the octonion algebra over the subspace of the zero trace elements O0 = {x ∈ O : t (x) = 0} defines a cross product on O0 as follows × : O0 × O0 → O0 ,

1 x × y = pr O0 (x y) = x y − t (x y)1. 2

That is, a binary product satisfying n(x × y, x) = n(x × y, y) = 0 and    n(x, x) n(x, y)  . n(x × y) =  n(y, x) n(y, y)  Equivalently we have a cross product in R7 given by the natural identification as 7 denotes the canonical basis of R7 ). vector spaces O0 → R7 , ei → ei (now {ei }i=1 Moreover, the trilinear map Ω : O0 × O0 × O0 → R defined by Ω(x, y, z) = n(x, y × z) = n(x, yz), is alternating and so defines a 3-form. It is frequently called the associative calibration on O0 [30].

2.3 The Automorphism Group F. Engel proved in [25] that the compact Lie group G 2 is the isotropy group of certain generic 3-form in 7 dimensions (for instance, Ω is such a generic 3-form). On the other hand, E. Cartan proved that G 2 is also the automorphism group of the octonion algebra [13]. We use here both approaches: the classification of the reductive homogeneous spaces of G 2 in [6] follows the viewpoint of the automorphism group of the octonion algebra, but we will profusely use the 3-form Ω to provide concrete descriptions of these homogeneous spaces in Sect. 3. So, we first think of G 2 as the automorphism group Aut(O) = { f ∈ GL(O) : f (x y) = f (x) f (y) ∀x, y ∈ O}. Since every automorphism preserves the norm, we have Aut(O) ⊂ SO(O, n). Moreover, every automorphism f satisfies f (1) = 1, and taking into account that O0 = 1 ⊥ , we get that f can be restricted to O0 . This restriction determines the action of f on O. Hence we can also consider Aut(O) as a subgroup of SO(O0 , n). Recall that GL(R7 ) ≡ GL(O0 ) acts on the set of alternating trilinear maps ω : O0 × O0 × O0 → R by ( f · ω)(x, y, z) = ω( f −1 (x), f −1 (y), f −1 (z)). The

Reductive Homogeneous Spaces of the Compact Lie Group G 2

35

orbit of Ω is open (in 3 O∗0 ) and the group G 2 = Aut(O) is isomorphic to the isotropy group { f ∈ GL(O0 ) : f · Ω = Ω}, by means of f → f |O0 .

2.4 The Exceptional Lie Algebra g2 The 14-dimensional simple Lie algebra g2 is the Lie algebra of derivations of the octonion algebra der(O) = {d ∈ gl(O) : d(x y) = d(x)y + xd(y) ∀x, y ∈ O}, endowed with the usual commutator. Similarly to the case of the group, the map d → d|O0 provides an isomorphism between der(O) and the Lie algebra {d ∈ gl(O0 ) : Ω(d(x), y, z) + Ω(x, d(y), z) + Ω(x, y, d(z)) = 0 ∀x, y, z ∈ O0 }. In general, the derivations of an algebra are not easy to describe. In the case of the octonion algebra, the concrete computations are carefully developed in [41, Chap. 8]. We will follow here this description. Let us denote by L x , Rx : O → O the left and right multiplication operators given by L x (y) = x y and Rx (y) = yx. They are not derivations but behave well with respect to the norm, that is, if x ∈ O0 , L x , Rx ∈ so(O, n) = { f ∈ gl(O) : n( f (x1 ), x2 ) + n(x1 , f (x2 )) = 0 ∀xi ∈ O}. Now, let us denote by Dx,y := [L x , L y ] + [L x , R y ] + [Rx , R y ]. Every Dx,y is a derivation of O such that Dx,y (z) = [[x, y], z] + 3(x, z, y). Moreover, these operators span the whole derivation algebra, that is,  der(O) =

k 

 Dxi ,yi : xi , yi ∈ O, k ∈ N .

i=1

Besides, the unique non-zero g2 -invariant map O0 × O0 → der(O), up to scalar multiple, is given precisely by (x, y) → Dx,y . The invariance means that, for any d ∈ der(O), we have (2) [d, Dx,y ] = Dd(x),y + Dx,d(y) .

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2.5 Reductive Subalgebras of g2 Recall that a subalgebra h of a Lie algebra g is said to be reductive if g is completely reducible as h-module. In particular there is an h-submodule m of g such that g = h ⊕ m, that is, (g, h) is a reductive pair. Take care because the converse is not necessarily true if h has radical, that is, there can be an h-module m such that g = h ⊕ m but where m does not decompose as a sum of irreducible submodules. In order to describe the reductive subalgebras of g2 , we consider the nondegenerate Hermitian form σ : O × O → C, σ(x, y) = n(x, y) − n(ix, y)i,

(3)

and the automorphism τ ∈ Aut(O) given by τ (q1 + q2 l) = q1 + (iq2 )l,

(4)

for qi ∈ H. Then, as a consequence of [6, Theorem 2.1, Corollary 3.5, Proposition 3.6], we get: Theorem 1 If h is a non-abelian proper reductive subalgebra of the Lie algebra g2 = der(O), then either (a) h is a 3-dimensional simple Lie algebra and O0 is an absolutely irreducible h-module, or (b) h is conjugated (by an automorphism of g2 ) to one and only one of the subalgebras in the following list: • • • • • • •

h1 h2 h3 h4 h5 h6 h7

= {d ∈ g2 : d(H) ⊂ H} ∼ = so(H⊥ , n) ∼ = so(4); = {d ∈ g2 : d(H) ⊂ H, d(C) = 0} ∼ = u(H⊥ , σ) ∼ = u(2); = {d ∈ g2 : d(H) = 0} ∼ = su(H⊥ , σ) ∼ = su(2); = {d ∈ g2 : dτ = τ d} ∼ = u(H, σ) ∼ = u(2); = centh1 (h3 ) = {d ∈ h1 : [d, h3 ] = 0} ∼ = su(H, σ) ∼ = su(2); ⊥ ∼ ∼ = {d ∈ g2 : d(C) = 0} = su(C , σ) = su(3); = {d ∈ g2 : d(H) ⊂ H, d(l) = 0} ∼ = so(H0 l, n) ∼ = so(3).

In case (a), g2 is the sum of h and an absolutely irreducible h-module of dimension 11. We only provide here a rough sketch of the proof jointly with several relevant features to be used later. Proof A subalgebra h of g2 turns out to be reductive if and only if O0 is a completely reducible h-module. The proof in [6] is based on the fact that, if O0 is not an irreducible h-module, then it has a submodule, and it can be checked (adding the unit) that a subalgebra isomorphic to either H or C remains invariant. Taking into account that such proof is realized in a more general context, we add a comment here on the precise isomorphisms:

Reductive Homogeneous Spaces of the Compact Lie Group G 2

37

d ∈ hi → d|H⊥ , d ∈ h j → (Rl−1 d Rl )|H , d ∈ h6 → d|C⊥ , d ∈ h7 → d|H0 l , for i = 1, 2, 3, j = 4, 5. (Recall that Rl denotes the right multiplication by l ∈ O.)  More explicit descriptions of the elements of the subalgebras in Theorem 1 in terms of our derivations Dx,y can be achieved as follows. First, note that the decomposition O = O0¯ ⊕ O1¯ , for O0¯ = H and O1¯ = Hl, is a Z2 -grading on O. In particular, this Z2 -grading induces a Z2 -grading on the Lie algebra g2 = der(O), with homogeneous components: der(O)0¯ = {d ∈ g2 : d(Oi¯ ) ⊂ Oi¯ ∀ i = 0, 1} = h1 = DO0¯ ,O0¯ + DO1¯ ,O1¯ , der(O)1¯ = {d ∈ g2 : d(Oi¯ ) ⊂ Oi+ ¯ 1¯ ∀ i = 0, 1} = DO0¯ ,O1¯ .

(5)

Taking into account that so(H, n) = L H0 ⊕ RH0 ∼ = 2su(2) is a sum of two simple ideals, we find that h1 = der(O)0¯ = hl ⊕ hr is also a sum of two simple ideals: hl = {dal : a ∈ H0 } and hr = {dar : a ∈ H0 }, where the derivations dal and dar are determined by dal |H⊥ = Rl L a Rl−1 and dar |H⊥ = Rl Ra Rl−1 . We can explicitly write down that dal (q1 + q2 l) = (aq2 )l,

dar (q1 + q2 l) = [q1 , a] + (q2 a)l.

(6)

The indices l and r simply refer to the respective left and right action on the odd part H⊥ .1 Now, we have that h3 = hl and h2 = hl ⊕ dir . Also, [hl , hr ] = 0 so we get that hr = h5 . As Eq. (2) tells that [dal , D p,q ] = 0 for all p, q ∈ H, then DH,H ⊂ h5 . Moreover, we have D p,q = −d[rp,q] for any p, q ∈ H, so that DH,H = hr = h5 . Since dil commutes with τ , we also obtain h4 = hr ⊕ dil . Finally, we have h7 = {dal − dar : a ∈ H0 }, since dal (l) = al = dar (l). Note that, in Theorem 1, the subalgebra of type (a) must be a maximal subalgebra of g2 , since if it were properly contained in another subalgebra, the complementary submodule would be reducible. Moreover, it corresponds to the so called principal subalgebra, which is related to some important topics in Lie theory. We have not provided an explicit description of such subalgebra in Theorem 1, but only of some of the properties which characterize it, because it is difficult to achieve a concrete description in terms of derivations of the octonions. Such description will be provided in Proposition 2, where we will explicitly define h8 . Perhaps, describing h8 in terms of derivations does not particularly help us to understand better the related homogeneous space, but we have added it by completeness. However, its existence (a general fact in Proposition 1) and its uniqueness up to conjugation (Proposition 3) will be highly relevant in Sect. 3.9 for studying the isotropy irreducible Wolf space. Due to the fact that these algebraic questions are not immediate at all, we will specifically devote 1

Note that l and r correspond to − and +, respectively, in the literature.

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Sect. 2.6 to deepen in the knowledge of the principal subalgebras, better well-known in the complex case. Remark 2 All semisimple subalgebras of the complex semisimple Lie algebras were determined by Dynkin in 1952 [22]. This paper introduces some important concepts, as the index of a subalgebra, an integer number which permits to distinguish different (non-conjugate) embeddings of the same algebra. As regards gC 2 , the four types of simple three-dimensional subalgebras jointly with their indices appear in [22, Table 16]. It also provides the classification of regular subalgebras, which reduces to a combinatorial problem related to root systems, introducing the notions of Rsubalgebras and S-subalgebras, corresponding in some sense to reducible/irreducible subalgebras respectively. Our algebra gC 2 has no any S-subalgebra and it has only one simple subalgebra of rank greater than 1 (of type A2 ), as listed in [22, Table 25]. Many of these results are summarized and revisited in Chap. 6 in the encyclopaedia [39]. Tables 5 and 6 give the two only maximal subalgebras of rank 2 of gC 2 , both of them semisimple, isomorphic to sl3 (C) and so4 (C), corresponding to the fixed subalgebra by an inner automorphism of order 3 and 2 respectively. The results agree with our situation in the real compact case. In spite of the very thorough study made by Dynkin, it does not contain an explicit description of all the subalgebras. In the gC 2 -case, such classification appears in a very recent reference: according to [35, Theorem 1.1.], there are 115 subalgebras of gC 2 up to conjugacy by an inner automorphism: 64 types of regular subalgebras, 2 nonregular semisimple subalgebras and 49 types of non-regular solvable subalgebras. The techniques are combinatorial (the paper proceeds by calculation in the Chevalley basis), and differ very much from our techniques in [6], taking into account that [6] studies only the reductive subalgebras and the only restriction on the ground field is having zero characteristic.

2.6 The Principal Three-Dimensional Subalgebra of g2 The subalgebra considered in item (a) in Theorem 1 cannot be so easily described as the others, that is, as the subalgebra of der(O) which leaves invariant some subalgebra or commutes with some automorphism. One could think that this is a weird subalgebra, but the situation is the opposite: this is the most “frequent”three-dimensional simple Lie subalgebra, corresponding to the so called principal subalgebra. Now we will recall these concepts in detail, not only by completeness, but because we will use the knowledge on principal subalgebras for describing the homogeneous space related to this case. Some of the information is extracted from [39, Chap. 6, Sect. 2.3]. A crucial reference is [33], where Kostant relates the 3-dimensional principal subalgebras to many other topics. If g is a complex semisimple Lie algebra, the classification of three-dimensional simple subalgebras of g is equivalent to the classification of nilpotent elements. A triple of elements {e, h, f } ⊂ g is called an sl2 -triple if [h, e] = 2e, [h, f ] = −2 f

Reductive Homogeneous Spaces of the Compact Lie Group G 2

39

and [e, f ] = h, that is, they form a canonical basis of a subalgebra of g isomorphic to sl2 (C). According to Morozov’s theorem, for each nilpotent element 0 = e ∈ g, there is some sl2 -triple of g containing e. The element h in such sl2 -triple is called a characteristic of e. It turns out that the set N = {e ∈ g : e nilpotent} is an algebraic variety of dimension equal to dim g − rank g. The group G of inner automorphisms of g acts on N producing a finite number of orbits. There is only one dense orbit, open in N in the Zarisky topology, which is called the principal orbit (being the biggest). The nilpotent elements in this orbit are also called principal, and all of them are obviously conjugated. So the condition for a nilpotent element to be principal is that the dimension of Z (e) = {σ ∈ G : σ(e) = e} coincides with the rank of g, or equivalently, the dimension of the centralizer z(e) = {x ∈ g : [x, e] = 0} coincides with the rank of g. An sl2 -triple {e, h, f } where e is a principal nilpotent element is called a principal sl2 -triple, and the subalgebra spanned by it is called a principal subalgebra. Not every semisimple element is contained in an sl2 -triple of g, that is, not every semisimple element is a characteristic of some nilpotent element. If h is a characteristic of e, the set of all the characteristics of e is just {σ(h) : σ ∈ Z (e)}. If we have fixed h a Cartan subalgebra of g and a system of simple roots {αi : i = 1, . . . , l} of the root system relative to h, then the characteristic of any nilpotent element of g is conjugated to some h ∈ h such that αi (h) ∈ {0, 1, 2} for all i, although the converse is not true. Example 1 In the complex exceptional algebra gC 2 , there are 9 pairs of elements in {0, 1, 2}2 , but not all of them are (α1 (h), α2 (h)) for h a characteristic of some nilpotent element. According to [22] ([18] for the real—of course non-compact— case), there are just 4 orbits of non-zero nilpotent elements, corresponding to the pairs (0, 1), (1, 0), (0, 2), (2, 2). As the set of positive roots is {α1 , α2 , α1 + α2 , 2α1 + α2 , 3α1 + α2 , 3α1 + 2α2 } (α1 short root), an easy computation with eigenvalues shows then that the decomposition of gC 2 as a sum of sl2 (C)-modules for the corresponding sl2 (C) is, respectively, 4V (1) ⊕ V (2) ⊕ 3V (0), 2V (3) ⊕ V (2) ⊕ 3V (0), 3V (2) ⊕ V (4), V (2) ⊕ V (10), where V (n) denotes here the unique irreducible sl2 (C)-module of dimension n + 1 up isomorphism. That computation is immediate by taking into consideration that  1 to 0 0 −1 ∈ sl2 (C) acts on V (n) with eigenvalues {n − 2k : k = 0, . . . , n}, each with multiplicity 1. Hence, the corresponding algebra in Theorem 1 is, respectively, (the complexification of) h3 , h5 , h7 and the three-dimensional algebra described in item (a). Once we know the above decomposition as a sum of irreducible submodules, it is very easy to compute the dimensions of the centralizers z(e) and z(h). For each V (n), the highest vector (that one of weight n) belongs to z(e), and no other independent element, while there is at most one vector of weight 0 (its existence linked to the

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parity of n), which belongs to z(h). Hence the dimension of z(e) coincides with the number of irreducible modules appearing in the decomposition, while the dimension of z(h) coincides with the number of irreducible modules V (n) with n even appearing in the decomposition. Thus, dim z(e) ≥ dim z(h),2 and, in our cases, respectively, dim z(e) = 8, 6, 4, 2;

dim z(h) = 4, 4, 4, 2.

In particular, the nilpotent element with characteristic h such that α1 (h) = 2 = α2 (h) belongs to the principal orbit, since dim z(e) = 2. This example illustrates the way of getting a characteristic of a principal nilpotent element. In general, we can construct a principal three-dimensional simple subalgebra as follows. Lemma 1 ([39, Chap. 6, Sect. 2.3]) Let h be a Cartan subalgebra of a semisimple complex Lie algebra g and {αi : i = 1, . . . , l} a system of simple roots of Φ, the root system relative to h. For each α ∈ Φ, denote by tα ∈ h the element determined by κ(tα , t) = α(t), being κ the Killing form of g, and take h α = κ(t2tα ,tα α ) . For any root space gα with α ∈ Φ + , and any 0 = eα ∈ gα , choose f α ∈ g−α such that [eα , f α ] = h α . As the Cartan matrix C = ( αi , α j ) is invertible, take {c j }lj=1 ⊂ C (in fact, a subset of Q) unique scalars such that for any i = 1, . . . , l, the equation l  αi , α j c j = 2

(7)

j=1

holds. Then {e, h, f } is a principal sl2 -triple for e = eα1 + · · · + eαl , h = c1 h α1 + · · · + cl h αl ,

f = c1 f α1 + · · · + cl f αl .

We include a proof since the same argument proves that {e, ˜ h, f˜} is a principal sl2 -triple too, for e˜ = γ1 eα1 + · · · + γl eαl

cl c1 f α + · · · + f αl , f˜ = γ1 1 γl

(8)

being {γi : i = 1, . . . , l} any choice of non-zero scalars. Proof Equation (7) says that αi (h) = 2 for any i. The fact [h, e] = 2e, [h, f ] = −2 f  and [e, f ] = h is a straightforward computation. Note  that z(h) = h since for α = i m i αi ∈ Φ and x ∈ gα , then [h, x] = α(h)x = 2( i m i )x = 0. In general, dim z(e) ≥ dim z(h), but both dimensions coincide if all the of h are   eigenvalues even. This is just the case since the set of eigenvalues is {2( i m i ) : m i αi ∈ Φ}. Hence dim z(e) = dim h = rank g, and e is principal, as required.  2

Be careful with the typo in that formula in [39, Proposition 2.4].

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41

Come back to our setting, real Lie algebras. As far as we know, it is not easy to find many references to principal subalgebras of real Lie algebras. We will say that a three-dimensional simple subalgebra (usually denoted by TDS in the literature) of a simple compact real Lie algebra g is principal if so is its complexification. Such TDS is necessarily isomorphic to su2 , since g does not possess nilpotent elements. As a consequence of the previous arguments, Proposition 1 Any simple compact real Lie algebra g has a principal three-dimensional subalgebra. Proof Take, as in the previous lemma, a Cartan subalgebra of the complex Lie algebra gC ≡ g ⊗R C, a set of simple roots {α1 , . . . , αl } and an sl2 -triple {eα , f α , h α } ⊂ gC adapted to the root decomposition such that α(h α ) = 2 for any root α. We can assume (see, for instance, [24, 1.3 Theorem]) that ih αi , eαi − f αi , i(eαi + f αi ) ∈ g for any i = 1, . . . , l. If C = ( αi , α j ) denotes the Cartan matrix of gC , then the entries in the inverse matrix C −1 are non-negative [34, 1.2.1. Proposition], and hence all ci ’s are positive (since the column vector (ci )li=1 is twice the vector obtained summing the columns of C −1 , and evidently the ith row of C −1 is non-zero). For γi ∈ R such that γi2 = ci , take s = span x, y, z ⊂ g for z :=

l 

ci ih αi , x :=

i=1

l    γi eαi − f αi ,

y :=

i=1

l    γi i(eαi + f αi ) . i=1

Now simply note that {e, ˜ h, f˜} is a principal sl2 -triple in sC as in Eq. (8), for h = −iz, 1 ˜  e˜ = 2 (x − iy) and f = − 21 (x + iy). In our concrete case g = g2 , we can go further and provide an explicit description of a principal subalgebra in terms of derivations of octonions, as we did for hi , i = 1, . . . , 7. The following is a straightforward computation. Lemma 2 The derivation h :=

 1 4Dj,k + 5Dl,il ∈ g2 6

(9)

acts as follows: i → 0, j → k, k → −j, l → 2il, il → −2l, jl → 3kl, kl → −3jl. C So the eigenvalues of h ⊗ 1 ∈ gC 2 acting on O0 = O0 ⊗R C are all different, namely, {0, ±i, ±2i, ±3i}.

Be careful with the confusing notation, since i denotes at the same time the element in O0 and the scalar in the field C that we are using for complexifying.

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Proposition 2 Take h8 := span h, x, y ⊂ g2 for h defined as in Eq. (9) and √ x := Di,k +

 15  Dj,l + Dk,il , y := −Di,j + 9

√

 15  − Dk,l + Dj,il . 9

Then [h, x] = y, [h, y] = −x, [x, y] =

8 h, 3

(10)

and h8 is a principal three-dimensional simple subalgebra of g2 = der(O). Proof The fact that h8 is a TDS is a direct consequence of Eq. (10), which can be checked in a fairly straightforward way. The algebra h8 is in the situation of item (a) in Theorem 1 by taking into account Lemma 2, which implies that the action of h C ∼ C  (≡ h ⊗ 1) on OC 0 is irreducible (O0 = V (6) as an h8 -module). Remark 3 This result is directly inspired in a more general and striking not published result [19, Theorem 21], which asserts that, for an arbitrary field of zero characteristic F, and a Cayley F-algebra C, then there exists a three-dimensional simple subalgebra s of der(C) such that der(C) can be decomposed as a sum of s with an irreducible s-module if and only if there is a zero trace element c ∈ C0 such that n(c) = 15. Clearly, this is the case if F = R and C = O.

3 Homogeneous Spaces of G 2 Assume G × M → M is an action of a Lie group G on a manifold M. The action is said to be transitive if for any points x, y ∈ M, there is σ ∈ G such that σ · x = y. That is, the action has only one orbit. In this case, we call M a G-homogeneous manifold. For any fixed point o ∈ M (the origin), the isotropy subgroup H = {σ ∈ G : σ · o = o} is a closed subgroup of G, and M can be identified with the set of left cosets G/H . The natural projection G → G/H becomes a principal fiber bundle with structure group H . The homogeneous manifold M ∼ = G/H is reductive if there is an Ad(H )-invariant subspace m of g = Te G that is a complement of the Lie subalgebra h. This condition always implies that [h, m] ⊂ m, and the converse holds whenever H is connected. The natural projection π : G → G/H is a submersion. Therefore, for a reductive homogeneous manifold M ∼ = G/H with fixed complement an h-module m, the differential map of π at e ∈ G induces an isomorphism between m and To M. A Riemannian manifold (M, g) is said to be homogeneous if the Lie group of all isometries Isom(M, g) acts transitively. If G is a subgroup of Isom(M, g) which also acts transitively, then the Riemannian manifold (M, g) is said to be G-homogeneous. In this case, for any fixed point o ∈ M, the isotropy subgroup is compact. The linear isotropy representation H → GL(To M) is given by f → ( f ∗ )o , where ( f ∗ )o denotes the differential map of f at the point o ∈ M.

Reductive Homogeneous Spaces of the Compact Lie Group G 2

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A connected Riemann manifold (M, g) is a symmetric space if, for any x ∈ M, there is a g-isometry ξ x : M → M such that ξ x (x) = x and (ξ∗x )x = −id Tx M . Every symmetric space is a G-homogeneous manifold M ∼ = G/H and the symmetry ξ o gives further structures. There is an involutive automorphism F : G → G such that m := {X ∈ g : (F∗ )e (X ) = −X } is an Ad(H )-invariant subspace of g that is a complement of the Lie subalgebra h. Even more, the decomposition g = h ⊕ m is a Z2 -grading with odd part m, and m is endowed with a Lie triple system structure, given by [x, y, z] = [[x, y], z]. In order to be used later, let us denote by Vn,k the Stiefel manifold of all orthonor. The set of all oriented mal k-frames in Rn and recall that dim Vn,k = nk − k(k+1) 2 n, p , and it is known as the Grassmann p-dimensional subspaces of Rn is denoted by Gr n, p = p(n − p). manifold of the oriented p-planes in Rn . Its dimension is dim Gr According to [42, Theorem 3.19], for every Lie group G with Lie algebra g and every Lie subalgebra h of g, there is a unique connected Lie subgroup H of G with corresponding Lie algebra h. Thus, for any i = 1, . . . , 8, let us denote by Hi the unique connected Lie subgroup of G 2 corresponding to the Lie subalgebra hi considered in Theorem 1. We are in position to give explicit geometric descriptions of each of the reductive homogeneous spaces G 2 /Hi , for i = 1, . . . , 8. We will use the notation Mi ∼ = G 2 /Hi for i = 0, . . . , 8 (H0 = {e}), but in general we will prefer to use labels to facilitate the reading of the text. Namely, – – – – – – –

M0 M1 M2 M4 M6 M7 M8

≡ Mt from torsor or total (see Eq. (11)); ≡ Ms from symmetric (see Eqs. (12) and (13)); ≡ Ml and M3 ≡ Pl from left (see Eq. (14) and Sect. 3.4); ≡ Mr and M5 ≡ Pr from right (see Eqs. (20) and (21)); ≡ S6 does not need any extra label; ≡ Min from intermediate (see Eq. (15)); ≡ Mirr from irreducible (see Eq. (24)).

In a general setting, for every Lie group G with closed subgroups K ⊂ H ⊂ G, the natural projection G/K −→ G/H, gK → g H is a fiber bundle with standard fiber the homogeneous manifold H/K . We will denote by πi j : G 2 /Hi −→ G 2 /H j whenever H j is a subgroup of Hi . We won’t be concerned about the homogeneous quotients appearing for not connected subgroups, because they are locally indistinguishable of those ones in our list.

3.1 G 2 as a Hypersurface of V7,3 Any automorphism f of the octonion algebra is determined by the triple of octonions ( f (i), f (j), f (l)), since the algebra generated by {i, j, l} is the whole O. In this way,

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see [20, Remark 5.13] or [3, 4.1], the group G 2 can be identified with the set of Cayley triples, that is, triples (X 0 , X 1 , X 2 ) of orthonormal vectors in R7 such that X 2 is orthogonal to the subalgebra generated by the other two elements, in other words, Ω(X 0 , X 1 , X 2 ) = 0. For any Cayley triple (X 0 , X 1 , X 2 ), there is a unique automorphism f ∈ G 2 such that ( f (i), f (j), f (l)) = (X 0 , X 1 , X 2 ). This comes from the fact that {X 0 , X 1 , X 2 , X 0 × X 1 , X 0 × X 2 , X 1 × X 2 , X 0 × (X 1 × X 2 )} is an orthonormal basis of R7 and it is very easy to reconstruct the image by f of all these basic elements starting from { f (X i ) : i = 0, 1, 2}. The map constructed in this way is an automorphism as a consequence of the properties of the cross product. Thus, G 2 can be viewed as the following hypersurface inside the Stiefel manifold V7,3 , (11) G2 ∼ = Mt := {(X 0 , X 1 , X 2 ) ∈ V7,3 : Ω(X 0 , X 1 , X 2 ) = 0}. For references to this description, see the problem (9c) proposed in [30, p. 121]. Thus, Mt is the principal homogeneous space (that is, a homogeneous space for G 2 in which the stabilizer subgroup of every point is trivial), also called a torsor of the group G 2 .

3.2 The Symmetric Space Ms ∼ = G 2 /H1 ∼ = G 2 /SO(4) We will consider Ms := {Q ≤ O : dim Q = 4, Q2 ⊂ Q},

(12)

the set of subalgebras of O of dimension 4. The Lie group G 2 acts on Ms by f · Q = f (Q). From Remark 1, every subalgebra Q ∈ Ms is in the orbit of H, therefore this action is transitive. For computing the isotropy subgroup of H ∈ Ms , note the isomorphism { f ∈ Aut(O) : f (H) ⊂ H} = H1 → SO(H⊥ , n),

f → f | H⊥ .

Hence, we conclude that Ms ∼ = G 2 /SO(4) and the natural submersion G 2 → G 2 /SO(4) is given by G 2 −→ Ms , f → f (H). Alternatively, in terms of the description in Eq. (11), we get Mt −→ Ms ,

(X 0 , X 1 , X 2 ) → 1, X 0 , X 1 , X 0 × X 1 .

A different description of the manifold Ms in terms of the 3-form Ω can be provided.

Reductive Homogeneous Spaces of the Compact Lie Group G 2

45

Lemma 3 A 4-dimensional vector subspace Q ≤ O is a subalgebra if and only if Q⊥ is a 4-dimensional subspace of O0 where Ω vanishes. Proof If Q ∈ Ms , necessarily Q is a subalgebra isomorphic to H as in Remark 1. Then O = Q ⊕ Q⊥ is a Z2 -grading, Q⊥ Q⊥ ⊂ Q is orthogonal to Q⊥ and the 3-form Ω vanishes on Q⊥ (the same happens to H). Conversely, take W a 4-dimensional subspace of O0 such that Ω(W, W, W ) = 0, and let us check that Q = R ⊕ W ⊥ is a subalgebra (⊥ denotes here the orthogonal subspace in O0 ). Take {X 0 , X 1 , X 2 , X 3 } an orthonormal basis of W . As Ω(X 0 , X 1 , X 2 ) = 0, we know that {X 0 , X 1 , X 2 , X 0 × X 1 , X 0 × X 2 , X 1 × X 2 , X 0 × (X 1 × X 2 )} is an orthonormal basis of R7 . But Ω(X 0 , X 1 , X 3 ) = 0 too, so X 3 is orthogonal to X 0 × X 1 and analogously X 3 is orthogonal to X 0 × X 2 and to X 1 × X 2 . This means that X 3 should be proportional to the seventh element in the above basis, X 0 × (X 1 × X 2 ), and then W ⊥ = X 0 × X 1 , X 0 × X 2 , X 1 × X 2 , which is of course closed for the cross product.  Hence we can consider Ms := {W ≤ R7 : dim W = 4, Ω(W, W, W ) = 0},

(13)

and the bijective correspondence Ms → Ms given by Q → Q⊥ is compatible with the G 2 -action, thus providing an alternative description of the symmetric space G 2 /SO(4) which does not make use of the octonionic product. The submersion in these terms is f → f (Hl). G 2 −→ Ms ,

Remark 4 One of the advantages of this approach is that it can be generalized to other 3-forms (there are two orbits of generic 3-forms in R7 ), providing a family of non-compact manifolds, quotients of the Lie group Aut(Os ), where Os denotes the split octonion algebra. Remark 5 Alternative descriptions appear in the literature. The most usual one consists of considering the 3-dimensional associative subspaces, where a 3-dimensional subspace V of O0 is said associative if the associator vanishes: (V, V, V ) = 0. This means that R ⊕ V is necessarily a subalgebra of O (isomorphic to H), hence belonging to Ms . Remark 6 This is the best known quotient of G 2 since it is a symmetric space. In fact, let us recall that G 2 /H1 ∼ = G 2 /SO(4) comes from considering the Z2 -grading on der(O) induced by the Z2 -grading on O = H ⊕ Hl, as in Eq. (5). This means that a model for the tangent space is given by the set of odd derivations, TH Ms ∼ =

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der(O)1¯ = DH0 ,Hl , which is a Lie triple system. An alternative nice description of this tangent space is found in [37] as { f : H0 → H linear : f (i)i + f (j)j + f (k)k = 0}. An explicit isomorphism of the above vector space with der(O)1¯ is provided by d → f d , where f d : H0 → H is determined by d(q) = f d (q)l for all q ∈ H0 . As an application, the approach of odd derivations is advantageous because each subtriple provides a totally geodesic submanifold of the symmetric space G 2 /SO(4) (see [14] and for instance [15, Chap. 11]), as in Eq. (17) in Sect. 3.6. The 8-dimensional symmetric space G 2 /SO(4) is a quaternion-Kähler symmetric Riemannian manifold, [7, Chap. 14]. This manifold appeared in the classification of quaternion-Kähler symmetric space with non-zero Ricci curvature by Wolf, [43]. After Wolf’s paper, quaternion-Kähler symmetric spaces are called Wolf spaces.

3.3 The 6-Dimensional Sphere S6 ∼ = G 2 /H6 ∼ = G 2 /SU(3) The description of the 6-dimensional sphere S6 ≡ M6 := {X ∈ R7 : n(X ) = 1} as a quotient of G 2 is very well-known too and it is possible to find it in detail in many references (for instance, [20, 23, 30]). For the sake of completeness, we briefly recall some details here. Again identifying R7 with O0 , the action of G 2 = Aut(O) restricts to S6 since every automorphism of O preserves the norm. The action is transitive. In fact, any element in O0 of norm 1 can be completed to a Cayley triple, even more: any pair of orthonormal vectors in R7 can be completed to a Cayley triple. Now, the map which sends (i, j, l) to a fixed Cayley triple is an automorphism of O. In particular we find an element of G 2 which sends i to any norm 1 element in O0 , which gives the transitively of the action of G 2 on S6 . The isotropy subgroup of the element i ∈ S6 is H6 = { f ∈ Aut(O) : f (i) = i} ∼ = SU(C⊥ , σ), where σ is the Hermitian form in Eq. (3), and the precise isomorphism is f → f |C⊥ . Thus, we get S6 ∼ = G 2 /SU(3) and the natural submersion reads as G 2 −→ S6 ,

f → f (i).

Alternatively, with the notations of Eq. (11), we get Mt −→ S6 ,

(X 0 , X 1 , X 2 ) → X 0 .

Reductive Homogeneous Spaces of the Compact Lie Group G 2

47

Remark 7 This description of S6 as G 2 -homogeneous manifold is closely related with the nearly Kähler structure J induced on S6 from the cross product ×. In fact, the group of automorphisms of this nearly Kähler structure on S6 is just G 2 (see [2] for a clear description of these facts with very interesting historical comments). Remark 8 Note that G 2 acts also in the projective space RP 6 = R7 \ {0}/ ∼, where for any x, y ∈ R7 \ {0}, we say that x ∼ y if there is λ ∈ R with x = λy. The class of x ∈ R7 \ {0} is denoted by [x]. Since the action of G 2 on R7 is linear, it induces an action on RP 6 which is obviously transitive. The only difference is that the isotropy subgroup of [i] ∈ RP 6 is { f ∈ Aut(O) : f (i) = ±i}, which is not connected, but a double covering of SU(3) with the same related Lie algebra, {d ∈ der(O) : d(i) = 0} = h6 .

3.4 The Unit Fiber Bundle Over the Six Dimensional Sphere V7,2 ∼ = G 2 /H3 ∼ = G 2 /SU(2)l ∼ = US6 Since G 2 is a subgroup of SO(O0 , n), we have a natural action on any Stiefel manifold V7,k ≡ {(X 1 , . . . , X k ) : X i ∈ O0 , n(X i , X j ) = δi j } for any k ≤ 7. This action is transitive for k = 1, 2, since any orthonormal k-frame (X 1 , . . . , X k ) can be completed to a Cayley triple. But it is not transitive for k = 3, since G 2 preserves Ω and not all the orthonormal 3-frames behave similarly for Ω. Of course V7,1 ∼ = S6 and we study now the Stiefel manifold Pl := V7,2 as homogeneous quotient of G 2 . The isotropy subgroup of (i, j) ∈ V7,2 is H3 = { f ∈ Aut(O) : f (i) = i, f (j) = j} = { f ∈ Aut(O) : f |H = id}. In a similar way to what happened with its related subalgebra h3 , we have the isomorphism H3 ∼ = SU(2), f → f |H⊥ . = SU(H⊥ , σ) ∼ Therefore, we can call H3 = SU(2)l and we get V7,2 ∼ = G 2 /SU(2)l . Remark 9 We would like to point out that there are several subgroups isomorphic to SU(2) into G 2 . The related homogeneous manifolds are completely different. Our copy of SU(2) into G 2 is achieved by means of Theorem 1 and [42, Theorem 3.19]. A mention to this quotient appears in [30, p. 121]. The Stiefel manifold V7,2 has another geometric interpretation. Namely, for any Riemannian manifold (M, g), the unit fiber bundle over M is described as UM = {u ∈ T p M : g p (u, u) = 1, p ∈ M}. For any n, the Stiefel manifold is diffeomorphic to the unit fiber bundle over the sphere

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Vn,2 = {(X 1 , X 2 ) : X i ∈ Rn , X i , X j = δi j } ∼ = USn−1 , since (X 1 , X 2 ) → X 2 ∈ TX 1 Sn−1 = X 1 ⊥ , which is a unit tangent vector to Sn−1 at the point X 1 . In particular, the above description as homogeneous manifold of V7,2 gives that US6 ∼ = G 2 /SU(2)l . Again, we can describe explicitly the natural submersions as follows (X 0 , X 1 , X 2 )

Mt π03

(X 0 , X 1 )

V7,2 π31

π36

S6

Ms

X0

1, X 0 , X 1 , X 0 × X 1

or, alternatively, G 2 → V7,2 → S6 , f → ( f (i), f (j)) → f (i), and in this way V7,2 → Ms , ( f (i), f (j)) → f (H). This projection V7,2 → Ms is a fiber bundle with standard fiber SO(4)/SU(2).

3.5 The Complex Projective Quadric  7,2 ∼ Gr = G 2 /H2 ∼ = G 2 /U(2)l ∼ = Q5 Consider now the set Ml := {(w, W ) : W ∈ Ms , w ∈ W, n(w) = 1}.

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Again f ∈ G 2 acts on Ml by means of f · (w, W ) = ( f (w), f (W )). This action is transitive. Namely, for any (w, W ) ∈ Ml , take w  ∈ W ∩ w ⊥ such that n(w  ) = 1. The element (w, w ) ∈ V7,2 can be completed to a Cayley triple (w, w , w  ) with / W = 1, w, w  , w × w  and it is clear that the Ω(w, w , w  ) = 0. Note that w  ∈ automorphism f ∈ Aut(O) which sends (i, j, l) to (w, w , w  ) satisfies f (H) = W . The isotropy subgroup of the element (i, H) ∈ Ml is H2 = { f ∈ Aut(O) : f (H) ⊂ H, f (i) = i}, whose corresponding Lie algebra is evidently h2 in Theorem 1. We also have the isomorphism H2 −→ U(H⊥ , σ) ∼ = U(2), f → f |H⊥ , and, then, as a consequence of this discussion, we get Ml ∼ = G 2 /U(2)l . Now, the l 6 ∼ /U(2) to S /SU(3) is a fiber bundle with natural projection π26 from Ml ∼ G G = 2 = 2 standard fiber the homogeneous manifold SU(3)/U(2); and the natural projection π21 : Ml → Ms is a fiber bundle with standard fiber the homogeneous manifold SO(4)/U(2); which are given by

Reductive Homogeneous Spaces of the Compact Lie Group G 2

(w, W )

Ml π26

S6

49

π21

Ms

w

W.

Remark 10 Let us note the appearance of the manifold Ml as

Fl 1,ass (O0 ) in [38]. There, F. Nakata takes into account the double fibration given by (π26 , π21 ) (with our notations) to prove in [38, Theorem 6.3] that, for any w ∈ S6 , π21 (π26 −1 (w)) is a totally geodesic submanifold of Ms isomorphic to CP 2 , and for any W ∈ Ms , π26 (π21 −1 (W )) is a totally geodesic submanifold of S6 isomorphic to S2 . We can also find this double fibration from the point of view of isoparametric hypersurfaces in [36, Main Theorem].

r 7,2 be the Grassmann More geometrical interpretations are possible for Ml . Let G manifold of the oriented planes in R7 . This manifold can be identified to Ml by

r 7,2 −→ Ml ,  = {X 1 , X 2 } → (X 1 × X 2 , 1, X 1 , X 2 , X 1 × X 2 ), G

r 7,2 . There is a natuwhere {X 1 , X 2 } is an oriented orthonormal basis of  ∈ G

r 7,2 and the above identification is ral action of G 2 on the Grassmann manifold G compatible with the G 2 -actions. The inverse map sends (w, W ) ∈ Ml to the plane W ∩ 1, w ⊥ with the orientation given by {X, w × X } for any X ∈ W ∩ 1, w ⊥ with n(X ) = 1. For instance (i, H) ∈ Ml corresponds to the oriented plane {j, k} ∈

7,2 . Gr

r 7,2 Let us recall the well-known diffeomorphism of the Grassmann manifold G with the following quadric of the complex projective space,  C7 \ {0} 2 6 2 : z1 + · · · + z7 = 0 . Q 5 = [z] ∈ CP = ∼ The identification proceeds as follows

7,2 −→ Q 5 , {X 1 , X 2 } → [X 1 + iX 2 ], Gr and the inverse map sends every [z] ∈ Q 5 to the oriented plane given by {Re(z), Im(z)} ≤ R7 . Hence, we have Q 5 ∼ = G 2 /U(2)l . Remark 11 Every complex quadric Q m inherits a Riemannian metric from the Fubini-Study metric on CP m+1 . Thus, each of these complex hyperquadrics Q m is a symmetric space, but viewed as Q m ∼ = SO(m + 2)/SO(2) × SO(m), which is not the decomposition considered here for m = 5. Recall that the same situation happened for the spheres Sm ∼ = SO(m + 1)/SO(m) and m = 6. By taking advantage of the techniques of Lie triple systems, this description of Q m as a symmetric space has been used to obtain its totally geodesic submanifolds, [16]. For m = 5, the diffeomorphism Q 5 ∼ = G 2 /U(2) was used by R. Bryant in [11]. Recall that A. Gray

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proved in [29] that every almost complex submanifold of the nearly Kähler manifold S6 is minimal and S6 has no 4-dimensional almost complex submanifolds with respect to this nearly Kähler structure. In this context, R. Bryant investigated almost complex curves in S6 , that is, non-constant smooth maps f : M 2 → S6 from a Riemann surface M 2 such that the differential map f ∗ is complex linear. Every complex curve is minimal and the ellipse of curvature {II(u, u) : u ∈ U p M 2 } describes a circle in (T p M 2 )⊥ for every p ∈ M 2 , where II is the second fundamental form. The almost complex curve f : M 2 → S6 is called superminimal when morever {(∇¯ u II)(u, u) : u ∈ U p M 2 } describes a circle too, where ∇¯ denotes the Levi-Civita connection of S6 . In order to study superminimal almost complex curves in S6 , Bryant’s construction in [11] just uses the natural projection Q5 ∼ = G 2 /U(2)l −→ S6 ∼ = G 2 /SU(3). See [15, Sect. 19.1] and references therein for more details.

3.6 The Unknown Quotient Min ∼ = G 2 /SO(3) We consider now the set {(w, W ) : W ∈ Ms , w ∈ W ⊥ , n(w) = 1}, which is in one-to-one correspondence with Min := {(w, W ) : W ∈ Ms , w ∈ W, n(w) = 1}

(15)

by means of the map (w, W ) → (w, W ⊥ ). This manifold has a quite similar description to the above one of Ml , but its geometry has nothing to do, not even the dimension. Again f ∈ G 2 acts on Min as f · (w, W ) = ( f (w), f (W )). This action of G 2 on Min is transitive. Indeed, take (l, Hl) ∈ Min . For any (w, W ) ∈ Min , let X 1 , X 2 be a pair of orthonormal vectors in W ⊥ (in W ⊥ ∩ 1 ⊥ if we think of the orthogonal in O instead of in R7 ). Thus, (X 1 , X 2 , w) is a Cayley triple. The automorphism f ∈ Aut(O) which sends (i, j, l) to (X 1 , X 2 , w) satisfies f (H) = R ⊕ W ⊥ , because {X 1 , X 2 } is a set of generators (as an algebra) of the subalgebra R ⊕ W ⊥ . Hence ( f (l), f (Hl)) = (w, W ). On the other hand, recall that an automorphism f leaves H invariant if and only if it leaves H⊥ = Hl invariant, so that the isotropy subgroup of (l, Hl) ∈ Min is H7 = { f ∈ Aut(O) : f (H) ⊂ H, f (l) = l}, whose related Lie algebra is evidently h7 . We have the isomorphism H7 −→ SO(H0 l, n) = SO(3),

f → f | H0 l ,

Reductive Homogeneous Spaces of the Compact Lie Group G 2

51

given by the restriction map. Therefore, Min is a manifold diffeomorphic to G 2 /SO(3), and the natural submersion reads as G 2 −→ Min ,

f → ( f (l), f (Hl)).

Alternatively, through Eq. (11), we get Mt −→ Min , (X 0 , X 1 , X 2 ) → (X 0 , X 0 , X 0 × X 1 , X 0 × X 2 , X 0 × (X 1 × X 2 ) ). The manifold M7 is a fiber bundle over the sphere S6 and over the symmetric space G 2 /H1 . The projections maps can again be described as follows Min π76

S6

(w, W )

π71

(16) Ms

w

W.

The standard fibers of π76 and π71 are SU(3)/SO(3) and SO(4)/SO(3) ∼ = S3 , respectively. Remark 12 As far as we know, this space Min ∼ = G 2 /SO(3) appears in the literature less frequently than the other G 2 -homogeneous manifolds. To be precise, the only ˜ appearance that we have found occurs recently in [26, 27] as the set M(0) of level t = 0 of the calibration Ω, ˜

r 7,3 : Ω(V ) = t}, M(t) = {V ∈ G where Ω(V ) refers to Ω(X 1 , X 2 , X 3 ) for any orthonormal basis {X 1 , X 2 , X 3 } of the three-dimensional oriented subspace V . Enoyoshi first proved in [26] that all these sets M(t) are diffeomorphic to G 2 /SO(3) for t ∈ (−1, 1) and he also computed

r 7,3 . Remarkably, the level sets M(1) their principal curvatures as hypersurfaces of G and M(−1) are diffeomorphic to the symmetric space G 2 /H1 . The double fibration (16) appears in [27] to prove that, for any w ∈ S6 , the submanifold π71 (π76 −1 (w)) is totally geodesic in Ms and isomorphic to SU(3)/SO(3), and for any W ∈ Ms , the submanifold π76 (π71 −1 (W )) is isomorphic to S3 and totally geodesic and Lagrangian in S6 [27, Theorem 4.2]. Let us recall that a submanifold of S6 is Lagrangian when the nearly Kähler structure of S6 applies the tangent bundle of the submanifold in its normal bundle. The existence of these 5-dimensional totally geodesic submanifolds of Ms agrees with the results of S. Klein [32, Theorem 5.4] on maximal totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2, obtained with very different arguments based on Lie triple systems. As we have concrete expressions of π76 and π71 , we can easily compute π71 (π76 −1 (l)), which coincides with N = {W ∈ Ms : l ∈ W }.

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So N is a totally geodesic submanifold of Ms diffeomorphic to SU(3)/SO(3). An independent algebraic proof is given by the fact that the 5-dimensional vector subspace n = D p, pl : p ∈ H0 ≤ DH,Hl = der(O)1¯ is closed for the triple product [x, y, z] = [[x, y], z]. Besides { f ∈ G 2 : f (l) = l} ∼ = SU(3) (isomorphic to H6 ) acts transitively on N and the isotropy subgroup of Hl ∈ N is of course H7 = { f ∈ G 2 : f (H) ⊂ H, f (l) = l} ∼ = SO(3). Note that, in turn, the manifold N is a symmetric space which has received quite attention in [8], where five-dimensional geometries modeled on N have been studied. An algebraic model for the tangent space of G 2 /SO(3) can be found in [5, Sect. 6], mainly based on linear algebra. Some work in progress is making use of this model in order to find good metrics in Min .

3.7 The Twistor Space of Complex Structures Mr ∼ = G 2 /U(2) r As mentioned above, the symmetric space Ms is an 8-dimensional well-known quaternion-Kähler manifold (see, for instance, [7, Chap. 14] for general notions on quaternion-Kähler manifolds). In particular, there is a subbundle Q ⊂ End T Ms , locally generated by three anticommuting fields of endomorphisms J1 , J2 , J3 = J1 J2 such that Ji2 = −id, which consists of skew-symmetric endomorphisms and such that the Levi-Civita connection preserves Q. The twistor space Z = {0 = A ∈ Q : A2 = −id}, endowed with its natural projection on Ms , is a fiber bundle with standard fiber the two-dimensional sphere S2 . It is a well-known space, for instance Z has a natural complex structure [40, Theorem 4.1] and a Kähler-Einstein metric of positive scalar curvature. We now describe the twistor space Z of the quaternion-Kähler symmetric manifold Ms from our concrete algebraic viewpoint. Recall that, for W a real vector space, an endomorphism J : W → W is said to be a complex structure of W if J 2 = −id W . In this case, W can be endowed with a complex vector space structure by taking as a scalar multiplication C × W → W , (α + iβ)w = αw + J (βw). If we have fixed a scalar product on W , we will say that the complex structure J is metric if J ∈ SO(W ).3 We consider N4 = {(W, J ) : W ∈ Ms , J metric complex structure on W }, where the subspaces W of R7 are endowed with the scalar product inherited from the usual one on R7 . It is evident that G 2 acts on N4 by f · (W, J ) = ( f (W ), f J f −1 ). 3

Note that, if J 2 = −id, then J ∈ SO(W ) if and only if J ∈ so(W ).

Reductive Homogeneous Spaces of the Compact Lie Group G 2

53

The derivation dil , considered in Eq. (6), restricts to a complex structure dil |H⊥ of H⊥ , since ql → (iq)l → −ql. Besides, dil is metric since n(ql) = n((iq)l). In other words, we have (Hl, dil ) ∈ N4 . The subgroup of G 2 which fixes this element in N4 is H(Hl,dil ) = { f ∈ Aut(O) : f (H⊥ ) ⊂ H⊥ , f dil |H⊥ = dil f |H⊥ }. Let us check that this group coincides with H4 = { f ∈ Aut(O) : f τ = τ f }, the unique connected Lie subgroup with related Lie algebra h4 . (Recall that the automorphism τ was defined in Eq. (4).) Consider f ∈ H(Hl,dil ) , and take into account that f (H) ⊂ H and τ |H = id to get f τ = τ f on H. Also, we have f τ = τ f in H⊥ , since dil |H⊥ = τ |H⊥ (funny fact, since dil ∈ der(O) but τ ∈ Aut(O)). Hence f τ = τ f . Conversely, if f is an automorphism commuting with τ , then f (H) ⊂ Fix(τ ) = H and hence f (H⊥ ) ⊂ H⊥ too. Again the fact dil |H⊥ = τ |H⊥ finishes the discussion. In particular the restriction f → f |H provides the isomorphism between the isotropy subgroup H4 = { f ∈ Aut(O) : f τ = τ f } and U(H, σ). In order to study whether the action of G 2 on N4 is transitive or not, the results in the subsequent lemma are useful. Lemma 4 Fix (W, J ) ∈ N4 . (a) For any X ∈ W with n(X ) = 1 and any Y ∈ W ∩ X, J (X ) ⊥ with n(Y ) = 1, we have W = X, Y, J (X ), J (Y ) , W ⊥ = X × Y, X × J (X ), Y × J (X ) .

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(b) For X, Y as above, there is α ∈ {±1} such that (X × Y ) × J (X ) = αJ (Y ). If α = 1, the automorphism of O which sends (i, l, jl) to the Cayley triple (X × J (X ), X, Y ) satisfies that f · (Hl, dil ) = (W, J ). Proof Recall (for instance, [20, Eqs. (3.2), (4.13) and (4.14)]) that the cross product is anticommutative and satisfies X Y = X × Y,

X × (X × Y ) = −n(X )Y,

X × (Y × Z ) = −Y × (X × Z ), (19) whenever X, Y, Z ∈ R7 ≡ O0 are pairwise orthogonal. Since J ∈ SO(W, n) and J 2 = −id, then n(X, J (X )) = n(J (X ), J 2 (X )) = −n(X, J (X )), so that J (X ) is orthogonal to X for any X ∈ W . (a) It is clear that we can choose Y ∈ W ∩ X, J (X ) ⊥ with n(Y ) = 1. Let us check that {X, Y, J (X ), J (Y )} is an orthonormal frame in W , in particular a basis. Indeed, n(J (Y )) = n(Y ) = 1 and J (Y ) is orthogonal to Y as above. Also, we have n(J (Y ), X ) = n(J 2 (Y ), J (X )) = −n(Y, J (X )) = 0 and n(J (Y ), J (X )) = n(J 2 (Y ), J 2 (X )) = n(−Y, −X ) = 0. As W ∈ Ms , we have

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Ω(W, W, W ) = 0 and so W × W ⊂ W ⊥ (orthogonal in R7 ≡ O0 ). To get (18), we only need to prove that {X × Y, X × J (X ), Y × J (X )} are linearly independent, for instance checking that they are orthogonal. This is straightforward: n(X × Y, X × J (X )) = n(X Y, X J (X )) = n(X )n(Y, J (X )) = 0 and the same argument applies to the other two cases. (b) Clearly, (X × J (X ), X, Y ) is a Cayley triple: Ω(X × J (X ), X, Y ) = n(X × J (X ), X × Y ) = n(X )n(J (X ), Y ) = 0, so that we can take the automorphism f ∈ Aut(O) determined by f (i) = X × J (X ), f (l) = X and f (jl) = Y . Now we use Eq. (19) to compute the images under f of the remaining basic elements: il → J (X ), j = −(jl)l → X × Y,

kl = j(il) → (X × Y ) × J (X ), k = −(il)(jl) → Y × J (X ).

Note that, again keeping in mind Eq. (19), the element (X × Y ) × J (X ) = X × (J (X ) × Y ) = −Y × (J (X ) × X ) is orthogonal to X × Y , X × J (X ) and Y × J (X ) so that it belongs to (W ⊥ )⊥ = W . But it is also orthogonal to X , Y and J (X ). Hence, there is α ∈ R such that (X × Y ) × J (X ) = αJ (Y ). Thus ( f (l), f (il), f (jl), f (kl)) = (X, J (X ), Y, αJ (Y )), and, in particular, f (H⊥ ) = W . Taking norms, α2 = n(X )n(Y )n(J (X )) = 1 and necessarily α = ±1. In case α = 1, we immediately check that f τ (l) = J (X ) = J f (l), f τ (jl) = J (Y ) = J f (jl),

f τ (il) = −X = J 2 (X ) = J f (il), f τ (kl) = −Y = J f (kl).

Hence, we get f τ |H⊥ = J f |H⊥ , as required.  Note that the complex structure J = dil |H⊥ satisfies the following additional property: J (X ) × J (Y ) = ((iq)l)((i p)l) = −(i p)(iq) = − p¯ ¯iiq = − pq ¯ = (ql)( pl) = X × Y,

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for any X = ql, Y = pl ∈ Hl = H⊥ . Nevertheless, (Hl, dir ) ∈ N4 does not preserve the cross product W × W ⊂ W ⊥ . This says that the action of G 2 on N4 is not transitive and leads us to consider  Mr :=

(W, J ) : W ∈ Ms , J ∈ SO(W, n), J 2 = −id, J (X ) × J (Y ) = X × Y ∀X, Y ∈ W

.

Since the Lie group G 2 preserves ×, the restriction of the action of G 2 on N4 to Mr is still an action, but now the action is transitive. Indeed, if (W, J ) ∈ Mr , take X and Y as in Lemma 4. Then (19)

(X × Y ) × J (X ) = (J (X ) × J (Y )) × J (X ) = J (Y ) and item b) allows us to find a concrete automorphism f ∈ G 2 with f · (Hl, dil ) = (W, J ). Hence, taking into account that the isotropy group at (Hl, dil ) is H4 ∼ = U(H, σ) ∼ = G 2 /U(2)r . According to [9, Chap. 13], the mani= U(2), we get Mr ∼ fold Mr can be identified with the twistor space Z of the quaternion-Kähler manifold G 2 /SO(4). In order to write the good complex structures involved in the definition of Mr without reference to an outer object, let us consider for any W ∈ Ms the ternary product given by { , , } : W × W × W → W, {X, Y, Z } := (X × Y ) × Z . Now, we consider Aut(W, n, { , , }) := {J ∈ SO(W, n) : {J (X ), J (Y ), −} = {X, Y, −} ∀X, Y ∈ W }, and then it is easy to check that Mr = {(W, J ) : W ∈ Ms , J ∈ Aut(W, n, { , , }), J 2 = −id}.

(20)

3.8 The Twistor Space of Quaternionic Structures Pr ∼ = G 2 /SU(2) r For any real vector space W , a pair (J, K ) of endomorphisms J, K : W → W is said to be a quaternionic structure on W if J 2 = K 2 = −id W and J K = −K J . In other words, J , K and J K are three complex structures which anticommute. In this case, W can be endowed with a quaternionic vector space structure by taking as a scalar multiplication W × H → W , w(α + βi + γj + δk) := α + β K J (w) + γ J (w) + δK (w).

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The group G 2 acts on the set ⎧ ⎫ ⎨ (W, (J, K )) : W ∈ Ms ⎬ (J, K ) quaternionic structure on W . Pr := ⎩ ⎭ J, K ∈ Aut(W, n, { , , })

(21)

The action of f ∈ Aut(O) is given by f · (W, (J, K )) = ( f (W ), ( f J f −1 , f K f −1 )), well defined since f preserves the ternary product { , , }. As dal dbl (q1 + q2 l) = l |H⊥ and therefore (H⊥ , (dil , djl )) ∈ Pr . Let us (abq2 )l, then we have dal dbl |H⊥ = dab check that the subgroup of G 2 which fixes the element (H⊥ , (dil , djl )) ∈ Pr has just h5 as a Lie algebra, so that it worths the name H5 . Indeed, every element f in the group { f ∈ Aut(O) : f (H⊥ ) ⊂ H⊥ , f dil |H⊥ = dil f |H⊥ , f djl |H⊥ = djl f |H⊥ } commutes with dkl |H⊥ = dil |H⊥ djl |H⊥ too, so that the Lie algebra is {d ∈ der(O)0¯ : ddal |H⊥ = dal d|H⊥ ∀a ∈ H0 } = {d ∈ der(O)0¯ : [d, hl ] = 0} = h5 , because of course any d ∈ der(O)0¯ commutes with dal |H ≡ 0. This tells us that we have the isomorphism H(H⊥ ,(dil ,djl )) = H5 ∼ = SU(H, σ),

f → f |H ,

as happened at the Lie algebra level. The transitivity of the action of G 2 on Pr is a consequence of the ensuing lemma. Lemma 5 Fix (W, (J, K )) ∈ Pr . (a) For any X ∈ W with n(X ) = 1, we have W = X, J (X ), K (X ), J K (X ) , W ⊥ = X × J (X ), X × K (X ), X × J K (X ) ;

(b) The automorphism of O which sends the Cayley triple (i, j, l) to the Cayley triple (X × J (X ), X × K (X ), X ) satisfies that f · (H⊥ , (dil , djl )) = (W, (J, K )). Proof In order to check that {X, J (X ), K (X ), J K (X )} is a basis of W , it is enough to check that it is an orthonormal set. Indeed, the four elements have norm 1 since J, K , J K ∈ SO(W, n); and X is orthogonal to the others as in Lemma 4. For another pair of elements, n(J (X ), K (X )) = n(J 2 (X ), J K (X )) = −n(X, J K (X )) = 0. Besides, as Ω(W, W, W ) = 0, then W × W ⊂ W ⊥ and we get (a). Retain Eq. (19) for the remaining computations. We have a Cayley triple since Ω(X × K (X ), X, X × J (X )) = n(−J (X ), X × K (X )) ∈ n(W, W ⊥ ) = 0,

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so that we can consider the automorphism f ∈ Aut(O) determined by f (i) = X × J (X ), f (j) = X × K (X ), f (l) = X . The remaining images necessarily are f (il) = (X × J (X )) × X = J (X ), f (jl) = K (X ), f (k) = f ((jl)(il)) = K (X ) × J (X ), f (kl) = X × (J (X ) × K (X )). Let us check that f (kl) = J K (X ). Of course there is α ∈ R such that f (kl) = αJ K (X ), since X × (J (X ) × K (X )) ∈ W is orthogonal to X , J (X ) and K (X ). In particular, we have f (Hl) = W . Taking norms gives α ∈ ±1, but, using besides that J ∈ Aut(W, n, { , , }) we can conclude α = 1 as a consequence of the following computation:   J (X ) × K (X ) = −X × X × (J (X ) × K (X )) = −X × αJ K (X ) = −J (X ) × αJ 2 K (X ). Having established f (kl) = J K (X ), it is easily deduced that f dil = J f and f djl =  K f in H⊥ . Hence, as H5 ∼ = SU(2), we call H5 = SU(2)r and the transitivity gives Pr ∼ = r G 2 /SU(2) . The natural projections can be explicitly given by Pr −→ Mr −→ Ms ,

(W, (J, K )) → (W, J ) → W,

which are fiber bundles with standard fiber U(2)/SU(2) and SO(4)/U(2), respectively. There is also a natural projection Mt → Pr that sends (X 0 , X 1 , X 2 ) ∈ Mt to (W, (J, K )) for W = X 0 , X 0 × X 1 , X 0 × X 2 , X 0 × (X 1 × X 2 ) ,

J = L X1 , K = L X2 ,

where L X : W → W denotes L X (Y ) = X × Y if X ∈ W ⊥ . ∼ Pr has a very rich geometric structure. Remark 13 The quotient G 2 /SU(2)r = Namely, G 2 /SU(2)r is a 3-Sasakian homogeneous manifold [9, Chap. 13]. A general study of invariant linear connections on 3-Sasakian homogeneous manifolds can be found in [21]. At this point, we would like to do a personal remark. The manifold G 2 /SU(2)r played a key role in the motivation of [21] and its algebraic structure inspired the notion of 3-Sasakian data there.

3.9 The Irreducible Isotropy A G-homogeneous manifold M = G/H is called an isotropy-irreducible space if the linear isotropy representation of H is irreducible. In this case, Wolf proved in [44] that M admits a unique G-invariant Riemannian metric (up to homotheties),

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which is necessarily an Einstein metric. In the same paper, Wolf classified the Ghomogeneous Riemannian manifolds G/H such that the connected component of the identity in H has an irreducible isotropy representation, called strongly isotropy irreducible spaces (equivalent if H is connected). This classification can be consulted also in [7, Chap. 7, Sect. H], and precisely Table 6, p. 203, contains our G 2 /SO(3), although no more geometric or topological information is provided, only a mention to [22] to explain that the fact that h is a maximal subalgebra of g is sufficient to characterize the embedding. Our search for another reference of G 2 /SO(3) has been unsuccessful. From an algebraic viewpoint, Lie-Yamaguti algebras (also called generalized Lie triple systems) are binary-ternary algebras (m, •, [ , , ]) satisfying a list of six identities, defined precisely to translate reductive homogeneous spaces to an algebraic structure. Similarly, the Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. The work [4] is devoted to study irreducible LieYamaguti algebras of generic type. The generic case occurs when both the inner derivation algebra h(m) and the standard enveloping algebra g(m) are simple. Of course this is our situation. Some general facts are that m coincides with h(m)⊥ , the orthogonal with respect to the Killing form of g(m), and that h(m) is a maximal subalgebra of g(m). Although the reductive pair (G 2 , A1 ) appears explicitly in [4, Theorem 5.1.ii)] and in [4, Table 9], the work does not provide a concrete description and the classification is transferred from the complex case. (Recall that we have provided in Proposition 2 a description of a principal subalgebra of g2 in terms of derivations of the octonions, but not a description of its -unique- invariant complement.) A concrete description (valid for the complex and the split case, but not for the compact one) appears in [17, Sect. 6], where Dixmier considers simultaneously several simple nonassociative algebras defined by the transvection of binary forms. The paper contains some typos in the scalars involved in the G 2 -construction, but a corrected version appears in [6, Theorem 4.6] as follows. Denote by Vn the complex vector space of the homogeneous polynomials of degree n in two variables X and Y . For any f ∈ Vn , g ∈ Vm , consider the transvection (n − q)! (m − q)!  (−1)i ( f, g)q = n! m! i=0 q

  ∂q f ∂q g q ∈ Vm+n−2q . i ∂ X q−i ∂Y i ∂ X i ∂Y q−i

The complex Lie algebra of type G 2 can be obtained as the standard enveloping algebra of the Lie-Yamaguti algebra m = V10 , with binary and ternary products given by f 1 • f 2 = ( f 1 , f 2 )5 ,

[ f1 , f2 , f3 ] =

25 (( f 1 , f 2 )9 , f 3 )1 , 378

if f i ∈ V10 . Here h(m) ∼ = (V2 , ( , )1 ) turns out to be a principal three-dimensional subalgebra of g(m). This Lie-Yamaguti algebra m = V10 has received some attention

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in [10], where its polynomial identities of low degree have been studied. Thus (V2 ⊕ V10 , [ , ]) is a exceptional Lie algebra of type G 2 for the bracket [g1 , g2 ] := (g1 , g2 )1 , [g, f ] := 5(g, f )1 , 5 [ f 1 , f 2 ] := 378 ( f 1 , f 2 )9 + ( f 1 , f 2 )5 ,

(22)

if g, gi ∈ V2 and f, f i ∈ V10 . The element h = 4X Y ∈ V2 is ad-diagonalizable with integer eigenvalues, since [h, X k Y 2−k ] = (2 − 2k)X k Y 2−k and [h, X k Y 10−k ] = (10 − 2k)X k Y 10−k . So, although the construction remains valid for the real field, the obtained algebra is the split algebra of type G 2 , not the compact one. The following result is key for our purposes. Proposition 3 There is, up to conjugation, only one principal three-dimensional subalgebra of the compact algebra g2 . This is very well-known for the complex case, but we include one proof of the real case for completeness, due to the lack of a suitable reference. It essentially arose from conversations with A. Elduque. Proof We dealt with the existence in Sect. 2.6. Now assume that we have two decompositions g2 = s ⊕ m = s ⊕ m with s and s principal subalgebras, [s, m] ⊂ m and [s , m ] ⊂ m , and we are going to provide an automorphism of g2 which sends s to s . Denote by πs and πm the projections of g2 = s ⊕ m on the subspaces s and m respectively, and similarly by πs and πm the projections with respect to the other decomposition. Write, for short, [x, y]t = πt ([x, y]), for any x, y ∈ g2 and any of our subspaces t of g2 . First, g2 is compact, so that the three-dimensional algebras s and s are necessarily isomorphic to su2 (there are no nilpotent elements in g2 ). So there exists an isomorphism of Lie algebras ϕ : s → s . Second, there is only one irreducible s-module of dimension 11. Hence, there is a bijective linear map ρ : m → m such that ρ([s, x]) = [ϕ(s), ρ(x)] for any s ∈ s and x ∈ m. Third, dimR Homs (m ⊗ m, m) = 1 since this is the situation after complexifying (where the set of homomorphisms is spanned by ( , )5 ). Two s-invariant bilinear maps from m × m to m are ρ−1 ◦ πm ◦ [ , ]|m ×m ◦ (ρ × ρ) and πm ◦ [ , ]|m×m , which differ into a scalar, so that there is α ∈ R such that [ρ(x), ρ(y)]m = αρ([x, y]m ) for all x, y ∈ m. Changing ρ by αρ , we can assume that α = 1. Fourth, dimR Homs (m ⊗ m, s) = 1 since this is the situation after complexifying (where the set of homomorphisms is spanned by ( , )9 ). Two s-invariant bilinear maps from m × m to s are ϕ−1 ◦ πs ◦ [ , ]|m ×m ◦ (ρ × ρ) and πs ◦ [ , ]|m×m , which differ into a scalar, so that there is β ∈ R such that [ρ(x), ρ(y)]s = βϕ([x, y]s ) for all x, y ∈ m. Now define  : g2 → g2 , (s + x) := ϕ(s) + ρ(x),

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for any s ∈ s and x ∈ m. Let us check that  is an automorphism of the algebra, with (s) = s . From the above it is clear that, for any s, s1 , s2 ∈ s, x, x1 , x2 ∈ m, ([s1 , s2 ]) = [(s1 ), (s2 )], ([s, x]) = [(s), (x)], ([x1 , x2 ]s ) = β −1 [(x1 ), (x2 )]s , ([x1 , x2 ]m ) = [(x1 ), (x2 )]m ;

(23)

so that the map  will be an automorphism if and only if β = 1. Fifth, use the Jacobi identity. 3 Denote, with the indices modulo 3, the Jacobian [[xi , xi+1 ], xi+2 ], which is 0 for any choice of eleoperator by J (x1 , x2 , x3 ) = i=1 ments xi in g2 . For any x1 , x2 , x3 ∈ m, Eq. (23) give πm ◦ ([[x1 , x2 ], x3 ]) = πm ◦ ([[x1 , x2 ]m , x3 ]) + πm ◦ ([[x1 , x2 ]s , x3 ]) = [[(x1 ), (x2 )]m , (x3 )]m + β −1 [[(x1 ), (x2 )]s , (x3 )].

At the same time we can apply Jacobi identity to get 0 = πm (J ((x1 ), (x2 ), (x3 )) − (J (x1 , x2 , x3 ))) 3 [[(xi ), (xi+1 )]s , (xi+2 )]. = (β −1 − 1) i=1 3 The expression i=1 [[(xi ), (xi+1 )]s , (xi+2 )] cannot be identically 0 in m, since it does not vanishes after complexification. Indeed, taking in mind Eq. (22), choosing (x1 ) = f 1 = X 10 , (x2 ) = f 2 = Y 10 and (x3 ) = f 3 = X Y 9 , the above expression coincides with  25  25  1 5 (X Y, X Y 9 )1 − (X 2 , Y 10 )1 = X Y 9  = 0. (( f i , f i+1 )9 , f i+2 )1 = 378 378 10 252 3

i=1

Thus β −1 − 1 = 0 and  is indeed an automorphism of g2 .



We are now prepared to give a model of the homogeneous space G 2 /SO(3)irr . Consider the set (24) Mirr := {s ≤ g2 : s principal subalgebra}. Recall that the group G 2 and the adjoint group Aut(g2 ) are isomorphic and a precise isomorphism is given by f ∈ G 2 → f˜ ∈ Aut(g2 ),

f˜(d) = f d f −1 for all d ∈ g2 .

Hence G 2 acts on Mirr by f · s = f˜(s) = { f s f −1 : s ∈ s}, which is a principal subalgebra of g2 if s so is. By Proposition 3, this action is transitive. If we denote by Hs the isotropy group of a fixed principal subalgebra s ∈ Mirr (for instance, s = h8 ), let us check that its Lie algebra hs coincides with s, which would prove that hs is principal, as required. As Hs = { f ∈ G 2 : f s f −1 ∈ s ∀s ∈ s}, then its Lie algebra is hs = {d ∈ g2 : [d, s] ∈ s ∀s ∈ s}, the normalizer of s, which obviously contains

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s. Besides there is an absolutely irreducible s-module m such that g2 = s ⊕ m. The fact [hs , s] ⊂ s implies that s acts trivially on hs /s, which is an s-submodule of the irreducible module g2 /s ∼ = m. It follows that hs /s = 0, that is, hs = s. To summarize, we can identify Mirr with G 2 /SO(3)irr for SO(3)irr = H8 , and hence, Mirr described in Eq. (24) is endowed with a manifold structure such that we can think of Mirr as the isotropy irreducible Wolf space.

4 Conclusions The main purpose in this work has been to provide a complete, concrete and unified panoramic of the reductive G 2 -homogeneous spaces. We have tried that these homogeneous spaces were explicit and the relations among them were clear, enclosing also detailed descriptions of the projections of the family of fiber bundles we can construct with the reductive G 2 -homogeneous spaces. We think the geometric prerequisites are modest. The numerous references do not intend to serve as a background, but only to understand the relations among the appearing manifolds and to enrich the project, illustrating the interplay between Algebra and Geometry. Some work in progress is in the following direction. Along this paper, we have focussed on the compact real form of the Lie group G 2 . As it is well-known, there is another real form of the complex Lie algebra g2 , namely, the split Lie algebra g2,2 of derivations of the split octonion algebra Os . Its Lie group of automorphisms, G 2,2 = Aut(Os ), is non-compact and not simply connected. Taking into account the wide generality of the results in [6], it is possible to find a family of G 2,2 8 described in manifolds closely related to the G 2 -homogeneous manifolds {Mi }i=1 this paper. This task is not direct, for instance there will be more than 8 reductive quotients, since Os possesses different kinds of quaternion subalgebras, not all of them conjugated. This new longer family is very different from ours. The quotients are a priori non-compact, but at the same time share some remarkable properties with our compact family. In fact, the involved homogeneous spaces are reductive, and when we decompose them as a sum of irreducible modules, these decompositions do not coincide with the ones for the compact case, but their complexifications do. Thus, the dimensions of the vector spaces providing invariant metrics, invariant connections, or invariant connections with additional properties, are equal. In general, the knowledge of the specific modules appearing in the decompositions of g2 as a sum of irreducible hi -modules for any i = 1, . . . , 8 (computed in [6]) provides a very useful tool to study the corresponding homogeneous manifolds, which is not even fully exploited for all our Mi ’s.

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27. Enoyoshi, K., Tsukada, K.: Lagrangian submanifolds of S 6 and the associative Grassmann manifold. Kodai Math. J. 43(1), 170–192 (2020) 28. Fernández, M., Gray, A.: Riemannian manifolds with structure group G 2 . Ann. Mat. Pura Appl. 132, 19–45 (1982) 29. Gray, A.: Six dimensional almost complex manifolds defined by means of three-fold vector cross products. Tohoku Math. J. 21(2), 614–620 (1969) 30. Harvey, F.R.: Spinors and calibrations. Perspectives in Mathematics, vol. 9, pp. xiv+323. Academic Press Inc., Boston (1990). ISBN: 0-12-329650-1 31. Hitchin, N.: The geometry of three-forms in six dimensions. J. Differ. Geom. 55, 547–576 (2000) 32. Klein, S.: Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2. Osaka J. Math. 47(4), 1077–1157 (2010) 33. Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959) 34. Leites, D., Lozhechnyk, O.: Inverses of Cartan matrices of Lie algebras and Lie superalgebras. Linear Algebr. Appl. 583, 195–256 (2019) 35. Mayanskiy, E.: The subalgebras of G 2 . arXiv:1611.04070 (2016) 36. Miyaoka, R.: Geometry of G 2 orbits and isoparametric hypersurfaces. Nagoya Math. J. 203, 175–189 (2011) 37. Nakata, F.: Homotopy groups of G 2 /Sp(1) and G 2 /U (2). Contemporary Perspectives in Differential Geometry and Its Related Fields, pp. 151–159, 2018. World Scientific Publishing, Hackensack (2017) 38. Nakata, F.: The Penrose type twistor correspondence for the exceptional simple Lie group G2 (Aspects of submanifolds and other related fields). Notes of the Institute of Mathematical Analysis, vol. 2145, pp. 54–68 (2020). https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/ 2433/255001/1/2145-08.pdf 39. Onishchik, A.L., Vinberg, È.B. (eds.): Lie groups and Lie algebras, III. Encyclopaedia of Mathematical Sciences, vol. 1, pp. iv+248. Springer, Berlin (1994). ISBN: 3-540-54683-9 40. Salamon, S.: Quaternionic Kähler manifolds. Inven. Math. 67(1), 143–171 (1982) 41. Schafer, R.D.: An introduction to nonassociative algebras. Pure and Applied Mathematics, vol. 22, pp. x+166. Academic Press, New York (1966) 42. Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, vol. 94, pp. ix+272. Springer, New York-Berlin (1983) 43. Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033–1047 (1965) 44. Wolf, J.A.: The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math. 120, 59–148 (1968) 45. Zhevlakov, K.A., Slin’ko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are nearly associative. Translated from the Russian by Harry F. Smith. Pure and Applied Mathematics, vol. 104, pp. xi+371. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1982)

On Certain Algebraic Structures Associated with Lie (Super)Algebras Noriaki Kamiya

Dedicated to the 60th birthday of Professor Alberto Elduque

Abstract In this paper we exhibit a survey of constructions of Lie (super)algebras associated with certain triple systems, several examples and a historical story in nonassociative algebras (in particular, Jordan algebras). Keywords Jordan algebras · Triple systems · Lie (super)algebras

1 Introduction This note is to deal with a certain survey based on our papers mainly, to give several examples of triple systems and Lie (super)algebras, furthermore to describe a history of Jordan rivers in nonassociative algebra from the author’s viewpoint. On the other hand, it seems that this work with respect to Jordan and Lie structures is in close contact with a symmetric (super)space equipped with complex structure, since the tangent space of the symmetric (super)space is a δ-Lie triple system (δ = ±1, the definition follows Sect. 2). From mathematical history’s viewpoint, the concept discussed here first appeared with a class of nonassociative algebras, that is commutative Jordan algebras, which was the defining subspace g−1 in the Tits-Kantor-Koecher (for short TKK) construction of 3-graded Lie algebras g = g−1 ⊕ g0 ⊕ g1 , such that [gi , g j ] ⊆ gi+ j . Nonassociative algebras are rich in algebraic structures, and they provide an important common ground for various branches of mathematics, not only for pure algebra and N. Kamiya (B) University of Aizu, Aizuwakamatsu 965-8580, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_4

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differential geometry, but also for representation theory and algebraic geometry (for example, [11, 46, 59, 61, 64]). Specially, the concept of nonassociative algebras such as Jordan and Lie (super)algebras plays an important role in many mathematical and physical subjects [5, 10–13, 15, 22, 26, 28], [29, 33–35, 45, 54, 55, 60, 65]. We also note that the construction and characterization of these algebras can be expressed in terms of the notion of triple systems [1–4, 6–8, 20, 23, 24, 40–45, 50–53, 56–58] by using the standard embedding method [22, 48, 49, 57, 63]. In particular, the generalized Jordan triple system of second order, or (−1, 1)-Freudenthal Kantor triple system (for short (−1, 1)-FKTS), is a useful concept [13–21, 41–44, 47] for the constructions of simple Lie algebras, while the (−1, −1)-FKTS plays the same role [6, 22, 25, 27, 30, 32, 33] for the construction of Lie superalgebras, while the δ-Jordan Lie triple systems act similarly for that of Jordan superalgebras [23, 24, 56]. Specially, we have constructed a model of basic Lie superalgebras D(2, 1; α), G(3) and F(4) [22, 25, 27]. As a final comment of this introduction, we provide well-known results due to O. Loos as follows; if A is a commutative Jordan algebra with an unital element e, that is, satisfying (x y)x 2 = x(y(x 2 )), and x y = yx, then the triple product given by {x yz} = (x y)z + x(yz) − y(x z) defines a Jordan triple system equipped with {xey} = x y, i.e., it satisfies the two relations {x y{abc}} = {{x ya}bc} − {a{yxb}c} + {ab{x yc}} (this relation is often called a fundamental identity), and {x yz} = {zyx} (this relation is called a commutative identity, since x y = {xey} = {yex} = yx) and next the new triple product [x yz] given by [x yz] = {x yz} − {yx z} defines a Lie triple system. Briefly summarizing this article, we will generalize these results and exhibit examples of Lie (super)algebras associated with generalized Jordan triple systems. Toward to its applications, in particular, we will give a construction of symmetric (super)spaces with an almost complex structure (i.e., equipped with Nijenhuis operator). Roughly describing, we have an illustration for our concept. Algebraic structures ⇐⇒ Geometric structures. For example, it seems that there are certain algebraic structures associated with symmetric, R-symmetric, homogeneous spaces, totally geodesic manifold, and symmetric domains, etc.

2 Definitions and Results In this paper triple systems have finite dimension being defined over a field  of characteristic =2 or 3, unless otherwise specified. In order to render the paper as

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self-contained as possible, we recall first the definition of a generalized Jordan triple system of second order (for short GJTS of 2nd order) [41–45]. A vector space V over a field  endowed with a trilinear operation V × V × V → V , (x, y, z) −→ (x yz) is said to be a GJTS of 2nd order if the following conditions are fulfilled: (ab(x yz)) = ((abx)yz) − (x(bay)z) + (x y(abz)),

(1)

K (K (a, b)x, y) − L(y, x)K (a, b) − K (a, b)L(x, y) = 0,

(2)

where L(a, b)c := (abc) and K (a, b)c := (acb) − (bca). A Jordan triple system (for short JTS) satisfies (1) and the following condition (abc) = (cba), i.e., K (a, c)b = 0.

(3)

The JTS is a special case in the GJTS of 2nd order since K (x, y) ≡ 0. We next can generalize the concept of GJTS of 2nd order as follows (see [13, 14, 18, 22, 28, 36, 63] and the earlier references therein). For ε = ±1 and δ = ±1, a triple product that satisfies the identities (ab(x yz)) = ((abx)yz) + ε(x(bay)z) + (x y(abz)),

(4)

K (K (a, b)x, y) − L(y, x)K (a, b) + εK (a, b)L(x, y) = 0,

(5)

where L(a, b)c := (abc),

K (a, b)c := (acb) − δ(bca),

(6)

is called an (ε, δ)−Fr eudenthal − K antor tri ple system (for short (ε, δ)-FKTS). An (ε, δ)-FKTS is said to be unitary if I d ∈ {K (a, b)}span . A triple system satisfying only the identity (4) is called a generalized FKTS (for short GFKTS), while the identity (5) is called the second order condition (this condition is needed to the construction of 5-graded Lie (super)algebras). Remark 1 From the relation (6), we note that K (b, a) = −δK (a, b).

(7)

A triple system is called a (α, β, γ)-triple system associated with a bilinear form if (x yz) = αx, y z + βy, z x + γz, x y, where x, y is a bilinear form such that x, y = κy, x , κ = ±1, α, β, γ ∈ . In the next Sects. 3 and 4, we will mainly consider this type of triple system. An (ε, δ)-FKTS is said to be balanced if there is a bilinear form x, y ∈ Φ ∗ such that K (x, y) = x, y I d, that is, dim {K (x, y)}span = 1 holds.

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Remark 2 We note that a balanced triple system (i.e., it fulfills K (x, y) = x, y I d) is unitary, since I d ∈ {K (x, y)}span . Triple products are denoted by (x yz), {x yz}, [x yz] and < x yz > upon their suitability. Remark 3 We note that the concept of GJTS of 2nd order coincides with that of (−1, 1)-FKTS. Thus we can construct the corresponding Lie algebras by means of the standard embedding method [6, 13–19, 21, 22, 25, 27, 43]. For δ = ±1, a triple system (a, b, c) → [abc], a, b, c ∈ V is called a δ-Lie triple system (for short δ-LTS) if the following three identities are fulfilled [abc] = −δ[bac], [abc] + [bca] + [cab] = 0, [ab[x yz]] = [[abx]yz] + [x[aby]z] + [x y[abz]],

(8)

where a, b, x, y, z ∈ V . An 1-LTS is a LTS while a −1-LTS is an anti-LTS, by [14]. Note that the set L(V, V ) of all left multiplications L(x, y) of V is a Lie subalgebra of Der V , where we denote by L(x, y)z = [x yz]. Proposition 1 ([13–16, 22]) Let (U (ε, δ), < x yz >) be an (ε, δ)-FKTS. If J is an endomorphism of U (ε, δ) such that J < x yz >=< J x J y J z > and J 2 = −εδ I d, then (U (ε, δ), [x yz]) is a LTS (if δ = 1) or an anti-LTS (if δ = −1) with respect to the product [x yz] :=< x J yz > −δ < y J x z > +δ < x J zy > − < y J zx > .

(9)

Remark 4 Note that for the case of ε = −1, δ = 1 and K (x, y) = 0, we have a special case in Prop.1.1, that is, it implies that J = I d, {x yz} is the JTS and [x yz] = {x yz} − {yx z} is the LTS described in the Introduction. Corollary 1 ([13]) Let U (ε, δ) be an (ε, δ)-FKTS. Then the vector space T (ε, δ) = U (ε, δ) ⊕ U (ε, δ) becomes a LTS (if δ = 1) or an anti-LTS (if δ = −1) with respect to the triple product        a c e e L(a, d) − δL(c, b) δK (a, c) = . (10) −εK (b, d) ε(L(d, a) − δL(b, c)) b d f f Thus we can obtain the standard embedding Lie algebra (if δ = 1) or Lie superalgebra (if δ = −1), L(U (ε, δ)) = D(T (ε, δ), T (ε, δ)) ⊕ T (ε, δ), associated with T (ε, δ) where D(T (ε, δ), T (ε, δ)) is the set of inner derivations of T (ε, δ);  D(T (ε, δ), T (ε, δ)) :=

L(a, b) δK (c, d) −εK (e, f ) εL(b, a)

 , span

On Certain Algebraic Structures Associated with Lie (Super)Algebras

T (ε, δ) :=

69

   x  x, y ∈ U (ε, δ) . y  span

We use the following notation: k := {K (x, y) ∈ End U (ε, δ)|x, y ∈ U (ε, δ)}span and {E F G} := E F G + G F E, ∀E, F, G ∈ k. Then, we may make the structure of a JTS k with respect to the triple product {E F G} ∈ k, hence [E F G] = {E F G} − {F E G} has a structure of LTS [20, 31]. We next introduce an analogue of Nijenhuis tensor in differential geometry defined by N (X, Y ) = [J X, J Y ] − J [J X, Y ] − J [X, J Y ] + J 2 [X, Y ], ∀X, Y ∈ T (ε, δ) 

 0 ε , that is, J 2 = −εδ I d. Hence the case of εδ = 1 has a structure −δ 0 of almost complex [13–15, 17, 36]. and J =

Proposition 2 ([36]) Let U be a (ε, δ)-FKTS, T (ε, δ) be the δ-LTS and L(U ) be the standard embedding Lie (super)algebra associated with U. Then the following are equivalent: (i) N (X, Y ) = 0, ∀X, Y ∈ T (ε, δ), (ii) εδL(y, x) − εL(x, y) = K (x, y), ∀x, y ∈ U (ε, δ). This J ∈ End T (ε, δ) may generalize on J˜ ∈ End L(U ) defined by J˜ := J D(X, Y )J −1 ⊕ J Z , ∀X, Y, Z ∈ T (ε, δ). Then we note that J˜ has an interesting property, for example, an automorphism of L(U ) associated with a (ε, δ)-FKTS U , the case of εδ = −1 is a para complex. Proposition 3 ([36]) For a (ε, δ)-FKTS U and L(U ) as in above Proposition,   01 assuming ε = δ and K (x, y) = L(y, x) − εL(x, y), then the elements e = , 00     00 1 0 f = ,h= ∈ sl(2) (i.e., [e, f ] = h, [e, h] = −2e, [ f, h] = 2 f ) 10 0 −1 are derivations of L(U ). Remark 5 We note that L(U ) = L(U (ε, δ)) := L −2 ⊕ L −1 ⊕ L 0 ⊕ L 1 ⊕ L 2 is the five graded Lie (super)algebra such that U (ε, δ) ⊕ U (ε, δ) = L −1 ⊕ L 1 =T (ε, δ) (δ-LTS), L −2 = k (JTS) and D(T (ε, δ), T (ε, δ)) = L −2 ⊕ L 0 ⊕ L 2 (the derivation of T (ε, δ)) equipped with [L i , L j ] ⊆ L i+ j and L −1 ⊕ L 1 = L(U )/L −2 ⊕ L 0 ⊕ L 2 . In the Introduction, we had used the notation g = g−1 ⊕ g0 ⊕ g1 instead of L −1 ⊕ L 0 ⊕ L 1 . This Lie (super)algebra construction, without using root systems, is one of

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reasons to study nonassociative algebras (in particular, Jordan algebras) and triple systems. (The definition of a Lie superalgebra refers to [9, 12, 60]). This section is, mainly, a survey of our papers with respect to the triple systems and the construction of Lie (super)algebras. If needed, the readers can see our earlier references therein.

3 Examples of (ε, δ)-JTS We will consider here examples of the special case defined by bilinear forms x, y , that is, exhibit an (ε, δ)-JTS of (α, β, γ)-triple systems equipped with K (x, y) ≡ 0. Moreover, we give two examples (Prop. 2.2 and Prop. 2.3) which are not (ε, δ)-JTS. Example 1 Let V be a vector space with a symmetric bilinear form x, y . Then < x yz >= x, y z + y, z x − z, x y defines on V a (−1, 1)-JTS. Note that (−1, 1)-JTS is the same as the JTS. Example 2 Let V be a vector space with an anti-symmetric bilinear form x, y . Then < x yz >= x, y z + y, z x − z, x y defines on V a (1, −1)-JTS. Example 3 Let V be a vector space with a symmetric bilinear form x, y . Then < x yz >= x, y z − y, z x defines on V a (−1, −1)-JTS. Example 4 Let V be a vector space with an anti-symmetric bilinear form x, y . Then < x yz >= x, y z − y, z x defines on V a (1, 1)-JTS. Example 5 Let V be a set of alternative matrices Asym(n, ) = {x|t x = −x}, where t x denotes the transpose matrix of x. Then < x yz >= x t yz − εz t yx, wher e ∀ x, y, z ∈ V defines on V a (ε, −ε)-JTS, that is, the case of ε = −1 ⇒ JTS.

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Remark 6 Let V be the set of p × q matrices Mat ( p, q; ). Then this vector space V is a JTS with respect to the product {x yz} = x t yz + z t yx, ∀x, y, z ∈ V . Proposition 4 ([14]) Let (U, < x yz >) be an (ε, δ)-JTS. Then the triple system is a δ-LTS with respect to the new product [x yz] =< x yz > −δ < yx z > .

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In the next Sect. 4 we study the case of an (ε, δ)-FKTS, but we give first two examples which are not (ε, δ)-JTS as it follows. Proposition 5 ([22, 27]) Let (U, < x yz >) be a triple system with < x yz >= y, z x and x, y = −εy, x . Then this triple system is an (ε, δ)-FKTS. Proposition 6 ([16, 18]) Let U be a balanced (1, 1)-FKTS satisfying x x x , x ≡ 0 (identically) and x, y is nondegenerate. Then U has a triple product defined by < x yz >=

1 (y, x z + y, z x + x, z y). 2

(12)

Note that the balanced (1, 1)-FKTS induced from an exceptional Jordan algebra is closely related to the 56-dimensional meta symplectic geometry due to H. Freudenthal ([13, 15, 16, 18] and the earlier references therein). Also the correspondence of a quaternionic symmetric space and the balanced (1,1)-FKTS has been studied in [5]. On the other hand, for (−1, −1)-FKTS, see [6, 7].

4 Examples of Lie (Super)Algebras Associated with (ε, δ)-Freudenthal-Kantor Triple Systems We will exhibit examples of some triple systems and Lie (super)algebras associated with their triple systems. Unless otherwise stated, all Lie (super)algebras considered here are complex and finite dimensional. Example (a) C(n + 1) type is of dimension dim C(n + 1) = 2n 2 + 5n + 1. Let U be the set of matrices M(1, 2n; ). Then, by Example 2, it follows that the triple product L(x, y)z =< x yz >:= x, y z + y, z x − z, x y such that the bilinear form fulfills x, y = −y, x , is a (1, −1)-JTS, since K (x, y) ≡ 0 (identically). Furthermore, the standard embedding Lie superalgebra is 3-graded and of C(n + 1) type. For the extended Dynkin diagram, we obtain  L −1 ⊕ L 0 ⊕ L 1 :=

     L(a, b) 0 e ∼  ε = 1 = −δ ⊕ =  0 εL(b, a) f span span

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⊗ α1

α2 α3

αn αn+1

 > ◦ − ◦ − − − − − ◦ ⊗ − ◦ − − − − − ◦ => ◦ = A1 ⊕ Bn t ype (α1 ⊗ deleted).

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Also, we obtain  L 0 :=

   L(a, b) 0 ∼  ε = −1 = δ = 0 εL(b, a)  span α2 α3

αn αn+1

◦ − ◦ − − − − − ◦ => ◦ = Bn ⊕ I d (α1 ⊗ and α0 ◦ deleted). Thus the last diagram is obtained from the extended Dynkin diagram of B(n, 1) type by deleting α1 ⊗ and α0 ◦. Similarly, for the case of D(n, 1) type we have L −2 ⊕ L 0 ⊕ L 2 ∼ =A1 ⊕Dn , L 0 ∼ = Dn ⊕ I d. We note that this triple system is balanced and with a complex structure of Nijenhuis tensor zero, since K (x, y) = x, y I d = L(x, y) + L(y, x) (c.f. [36]). In the rest of this section, we will consider the construction of simple B3 - type Lie algebra associated with a JTS (the case of ε = −1 and δ = 1) more easily. Example (c) B3 -type is of dimension B3 = 21. Let U be the set of matrices Mat (1, 5; ), then (U, < x yz >) is a JTS with respect to the product < x yz >= x t yz + y t zx − z t x y, ∀x, y, z ∈ U. By straightforward calculations, we obtain the standard embedding Lie algebra L1 ∼ L(U ) = L −1 ⊕ = B3 and the LTS T (U ) = L −1 ⊕ L 1 = U ⊕ U. Note that  L0 ⊕  1 0 L0 ∼ with dim L 0 = 11. Furthermore, dim T (U ) = 10 and = B2 ⊕  0 −1 L −2 = L 2 = {0}, since K (x, y) = 0. Remark 7 We note that for δ = 1 the case of balanced is discussed in [18, 28]. On the other hand, for the constructions of simple exceptional Lie algebras G 2 , F4 , E 6 , E 7 , E 8 , refer to [16, 18, 21]. Also, for the construction of simple Lie superalgebras G(3), F(4), D(2, 1, α), P(n), Q(n), H (n), S(n) and W (n), refer to [22, 25, 27, 31]. Of course, these construction are created from the concept of triple systems without using systems of roots. Thus, moreover, these examples (a) and (b) imply that our methods may apply the symmetric superspace (the case of δ = −1) as well as the structures (see, [5, 46]) of the symmetric spaces (the case of δ = 1) corresponding to the LTS. However we will not go into the details and in future, we will discuss it.

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5 Concluding Remarks In this section, we will give several references of mathematical physics in our works. We note that there are applications toward the Yang-Baxter equations associated with triple systems [26, 39, 57] and also toward the field theory associated with Hermitian triple systems [37, 38]. For other mathematical physics, it seems that the books [28, 33, 55] are useful.

6 History from a Certain Personal Viewpoint For a mathematical history, in particular for Jordan rivers, we describe belows: This brief history (with respect to nonassociative algebras) is a story from author’s personal aspect (judgement). Triple systems (ternary algebras) have first been appeared from Prof. N. Jacobson and continued by Profs. O. Loos, K. Meyberg and E. Neher of students of Prof. M. Koecher in Germany. Also certain triple systems associated with the geometry of 56 dimensional due to Prof. Freudenthal have been studied by Prof. J. Faulkner (resp. K. Meyberg) of the student of Prof. N. Jacobson (resp. Prof. M. Koecher). On the other hand, for generalized algebraic concepts with respect to the triple system of H. Freudenthal, there is a history; H. Freudenthal (Netherlands) −− >K. Yamaguti (Japan) or I. L. Kantor (Russian and Sweden, he was born in Belarus) −− > Author (N. Kamiya) −− > D. Mondoc (but these arrows are no students), however, Dr. Mondoc is only a student of Prof. Kantor in Sweden. Profs. O. Loos and E. Neher in the students of Prof. M. Koecher in Germany are working in Jordan triple systems and Jordan pairs. Profs. Kantor, Yamaguti, S. Okubo and author(N. Kamiya) are studying in their generalizations, for example, refer to N. Kamiya and S. Okubo “Representation of (α, β, γ) triple systems,” Linear and Multilinear Algebras, 58 no.5-6 (2010) 617-643. This history is a story without using concept of root systems and Cartan matrix in Lie algebras, in particular, is a study for triple systems. Note that there are a lot of mathematicians in nonassociative algebras (for Lie algebras), but a little groups in triple systems or Jordan algebras. For example, about the group’s persons, it seems that there are Profs. E. Zelmanov, K. McCrimmon, B. Allison, V. Kac, I. Shestakov, W. Hein, H. Petersson, M. Racine, H. Asano, I. Satake, M. C. Myung, A. Elduque, C. Martinez, S. González, S. Okubo and author, may be, only a few. Furthermore in addition, the book “A Taste of Jordan Algebras” (Springer, 2003) written by Prof. K. McCrimmon of a student in N. Jacobson is described about a history of the Jordan river. It here emphasizes that this historical survey of certain Jordan algebras is to give from the end of the 20th century to the beginning of 21th century by my (author) aspect (viewpoint). To describe more details about the above river, for a certain example, considering for our imaginative illustrations in

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constructions of exceptional Lie algebras with respect to a generalization of numbers; () R → C → H → O(octonion) → H3 (O)(J or dan algebra o f 27 dim) → M(H3 (O))(metasymplectic geometr y o f 56 dim) → T(H3 (O))(symmetric space o f 112 dim) → E8 (exceptional simple Lie algebra o f 248 dim). On the other hand, there is other river also, () O → C ⊗ O, H ⊗ O and O ⊗ O (Fr eudenthal  s magic square) → T(O ⊗ O)(symmetric space o f 128 dim) → E8 . Note that Der O ∼ = F4 , where these are simple Lie algebras = G 2 and Der H3 (O) ∼ of 14 and 52 dimension respectively. For another way, there is a river of Prof. Tits (called Tits’ construction) as follows [62]. () The case A = A0 ⊗ J0 : Here A0 denote {x ∈ O|trace x = 0} and J0 = {x ∈ H3 (O)|T race x = 0}, and with dim A = 7 × 26, dim Der (A) = 66, where the base field  is an algebraically closed field of characteristic 0. Then we have L(A) = Der (A) ⊕ A ∼ = G2 ⊕ = E 8 , Der (A) = Der A ⊕ Der J ∼ F4 =< D(X, Y ) >span . For the product of A, X ◦ Y = (a ∗ b) ⊗ (x ∗ y) and with respect to the Lie product of L(A), [X, Y ] = D(X, Y ) + X ◦ Y , the vector space (A, ◦) has an algebraic structure of satisfying D(X ◦ Y, Z ) + D(Y ◦ Z , X ) + D(Z ◦ X, Y ) = 0, where X, Y, Z ∈ A ([19] and see the earlier references therein). If we set J = H3 (O) → H3 (A) (= B), then we have the following table; dim dim dim dim

A=1 A=2 A=4 A=8

dim B = 1 0 0 A1 G2

dim B = 6 A1 A2 C3 F4

dim B = 9 A2 A2 ⊕ A2 A5 E6

dim B = 15 C3 A5 D6 E7

dim B = 27 F4 E6 E7 E8

Here the case of dim B = 1 means B ∼ =  and note that L(A)/(G 2 ⊕ F4 ) is a reductive homogeneous space with 182 dimension. It seems that there is several researchers group’s tradition for these studies and furthermore, for a nonassociative world of 21th century, Spanish, Portuguese and middle Europe scholars groups will glow up a development with respect to the study (may be, Prof. Elduque’s group mainly). For algebraic structures of nonassociative subject (AMS classification 17) related with geometry, about 20th century, roughly speaking, we may describe as follows, for example (in my opinion);

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Jordan algebras researchers (E. Artin origin), Lie algebras researchers (N. Jacobson origin). In summarizing about Jordan algebras or triple systems, we have the following diagrams (a generalization of complex and quaternionic numbers): octonion, pseudo octonion algebras and triple systems =⇒ Jordan algebras +Lie (super)algebras +symmetric composition algebras =⇒mathematical algebras (author’s new phrase) In final comments (although they had described in the introduction), also we emphasize that nonassociative algebras are rich in algebraic structures, and they provide important common ground for various branches of mathematics, not only pure algebra and mathematical physics (for example, Pierce decompositions, Yang-Baxter equations and quark theory), but also analysis (Jordan C ∗ algebras or J B ∗ triple), topology (racks or quandles) and geometries (generalized symmetric spaces, convex cones or bounded symmetric domains, in particular). Hence, in future aspect, it seems that the triple systems (or ternary product) without using unit elements are useful concepts for several subjects of sciences as well as the situation of symmetric spaces. Acknowledgements The author would like to thank the organizers of NAART II, in particular Prof. Dr. Jose Brox for exchanging several email messages. Due to COVID-19 and the travel restrictions imposed by the government of Japan, the author was unable to travel to Portugal to participate in the NAART II Conference dedicated to honor Alberto Elduque on the occasion of his 60th birthday, but he is grateful to have been able to contribute to the book dedicated to him. The author is also grateful for the referee’s comments and suggestions.

References 1. Allison, B.N.: A class of nonassociative algebras with involution containing the class of Jordan algebras. Math. Ann. 237(2), 133–156 (1978) 2. Allison, B.N.: Models of isotropic simple Lie algebras. Commun. Algebr. 7(17), 1835–1875 (1979) 3. Asano, H., Yamaguti, K.: A construction of Lie algebras by generalized Jordan triple systems of second order. Ned. Akad. Wet. (Indag. Math.) 42(3), 249–253 (1980) 4. Asano, H.: Classification of non-compact real simple generalized Jordan triple systems of the second kind. Hiroshima Math. J. 21, 463–489 (1991) 5. Bertram, W.: Complex and quaternionic structures on symmetric spaces - correspondence with Freudenthal-Kantor triple systems. In: Miyaoka, R., Tamaru, H. (eds.) Theory of Lie Groups and Manifolds. Sophia Kokyuroku in Math. 45, 61–80 (2002) 6. Elduque, A., Kamiya, N., Okubo, S.: Simple (−1, −1) balanced Freudenthal Kantor triple systems. Glas. Math. J. 11(2), 353–372 (2003) 7. Elduque, A., Kamiya, N., Okubo, S.: (−1, −1) balanced Freudenthal Kantor triple systems and noncommutative Jordan algebras. J. Algebr. 294(1), 19–40 (2005) 8. Faulkner, J.R.: Structurable triples, Lie triples, and symmetric spaces. Forum Math. 6, 637–650 (1994)

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9. Frappat, L., Sciarrino, A., Sorba, P.: Dictionary on Lie Algebras and Superalgebras. Academic Press, San Diego, California 92101–4495 (2000) 10. Jacobson, N.: Lie and Jordan triple systems. Am. J. Math. 71, 149–170 (1949) 11. Jacobson, N.: Structure and representations of Jordan algebras. Am. Math. Soc. Colloq. Publ. XXXIX. American Mathematical Society, Providence (1968) 12. Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977) 13. Kamiya, N.: A structure theory of Freudenthal-Kantor triple systems. J. Algebr. 110(1), 108– 123 (1987) 14. Kamiya, N.: A construction of anti-Lie triple systems from a class of triple systems. Mem. Fac. Sci. Shimane Univ. 22, 51–62 (1988) 15. Kamiya, N.: A structure theory of Freudenthal-Kantor triple systems II. Comment. Math. Univ. St. Paul. 38(1), 41–60 (1989) 16. Kamiya, N.: A structure theory of Freudenthal-Kantor triple systems III. Mem. Fac. Sci. Shimane Univ. 23, 33–51 (1989) 17. Kamiya, N.: On (, δ)-Freudenthal-Kantor triple systems. Nonassociative algebras and related topics (the conference of Hiroshima, 1990), pp. 65–75. World Scientific Publishing, River Edge (1991) 18. Kamiya, N.: The construction of all simple Lie algebras over C from balanced FreudenthalKantor triple systems. Contrib. Gen. Algebr. 7 (Vienna, 1990), 205–213, Hölder-PichlerTempsky, Vienna (1991) 19. Kamiya, N.: On Freudenthal-Kantor triple systems and generalized structurable algebras. Nonassociative algebra and its applications (Oviedo, 1993), pp. 198–203. Math. Appl. 303. Kluwer Academic Publishers, Dordrecht (1994) 20. Kamiya, N.: On the Peirce decompositions for Freudenthal-Kantor triple systems. Commun. Algebr. 25(6), 1833–1844 (1997) 21. Kamiya, N.: On a realization of the exceptional simple graded Lie algebras of the second kind and Freudenthal-Kantor triple systems. Bull. Pol. Acad. Sci. Math. 46(1), 55–65 (1998) 22. Kamiya, N., Okubo, S.: On δ-Lie supertriple systems associated with (, δ)− FreudenthalKantor supertriple systems. Proc. Edinb. Math. Soc. 43(2), 243–260 (2000) 23. Kamiya, N., Okubo, S.: A construction of Jordan superalgebras from Jordan-Lie triple systems. In: Costa, Peresi, etc. (eds.) Lecture Notes in Pure and Applied Mathematics, vol. 211. NonAssociative Algebra and Its Applications, pp. 171–176. Marcel Dekker Inc (2002) 24. Kamiya, N., Okubo, S.: A construction of simple Jordan superalgebra of F type from a JordanLie triple system. Ann. Mate. Pura Appl. 181, 339–348 (2002) 25. Kamiya, N., Okubo, S.: Construction of Lie superalgebras D(2, 1; α), G(3) and F(4) from some triple systems. Proc. Edinb. Math. Soc. 46(1), 87–98 (2003) 26. Kamiya, N., Okubo, S.: On generalized Freudenthal-Kantor triple systems and Yang-Baxter equations. In: Proceedings of the XXIV International Colloquium Group Theoretical Methods in Physics, IPCS, vol. 173, pp. 815–818 (2003) 27. Kamiya, N., Okubo, S.: A construction of simple Lie superalgebras of certain types from triple systems. Bull. Aust. Math. Soc. 69(1), 113–123 (2004) 28. Kamiya, N.: Examples of Peirce decomposition of generalized Jordan triple system of second order-Balanced cases. In: Fuchs , J. (ed.) Noncommutative Geometry and Representation Theory in Mathematical Physics, pp. 157–165. Contemp. Math. 391. AMS, Providence, RI (2005) 29. Kamiya, N., Okubo, S.: Composition, quadratic, and some triple systems. Non-associative algebra and its applications. Lecture Notes in Pure and Applied Mathematics, vol. 246, pp. 205–231. Chapman & Hall/CRC, Boca Raton (2006) 30. Kamiya, N., Mondoc, D.: A new class of nonassociative algebras with involution. Proc. Jpn. Acad., Ser. A 84(5), 68–72 (2008) 31. Kamiya, N., Mondoc, D., Okubo, S.: A structure theory of (−1, −1)-Freudenthal Kantor triple systems. Bull. Aust. Math. Soc. 81, 132–155 (2010) 32. Kamiya, N., Mondoc, D., Okubo, S.: A characterization of (-1,-1)-Freudenthal-Kantor triple systems. Glas. Math. J. 53, 727–738 (2011)

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33. Kamiya, N., Mondoc, D., Okubo, S.: A review of Peirce decomposition for unitary (-1,-1)Freudenthal-Kantor triple systems. In: Makhlouf, A., Paal, E., Silvestarov, S., Stolin, A. (eds.) Proceedings in Mathematics and Stastics, vol. 85, pp. 145–155. Springer (2014) 34. Kamiya, N., Okubo, S.: On triality of structurable and pre-structurable algebras. J. Algebr. 416, 58–88 (2014) 35. Kamiya, N., Okubo, S.: Algebras and groups satisfying triality relations monograph(book). University of Aizu (2015). Algebras, Groups and Geometries, vol. 33, pp. 1–92 (2016) and Algebras, Groups and Geometries, vol. 35, pp. 113–168 (2018) 36. Kamiya, N., Okubo, S.: Symmetry of Lie algebras associated with (.δ) Freudenthal-Kantor triple systems. Proc. Edinb. Math. Soc. 89, 169–192 (2016) 37. Kamiya, N., Sato, M.: Hermitian (ε, δ)-Freudenthal-Kantor triple systems and certain applications of ∗ -Generalized Jordan triple systems to field theory. Adv. High Energy Phys. (2014). https://doi.org/10.1155/2014/310264 38. Kamiya, N., Sato, M.: Hermitian triple systems associated with bilinear forms and certain applications to field theory. Hadron. J. 37(2), 131–147 (2014) 39. Kamiya, N., Shibukawa, Y.: Dynamical Yang-Baxter maps and weak Hopf algebras associated with quandles. In: Yamane, Kogiso, Koga, Kimura (eds.) Proceedings of the Meeting for Study of Number theory, Hopf Algebras and Related topics, pp. 1–23. Yokohama Publishers (2019) 40. Kaneyuki, S., Asano, H.: Graded Lie algebras and generalized Jordan triple systems. Nagoya Math. J. 112, 81–115 (1988) 41. Kantor, I.L.: Graded Lie algebras. Tr. Sem. Vect. Tens. Anal. 15, 227–266 (1970) 42. Kantor, I.L.: Some generalizations of Jordan algebras. Tr. Sem. Vect. Tens. Anal. 16, 407–499 (1972) 43. Kantor, I.L.: Models of exceptional Lie algebras. Sov. Math.-Dokl. 14(1), 254–258 (1973) 44. Kantor, I.L.: A generalization of the Jordan approach to symmetric Riemannian spaces. The Monster and Lie Algebras (Columbus, OH, 1996), pp. 221–234. Ohio State University Mathematics Research Institute Publications, vol. 7. de Gruyter, Berlin (1998) 45. Kantor, I.L., Kamiya, N.: A Peirce decomposition for generalized Jordan triple systems of second order. Commun. Algebr. 31(12), 5875–5913 (2003) 46. Loos, O.: Symmetric Spaces. I: General Theory. W. A. Benjamin, Inc., New York-Amsterdam (1969) 47. Koecher, M.: Embedding of Jordan algebras into Lie algebras I. II. Am. J. Math. 89, 787–16 (1967) and 90, 476–510 (1968) 48. Lister, W.G.: A structure theory of Lie triple systems. Trans. Am. Math. Soc. 72, 217–242 (1952) 49. Meyberg, K.: Lectures on algebras and triple systems. Lecture Notes. The University of Virginia, Charlottesville (1972) 50. Mondoc, D.: Models of compact simple Kantor triple systems defined on a class of structurable algebras of skew-dimension one. Commun. Algebr. 34(10), 3801–3815 (2006) 51. Mondoc, D.: On compact realifications of exceptional simple Kantor triple systems. J. Gen. Lie Theory Appl. 1(1), 29–40 (2007) 52. Mondoc, D.: Compact realifications of exceptional simple Kantor triple systems defined on tensor products of composition algebras. J. Algebr. 307(2), 917–929 (2007) 53. Mondoc, D.: Compact exceptional simple Kantor triple systems defined on tensor products of composition algebras. Commun. Algebr. 35(11), 3699–3712 (2007) 54. Neher, E.: Jordan triple systems by the grid approach. Lecture Notes in Mathematics, vol. 1280. Springer, Berlin (1987) 55. Okubo, S.: Introduction to octonion and other non-associative algebras in physics. Montroll Memorial Lecture Series in Mathematical Physics, vol. 2. Cambridge University Press, Cambridge (1995) 56. Okubo, S., Kamiya, N.: Jordan-Lie superalgebra and Jordan-Lie triple system. J. Algebr. 198(2), 388–411 (1997) 57. Okubo, S., Kamiya, N.: Quasi-classical Lie superalgebras and Lie supertriple systems. Commun. Algebr. 30(8), 3825–3850 (2002)

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58. Okubo, S.: Symmetric triality relations and structurable algebras. Linear Algebr. Appl. 396, 189–222 (2005) 59. Satake, I.: Algebraic Structures of Symmetric Domains. Princeton University (1980) 60. Scheunert, M.: The theory of Lie superalgebras. An introduction. Lecture Notes in Mathematics, vol. 716. Springer, Berlin (1979) 61. Springer, T.: Jordan Algebras and Algebraic Groups. Springer (1973) 62. Tits, J.: Alg`ebres alternatives, alg`ebres de Jordan, et alg`ebres de Lie exceptionnelles. Ned. Acad. Wet. Proc. Ser. A 69, 223–237 (1966) 63. Yamaguti, K., Ono, A.: On representations of Freudenthal-Kantor triple systems U (, δ). Bull. Fac. Sch. Educ. Hiroshima Univ. 7, no. II, 43–51 (1984) 64. Zhevlakov, K.A., Slinko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative. Academic Press Inc, New York-London (1982) 65. Zelmanov, E.: Primary Jordan triple systems. Sib. Mat. Zh. 4, 23–37 (1983)

Schreier’s Type Formulae and Two Scales for Growth of Lie Algebras and Groups Victor Petrogradsky

Abstract Let G be a free group of rank n and H ⊂ G its subgroup of finite index. Then H is also a free group and the rank m of H is determined by Schreier’s formula m − 1 = (n − 1) · |G : H |. Any subalgebra of a free Lie algebra is also free. But a straightforward analogue of Schreier’s formula for free Lie algebras does not exist, because any subalgebra of finite codimension has an infinite number of generators. But the appropriate Schreier’s formula for free Lie algebras exists in terms of formal power series. There exists also a version in terms of exponential generating functions. This is a survey on how these formulas are applied to study (1) growth of finitely generated Lie algebras and groups and (2) the codimension growth of varieties of Lie algebras. First, these formulae allow to specify explicit formulas for generating functions of respective types for free solvable (or more generally, polynilpotent) Lie algebras. Second, these explicit formulas for generating functions are used to derive asymptotic for these two types of the growth. These results can be viewed as analogues of the Witt formula for free Lie algebras and groups. In case of Lie algebras, we obtain two scales for respective types of growth. We also shortly mention the situation on growth for other types of linear algebras. Keywords Identical relations · Growth · Generating functions · Codimension sequence · Solvable Lie algebras · Polynilpotent Lie algebras

1 Analogue of Schreier’s Formula for Free Lie Algebras Denote the ground field by K . Let X be an at most countable set supplied with a weight function wt : X → N, namely we assume that V. Petrogradsky (B) Department of Mathematics, University of Brasilia, Brasilia, DF 70910-900, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_5

81

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V. Petrogradsky ∞

X = ∪ Xi ; i=1

X i = {x ∈ X | wt x = i}, |X i | < ∞, i ∈ N.

We say that X is finitely graded. Assume that an algebra A is generated by X . For simplicity, we initially consider that A is multigraded with respect to the generating set X . Then we naturally define the weight of a monomial a ∈ A. Let Y be a set of monomials in X , then one defines the Hilbert-Poincaré series of Y (with respect to X ), see e.g. [4] H X (Y ) = H X (Y, t) :=

∞ 

|Yi |t i ;

Yi = {y ∈ Y | wt y = i}, i ∈ N.

i=1

Now consider an arbitrary algebra A generated by a finitely graded set X . Consider a subspace V ⊂ A, then we define H X (V ) using a homogeneous basis of gr V ; where gr V is the associated graded space with respect to the filtration determined by the total weight degree in X . In particular, this definition applies to the algebra A. ∞  bn t n , bn ∈ {0, 1, 2, Next, we introduce the operator E on power series φ(t) = n=1

. . . } (see [25, 27]): E : φ(t) =

∞ 

bn t

n

−→ E(φ(t)) :=

n=1

∞ 

an t = n

n=0

∞ 

1 . (1 − t n )bn n=1

Assume that L is a Lie algebra generated by X and U (L) its universal enveloping algebra. Let H X (L , t) =

∞  n=1

bn t n ,

H X (U (L), t) =

∞ 

an t n .

n=0

One has a well-known formula that explains importance of the operator above H X (U (L)) = E(H X (L)) [36]. The following is the natural analogue of Schreier’s formula, introduced by the author. For basic facts on free Lie (super)algebras see [1, 2]. Theorem 1 ([27]) Assume that L is a free Lie algebra generated by a finitely graded set X . Let H be a subalgebra and Y is a set of its free generators. Then H(Y, t) − 1 = (H(X, t) − 1) · E(H(L/H ), t). Let L be a Lie algebra. One defines the lower central series as L 1 = L, L i+1 = [L , L i ] for i = 1, 2, . . . . Now, L is nilpotent of class s iff L s+1 = {0} while L s = {0}. All Lie algebras nilpotent of class at most s form the variety denoted by Ns . A Lie algebra L is called polynilpotent with a tuple of integers (sq , . . . , s2 , s1 ) iff there exists a chain of ideals

Schreier’s Type Formulae and Two Scales for Growth of Lie Algebras and Groups

{0} = L q+1 ⊂ L q ⊂ · · · ⊂ L 2 ⊂ L 1 = L ,

83

L n /L n+1 ∈ Nsn , n = 1, . . . , q.

All polynilpotent Lie algebras with a fixed tuple form the variety denoted by Nsq · · · Ns2 Ns1 . In the particular case sq = · · · = s1 = 1, one obtains the variety Aq , consisting of solvable Lie algebras of length at most q. On the other hand, a polynilpotent Lie algebra is solvable. Moreover, the free polynilpotent Lie algebras, the tuple being fixed, provide interesting examples of solvable Lie algebras. The definitions in case of group theory are similar. Let G be a group, denote by {γn (G) | n = 1, 2, . . .} the terms of the lower central series (warning: below γ has also a different meaning!). Suppose that L is the free Lie algebra of rank k, L = ⊕∞ n=1 L n its natural grading, and G the free group of rank k. Then the lower central series factors γn (G)/γn+1 (G) are free abelian groups and their ranks are given by the classical Witt formula [1]: ψk (n) := rank Z γn (G)/γn+1 (G) = dim K L n =

1  a  n  kn ≈ . k μ n a|n a n

(1)

where μ(∗) is the Möbius function. Theorem 1 allows to derive the following explicit formulas. Theorem 2 ([27–29]) Let L = F(Nsq · · · Ns1 , k) be the free polynilpotent Lie algebra of finite rank k ≥ 2, where q ≥ 1. Set β0 (z) := 0, α0 (z) := kz, and define the following functions recursively βi (z) := βi−1 (z) +

si  a 1  m   αi−1 (z m/a ) , μ m a|m a m=1

αi (z) := 1 + (kz − 1) · E(βi (z)),

1 ≤ i ≤ q.

Then H(L , z) = βq (z). For example, we have a particular case. Corollary 1 ([27, 29]) Let L := F(ANd , k). Then H(L , z) = ψk (1)z + · · · + ψk (d)z d + 1 +

kz − 1 . (1 − z)ψk (1) · · · (1 − z d )ψk (d)

2 Scale 1 and Growth of Free Solvable (Polynilpotent) Finitely Generated Lie Algebras and Groups Now we describe applications of the analogue of Schreier’s formula for free Lie algebras (Theorem 1 and its application Theorem 2) to specify the growth of free solvable (more generally, polynilpotent) Lie algebras and groups of finite rank. Assume that L is a relatively free algebra of some multihomogeneous variety of (associative, or Lie) algebras, generated by X = {x1 , . . . , xk }. Then we have a natural

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grading L = ⊕∞ by degree in X . One defines the growth function with respect n=1 L n to X as γ L (X, n) := ns=1 dim K L s . Theorem 3 (Berele, [3]) The growth function of a finitely generated associative PI-algebra is bounded by a polynomial function. For more details on proofs of this important result see [4, 16]. But the growth of finitely generated Lie PI-algebras can be more complicated [37]. In this case, the author introduced scale 1 of functions of intermediate growth (2) and suggested that it is complete in the sense of Conjecture 1 below. Define functions ln(0) x := x,

ln(s+1) x := ln(ln(s) x),

s = 0, 1, 2, . . .

exp(0) x := x, exp(s+1) x := exp(exp(s) x),

We use notations f (x) ≈ g(x), x → ∞, denotes that lim x→∞ f (x)/g(x) = 1; f (x) = o(g(x)) means that beginning with some number f (x) = α(x)g(x) where lim x→∞ α(x) = 0. Also, ζ(x) is the zeta function. Consider the scale 1 consisting of a series of functions qα (n), q = 1, 2, 3, . . . of a natural argument with a parameter α ∈ R+ :

scale 1:

1α (n) := α, 2α (n) := n α , α/(α+1) ), 3α (n) := exp(n

n q , α (n) := exp (ln(q−3) n)1/α

(2) q = 4, 5, . . . .

Now, we specify the growth of the free solvable (more generally, polynilpotent) Lie algebras of finite rank with respect to the scale 1 by giving the following asymptotic. Theorem 4 ([25]) Let L := F(Nsq · · · Ns1 , k) be the free polynilpotent Lie algebra of finite rank k ≥ 2, where q ≥ 2, generated by X = {x1 , . . . , xk }. Then ⎧ A + o(1) N ⎪ ⎪ n , q = 2, ⎪ ⎪ N! ⎪ ⎪ ⎪   ⎨ N /(N +1) , q = 3, γ L (X, n) = exp (C+o(1)) n ⎪ ⎪

⎪ ⎪ ⎪ n ⎪ 1/N ⎪ exp (B , q ≥ 4, +o(1)) ⎩ (ln(q−3) n)1/N

n → ∞;

where the constants are: N := s2 dim K F(Ns1 , k), B := s3 Aζ(N +1),

1 A := s2



k−1 s1 ψk (q) q=2 q

s2 , 1

C := (1 + 1/N )(B N ) 1+N .

Schreier’s Type Formulae and Two Scales for Growth of Lie Algebras and Groups

85

This result has the following application to group theory. Let G be a group. Due to Lazard, one obtains the related Lie algebra [19]: ∞

L K (G) := ⊕ (γn (G)/γn+1 (G)) ⊗Z K . n=1

If G is a free polynilpotent group, then L K (G) is the free polynilpotent Lie algebra of the same rank and with the same tuple, moreover, the lower central series factors are free abelian groups (A. Shmelkin [35]). Corollary 2 ([25, 29]) Let G = G(Nsq · · · Ns1 , k), q ≥ 2, be the free polynilpotent group of finite rank k ≥ 2. Consider ranks of the lower central series factors, denoting bn := rank Z γn (G)/γn+1 (G), n ≥ 1. Then the asymptotic holds: ⎧ A + o(1) N −1 ⎪ ⎪ , q = 2, n ⎪ ⎪ (N − 1)! ⎪ ⎪ ⎪   ⎨ N /(N +1) , q = 3, bn = exp (C+o(1)) n ⎪ ⎪

⎪ ⎪ ⎪ n ⎪ 1/N ⎪ ⎩ exp (B +o(1)) (q−3) 1/N , q ≥ 4, (ln n)

n → ∞;

where N , A, B, C are the same as in Theorem 4. Observe that just by setting sq = · · · = s1 = 1, Theorem 4 and its Corollary 2 are turned into results on the free solvable Lie algebra and group of rank k and length q. A similar observation is valid concerning the results (e.g. Theorem 14) on the codimension growth below. M.I.Kargapolov raised problem 2.18 in [17] to specify the lower central series ranks for free polynilpotent finitely generated groups. Exact recursive formulae were given by Egorychev [6]. We suggest another answer to this problem by specifying the asymptotic behaviour of that ranks. We consider to view the asymptotic of Theorem 4 and its Corollary 2 as the analogue of the Witt formula (1), now for the free solvable (more generally, polynilpotent) Lie algebras and groups. Since the growth of finitely generated solvable Lie algebras is intermediate [20], the respective generating function converges in the unit circle. It is important to study an asymptotic of the generating function when we approach the unit circumference from inside. Since the coefficients of the series are real nonnegative numbers, it is sufficient to study this behaviour while z = t → 1 − 0. The crucial step to prove Theorem 4 is the following asymptotic of the generating function H X (L , t). Theorem 5 ([25]) Let L := F(Nsq · · · Ns1 , k) be the free polynilpotent Lie algebra of finite rank k ≥ 2, where q ≥ 2, generated by X = {x1 , . . . , xk }. Then lim (1 − t) N H X (L , t) = A,

t→1−0

q = 2,

lim (1 − t) N ln(q−2) (H X (L , t)) = s3 ζ(N +1) A, q ≥ 3;

t→1−0

where N , A are the same as in Theorem 4.

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Now, we describe an important idea to prove Theorem 5. The explicit formula of the generating function (Theorem 2) shows that H X (L , t) is “roughly speaking” a (q − 1)-iteration of the operator E applied to β1 (z). We can easily describe the first application: β1 (z) = ψk (1)z + ψk (2)z 2 + · · · + ψk (s1 )z s1 ; 1 μ E(β1 (z)) = ≈ , as t → 1−0; ψ (1) s ψ (s ) k 1 k 1 (1 − z) · · · (1 − z ) (1 − t) M s1  where M = ψk (1) + · · · + ψk (s1 ) = dim K F(Ns1 , k), μ= q −ψk (q) . q=2

Next, we use the fact that E is “approximately” the exponent, thus H X (L , t) behaves like

¯ + o(1) (q−2) μ , t → 1−0 . exp (1 − t) N Thus, we arrive at the asymptotic of Theorem 5. Another idea in the proof of Theorem 4 is to specify a connection between the growth of a function analytic in the unit circle with asymptotic of its coefficients [25]. A similar version of Schreier’s formula for free Lie superalgebras was established as well [27, 29]. Also, the asymptotic of Theorem 4 was extended to the case of the free solvable (polynilpotent) Lie superalgebras of finite rank [13]. Below we see that the scale (4) for the superexponential codimension growth for varieties of Lie algebras is rather complete (Theorem 10). So, we conjecture that the scale (2) for the intermediate growth of finitely generated Lie PI-algebras is complete as well. Conjecture 1 ([22]) Let L be a finitely generated Lie PI-algebra. Then there exist numbers q, N0 such that γ L (n) ≤ exp

n ln(q) n

, n ≥ N0 .

This bound was confirmed for almost solvable (more generally, almost polynilpotent) Lie algebras [14, 15].

3 Exponential Analogue of Schreier’s Formula for Free Lie Algebras Let A := F(V, X ) be the free algebra of a variety V of linear algebras generated by a countable set X = {xi |i ∈ N}. Consider a set of different elements X˜ n = {xi1 , . . . , xin } ⊂ X and the space of all multilinear elements Pn ( X˜ n ) ⊂ A

Schreier’s Type Formulae and Two Scales for Growth of Lie Algebras and Groups

87

on X˜ n of degree n, for all n ≥ 0. Then, the dimension of this space does not depend on the choice of a set X˜ n ⊂ X of fixed cardinality n and is denoted as cn (V) = cn (A) := dim Pn ( X˜ n ) and is called the codimension growth sequence of the variety of algebras V and its free algebra A as well [10]. This is a particular case that helps to understand a more general setting that we need. We shall consider complexity functions, refereed to also as exponential generating functions in combinatorics. We provide definitions for sets of monomials, the case of spaces is similar. Assume that we are given a set A of monomials in X = {xi1 , . . . , xin } ⊂ X , denote X = {xi | i ∈ N}. Consider a set of distinct elements  X ) the subset of all multilinear elements of degree n on  X belonging to A. by Pn (A,  X ) does not depend on the choice Suppose that the number of these elements cn (A,  X ) and say that of  X , but depends only on n. In this case, we denote cn (A) := cn (A,  A is X -uniform and define the complexity function with respect to X : C X (A, z) :=

∞  cn (A) n=1

n!

z n , z ∈ C.

(3)

(the sum is taken from n = 0, c0 = 1 for associative algebras and groupoids with unity). Remark that A need not consist of multilinear elements. We also omit the variable z and (or) the set X and write C X (A, z) = C(A). These definitions are naturally extended to algebras, subspaces, and their dimensions with respect to their generating sets. We illustrate the notions above by computing the complexity functions for the following varieties of algebras. Let X be a countable set and A = A(X ), L = L(X ) the free associative and Lie algebras, respectively. Then C X (A, z) =

∞ 

zn =

n=0

1 , 1−z

∞  zn = − ln(1 − z), C X (L , z) = n n=1

C X (Ns , z) =

s  zn . n n=1

The author established an exponential analogue of Schreier’s formula for free Lie algebras as follows. In this case we need our more general setting than that for varieties of linear algebras. Theorem 6 ([26]) Assume that L is the free Lie algebra generated by a countable generating set X . Assume that H is an X -uniform subalgebra. Then H has an X -uniform set of free generators Y and C X (Y, z) − 1 = (z − 1) · exp(C X (L/H, z)).

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4 Scale 2 for the Codimension Growth of Lie PI-Algebras For the theory of varieties of associative and Lie algebras see [1, 4, 10]. Let V be a variety of Lie algebras, and F(V, X ) its free algebra generated by X = {xi | i ∈ N}. Let Pn (V) ⊂ F(V, X ) be the subspace of all multilinear elements in {x1 , . . . , xn } and consider the codimension growth sequence cn (V) = cn (F(V, X ), X ) := dim K Pn (V), n = 1, 2, . . . . In case of associative algebras the fundamental fact is as follows: Theorem 7 (Regev, [34]; Latyshev [18]) Let an associative algebra A satisfies a nontrivial identical relation of degree d. Then cn (A) ≤ C n , n ≥ 1; where C := (d − 1)2 . Another crucial fact on the codimension growth of associative √ algebras in characteristic zero is that the exponent, defined as Exp(A) := lim n cn (A) always exists and n→∞ is integral [8]. Now we start discussing the codimension growth for Lie algebras. The integrality of the exponent of the codimension growth for finite dimensional Lie algebras over a field of characteristic zero was proved by Zaitsev [38]. In general, the exponents for Lie algebras are not always integral [9, 39]. Moreover, the codimension growth in case of Lie algebras is more versatile. Unlike the associative case, the codimension growth of a rather small variety AN2 is overexponential (Volichenko [37]). On the other hand, the following upper bound was found. Theorem 8 (Grishkov [12]) Let L be a Lie algebra satisfying a nontrivial identity. Then for any r > 1 there exists N0 such that cn (L) ≤

n! , n ≥ N0 . rn

Razmyslov introduced the complexity functions (3) and reformulated the upper bound as follows. Theorem 9 (Razmyslov [33]) Let V be a nontrivial variety of Lie algebras. Then the complexity function C(V, z) is an entire function of complex variable. The author established a better and optimal general bound for the series that allowed to prove an upper bound for the codimension growth sequence as well. The estimate of the first item was recently established in [31]. Theorem 10 ([21, 23, 24, 31]) Let L be a Lie algebra satisfying a nontrivial identity of degree m ≥ 4. Then 1. The following coefficientwise bound for the series holds: C(L , z) ≺ z exp(z exp(. . . (z exp(z exp(z))) . . .)) .    m−2 times exp

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2. There exists an infinitesimal such that cn (L) ≤

n! (1 + o(1))n , n → ∞. (ln(m−3) n)n

Thus, we have a vast area of overexponential growth for Lie algebras, lying between the exponent and the factorial functions. To describe such a growth we introduce the scale 2 consisting of a series of functions αq (n), q = 2, 3, . . . , with a real parameter α [21]: ⎧   α−1 ⎪ ⎨ n! α , α > 1, q = 2; q α (n) := n! ⎪ ⎩ (q−2) n/α , α > 0, q = 3, 4 . . . (ln n)

scale 2:

(4)

The upper bounds of Theorem 10 are “adequate” and the scale 2 (4) for the codimension growth of Lie PI-algebras is complete, because the free solvable Lie algebras do have such an asymptotic behaviour, see Theorem 14 below. Actually, we obtain a more fine scale formed by a series of functions with two real parameters α, β:

scale 2 :

q

α,β (n) :=

⎧   α−1 ⎪ ⎪ n! α β n/α , α ≥ 1, β > 0; q = 2; ⎨ n! · (β/α)n/α ⎪ ⎪ ⎩ (q−2) n/α , α > 0, β > 0; q = 3, 4 . . . (ln n)

(5)

Observe that in terms of scale 2 (5), the exponential growth is just a particular subcase of level q = 2 in case of the parameter α = 1.

4.1 The Codimension Growth for Another Classes of Linear Algebras The codimension growth of arbitrary linear algebras can be weird [7]. The varieties of absolutely free (and free commutative, or anticommutative) algebras have Schreier’s type formulae in terms of generating function of both types, the regular generating functions and exponential generating functions (i.e. complexity functions), i.e. we have natural analogues of both, Theorems 1 and 6, see [30]. We describe both, the generating functions and the growth functions, for two kinds of growth, for different versions of nilpotency and solvability for three types of linear algebras above [30]. But we do not get an analogue of Theorem 11 and these results do not lead us to something like scale 1 and scale 2 for the respective types of growth [30]. It was recently shown that the same scale (4) stratifies the ordinary codimension growth of Poisson PI-algebras, but here it is essential to assume that a Poisson algebra

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satisfies a nontrivial Lie identical relation [31]. If a Poisson algebra is satisfying so called mixed identities only, then the ordinary codimension growth has a factorial behaviour [31, 32]. The codimension growth for Jordan PI-algebras can be overexponential [5, 11]. We conjecture that something like scale 2 (see (4)) should appear in case of Jordan PI-algebras as well.

5 Explicit Formulae for Complexity Functions and Asymptotic for Lie PI-Algebras Let M, V be varieties of Lie algebras. Their product M · V is the class of all Lie algebras L such that there exists an ideal H ⊂ L satisfying H ∈ M and L/H ∈ V, see [1]. Using the exponential Schreier’s formula (Theorem 6) the following explicit formula was proved. Theorem 11 ([26]) Let M · V be the product of varieties of Lie algebras, where M is multihomogeneous. Then C(M · V, z) = C(V, z) + C(M, 1 + (z − 1) exp(C(V, z))). Roughly speaking, the formula says that C(M · V, z) is “almost” a composition of three functions C(M) ◦ exp ◦ C(V). The variety V = Nsq · · · Ns1 can be viewed as the product V = Nsq · · · Ns2 · Ns1 . As application, the following explicit formula was derived. Theorem 12 ([26, 28, 29]) Consider the variety of polynilpotent Lie algebras V = Nsq · · · Ns1 , q ≥ 1. Define functions β1 (z) :=

s1  zm , m m=1

βi (z) := βi−1 (z) +

si  (1 + (z − 1) exp(βi−1 (z)))m , m m=1

2 ≤ i ≤ q.

Then C(V, z) = βq (z). Consider some particular cases. Corollary 3 Fix d ∈ N. Then C(ANd , z) = z +



zd z2 zd z2 + ··· + + 1 + (z − 1) exp z + + ··· + . 2 d 2 d

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Corollary 4 ([23]) Fix q ∈ N. Consider the variety of solvable Lie algebras of length q, denoted as Aq . Set β1 (z) := z, and βi+1 (z) := βi (z) + 1 + (z − 1) exp(βi (z)),

i = 1, 2, . . . , q − 1.

Then C(Aq , z) = βq (z). Corollary 5 ([23]) Fix c ∈ N. Then C(Nc A, z) = z +

c m  1 1 + (z − 1) exp(z) . m m=1

Complexity functions are useful for computation of the codimension growth. For example, the last result yields an asymptotic. Corollary 6 ([23]) cn (Nc A) ≈ cn−c−1 n c , as n → ∞. Let f (z) be an entire function of complex variable, we use standard notation M f (r ) := max|z|=r | f (z)|. Observe that in case of complexity functions, we have M f (r ) = f (r ), r ∈ R+ , since all coefficients are nonnegative. By Theorem C(Nsq · · · Ns1 , z) is “almost” q − 1 iterations of exp(∗) applied to  1 12, β1 (z) = sm=1 z m /m, thus one has something like exp(q−1) (z s1 /s1 ). More, precisely, we derive the following asymptotic. Theorem 13 ([23]) Consider the variety of polynilpotent Lie algebras V := Nsq · · · Ns1 , q ≥ 2, and its complexity function f (z) := C(V, z). Then ln(q−1) M f (r ) s2 = . s 1 r →∞ r s1 lim

Next, we establish a relation between the growth of fast growing entire functions and an asymptotic of their coefficients. This fact helped us to match the upper bounds in the next result. But a connection between the lower bounds require more direct estimates. Theorem 14 ([23]) Consider the variety of polynilpotent Lie algebras V := Nsq · · · Ns1 , q ≥ 2. Then there exists an infinitesimal such that

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cn (V) =

⎧ n/s1 s1 −1  ⎪ s , ⎪ ⎨ (n!) 1 s2 + o(1) ⎪ ⎪ ⎩

q = 2;

 s + o(1) n/s1 2 , q = 3, 4, . . . ; (q−2) n/s 1 s1 (ln n) n!

n → ∞.

In the following two cases we obtain somewhat more precise asymptotic, but they look rather complicated. Theorem 15 ([26]) Fix d ∈ N. Then  1−1/d cn (ANd ) ≈ μ n! exp

 d−1

λk n

1−k/d

n

3−d 2d

, n → ∞, where

k=1

   + 1 · · · dk + k − 1 , k = 1, . . . , d − 1; λk : = k!(d − k)

d 11 1−d (2π) 2d d −1/2 . μ : = exp − d k=2 k k

d

It is well known that cn (A2 ) = n − 1 ≈ n, this coincides with our asymptotic. Consider some particular cases.

cn (AN2 ) ≈

√

n!

√  √ n 4n , √ 4 8πe   exp 21 n 2/3 + 56 n 1/3 , √ √ 3 3 2πe5/18

exp

 2/3 cn (AN3 ) ≈ n!

n → ∞.

Theorem 16 ([26]) Consider the variety A3 of solvable Lie algebras of length 3 and its codimension growth sequence cn := cn (A3 ). Then cn =

2(ln ln n)2 − 2 ln ln n − 1 n! n 2 ln ln n + 1 + exp n (ln n) ln n ln n  1 

, n → ∞. +o ln n

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References 1. Bahturin, Yu.A.: Identical Relations in Lie Algebras. VNU Science Press, Utrecht (1987) 2. Bahturin, Yu.A., Mikhalev, A.A., Petrogradsky, V.M., Zaicev, M.V.: Infinite Dimensional Lie Superalgebras. de Gruyter Expositions in Mathematics, vol. 7. de Gruyter, Berlin (1992) 3. Berele, A.: Homogeneous polynomial identities. Israel J. Math. 42(3), 258–272 (1982) 4. Drensky, V.: Free Algebras and PI-Algebras. Graduate Course in Algebra. Springer-Verlag Singapore, Singapore (2000) 5. Drensky, V.: Polynomial identities for the Jordan algebra of a symmetric bilinear form. J. Algebra 108, 66–87 (1987) 6. Egorychev, G.P.: Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, vol. 59. American Mathematical Society, Providence, RI (1984) 7. Giambruno, A., Mishchenko, S., Zaicev, M.: Codimensions of algebras and growth functions. Adv. Math. 217(3), 1027–1052 (2008) 8. Giambruno, A., Zaicev, M.: Exponential codimension growth of PI algebras: an exact estimate. Adv. Math. 142(2), 221–243 (1999) 9. Giambruno, A., Zaicev, M.: Non-integrality of the PI-exponent of special Lie algebras. Adv. Appl. Math. 51(5), 619–634 (2013) 10. Giambruno, A., Zaicev, M.: Polynomial Identities and Asymptotic Methods. Mathematical Surveys and Monographs, vol. 122. American Mathematical Society (AMS), Providence, RI (2005) 11. Giambruno, A., Zelmanov, E.: On growth of codimensions of Jordan algebras. In: Groups, Algebras and Applications. Contemporary Mathematics, vol. 537, pp. 205–210. AMS, Providence, RI (2011) 12. Grishkov, A.N.: On growth of varieties of Lie algebras. Mat. Zametki 44(1), 51–54 (1988). Engl. transl., Math. Notes 44(1–2), 515–517 (1988) 13. Klementyev, S.G., Petrogradsky, V.M.: Growth of solvable Lie superalgebras. Comm. Algebra. 33(3), 865–895 (2005) 14. Klementyev, S.G., Petrogradsky, V.M.: On growth of almost solvable Lie algebras. Uspekhi Mat. Nauk 60(5), 165–166 (2005). translation in Russian Math. Surveys 60(5), 970–972 (2005) 15. Klementyev, S.G., Petrogradsky, V.M.: On growth of almost polynilpotent Lie algebras. In: Groups, Rings and Group Rings, Giambruno, A., Polcino Milies, C., Sehgal, S.K. (eds.) Contemporary Mathematics, vol. 499, pp. 173–180. AMS, RI (2009) 16. Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. Pitman, Boston (1985) 17. Kourovskaya tetrad, Unsolved problems in group theory, Nauka, Novosibirsk (1967) 18. Latyshev, V.N.: Two remarks on P I -algebras. Sibirsk. Mat. Ž. 4, 1120–1121 (1963) 19. Lazard, M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. École Norm. Sup. 71, 101–190 (1954) 20. Lichtman, A.I.: Growth in enveloping algebras. Israel J. Math. 47(4), 297–304 (1984) 21. Petrogradsky, V.M.: On types of overexponential growth of identities in Lie PI-algebras. (Russian) Fundam. Prikl. Mat. 1(4), 989–1007 (1995) 22. Petrogradsky, V.M.: Intermediate growth in Lie algebras and their enveloping algebras. J. Algebra 179, 459–482 (1996) 23. Petrogradsky, V.M.: Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions. Mat. Sb. 188(6), 119–138 (1997); translation in Russian Acad. Sci. Sb. Math. 188(6), 913–931 (1997) 24. Petrogradsky, V.M.: Exponential generating functions and complexity of Lie varieties. Israel J. Math. 113, 323–339 (1999) 25. Petrogradsky, V.M.: Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle. Int. J. Algebra Comput. 9(2), 179–212 (1999)

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26. Petrogradsky, V.M.: Exponential Schreier’s formula for free Lie algebras and its applications. Algebra, 11. J. Math. Sci. (New York) 93(6), 939–950 (1999) 27. Petrogradsky, V.M.: Schreier’s formula for free Lie algebras. Arch. Math. (Basel) 75(1), 16–28 (2000) 28. Petrogradsky, V.M.: On growth of Lie algebras, generalized partitions, and analytic functions. (Formal power series and algebraic combinatorics, Vienna, 1997). Discrete Math. 217(1–3), 337–351 (2000) 29. Petrogradsky, V.M.: On generating functions for subalgebras of free Lie superalgebras. Discrete Math. 246(1–3), 269–284 (2002) 30. Petrogradsky, V.M.: Enumeration of algebras close to absolutely free algebras and binary trees. J. Algebra 290(2), 337–371 (2005) 31. Petrogradsky, V.: Scale for codimension growth of Poisson PI-algebras. Israel J. Math. 254, 201–227 (2023) 32. Ratseev, S.M.: Correlation of Poisson algebras and Lie algebras in the language of identities. Math. Notes 96(3–4), 538–547 (2014); Translation of Mat. Zametki 96(4), 567–577 (2014) 33. Razmyslov, Yu.P.: Identities of Algebras and Their Representations. AMS, Providence, RI (1994) 34. Regev, A.: Existence of identities in A ⊗ B. Israel J. Math. 11, 131–152 (1972) 35. Shmel’kin, A.L.: Free polynilpotent groups. Izv. Akad. Nauk SSSR Ser. Mat., vol. 28(1), pp. 91–122 (1964); English transl., Amer. Math. Soc. Transl. ser. 2, vol. 55, pp. 270–304. American Mathematical Society, Providence, RI (1966) 36. Ufnarovskiy, V.A.: Combinatorial and asymptotic methods in algebra, Itogi Nauki i Tekhniki, Sovrem. Probl. Mat. Fund. Naprav. vol. 57, Moscow (1989); Engl. transl., Encyclopaedia Math. Sci., vol. 57, Algebra VI. Springer, Berlin (1995) 37. Volichenko, I.B.: On variety AN2 over field of zero characteristic. Dokl. Akad. Nauk Belarusi XXV 12, 1063–1066 (1981) 38. Zaitsev, M.V.: Integrality of exponents of growth of identities of finite-dimensional Lie algebras. Izv. Ross. Akad. Nauk Ser. Mat. 66, no. 3, 23–48 (2002); translation in Izv. Math. 66, no. 3, 463–487 (2002) 39. Zaicev, M., Mishchenko, S.P.: An example of a variety of Lie algebras with a fractional exponent. J. Math. Sci. 93, 977–982 (1999)

Leibniz Algebras

Universal Central Extensions of Compatible Leibniz Algebras José Manuel Casas Mirás and Manuel Ladra

This paper is dedicated with all our fondness to our great friend and excellent person Alberto Elduque on the occasion of his 60th birthday.

Abstract We show the interplay between compatible Leibniz algebras and compatible associative dialgebras by means of a commutative diagram. We construct a homology with trivial coefficients for compatible Leibniz algebras and use it to construct the universal central extension of a perfect compatible Leibniz algebra. Furthermore, we conjecture that the category of compatible Leibniz algebras does not satisfy the UCE condition, namely, the composition of central extensions (the middle term in one of them must be perfect) is also a central extension. Keywords Compatible Leibniz algebra · Compatible associative dialgebra · Homology · Universal central extension

1 Introduction Two algebraic structures of the same type (i.e. both are associative algebras, Lie algebras, etc.) (V, ◦) and (V, ∗) with the same underlying vector space are said to J. M. Casas Mirás Departamento de Matemática Aplicada I & CITMAga, Universidade de Vigo, 36005 Vigo, Spain e-mail: [email protected] M. Ladra (B) Departamento de Matemáticas & CITMAga, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_6

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be compatible if (V, λ · ◦ + μ · ∗) has the same algebraic structure as the first ones for any scalars λ, μ. When λ = 1, the product x  y = x ◦ y + μ (x ∗ y) can be considered as a deformation of the product ◦ in the parameter μ. For instance, a compatible Lie algebra [17] is a vector space g endowed with two skew-symmetric bracket operations [−, −] and (−, −) such that the following identity holds for any x, y, z ∈ g ([x, y], z) + ([y, z], x) + ([z, x], y) + [(x, y), z] + [(y, z), x] + [(z, x), y] = 0. (1) This compatible Jacobi identity (1) is derived by imposing on the bracket {x, y} = λ[x, y] + μ(x, y), for any λ, μ ∈ K, the Jacobi identity, bearing in mind that (g, [−, −]) and (g, (−, −)) are Lie algebras. Other compatible structures and its operadic study were considered for example in [6, 17]. Compatible algebraic structures are considered in a lot of fields in mathematics and mathematical physics. For instance, compatible Lie algebras appear in the context of integrable Hamiltonian equations [14] or in the context of the Yang-Baxter equation and principal chiral field [7] or elliptic theta functions [8]. Bi-hamiltonian structures play an important role in the theory of integrable systems from mathematical physics. Such structures correspond to pairs of compatible Poisson brackets defined on the same manifold. The operads of Lie compatible algebras and bi-Hamiltonian algebras have been studied by V. Dotsenko and A. Khoroshkin (see [6]), and free objects in the category of compatible associative algebras were studied by V. Dotsenko in [5]. The notion of semi-abelian category (see [9]) is designed to capture typical algebraic properties of categories such as that of groups, associative algebras without unit, Lie algebras, etc.; as well as abelian categories allow a generalized treatment of abelian-group and module theory. A semi-abelian category is a Barr-exact (i.e. a regular category in which every congruence is a kernel pair), has a zero object, finite coproducts and is protomodular (there is an intrinsic notion of normal subobject). The general theory of universal central extensions is well suited when the underlying semi-abelian category satisfies an additional requisite, called the universal central extension condition or (UCE) in [3], that is: → → C are central extensions in If B is a perfect object and f : A − → → B and g : B − a semi-abelian category, then the extension f

g

A −→ → B −→ →C is also central. The condition UCE holds for groups, Lie algebras, Leibniz algebras, associative dialgebras (all of them are categories of interest). On the other hand, the categories of non-associative algebras over a field and Hom-Lie algebras are semi-abelian but do not satisfy UCE (see [4] and [2]), which shows that this condition does not hold in an arbitrary semi-abelian category.

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Indeed, for any perfect object U of a semi-abelian category, the statements 1. 2. 3. 4.

each central extension U − → → X is universal; each central extension of U splits; each universal central extension of U splits; H2 (U ) = 0;

are equivalent if and only if UCE holds. In this paper, after the preliminaries section (Sect. 2) where we recall the concepts of Leibniz algebras and associative dialgebras, we introduce the structures of compatible associative dialgebras and compatible Leibniz algebras, showing their interconnection through the commutativity of the following diagram:

where CDias, CAss, CLie and CLeib denote the categories of compatible associative dialgebras, compatible associative algebras, compatible Lie algebras and compatible Leibniz algebras, respectively (see Sect. 3). In Sect. 4, we construct a homology with trivial coefficients for compatible Leibniz algebras (g, [−, −], (−, −)) and show its relation to the homology with trivial coefficients of the Leibniz algebras (g, [−, −]) and (g, (−, −)) through the long exact sequence (6). Finally, in Sect. 5 we study universal central extensions of compatible Leibniz algebras, where the most relevant results we get are: 1. A compatible Leibniz algebra admits a universal central extension if and only if it is perfect. Moreover, the kernel of the universal central extension is the first homology with trivial coefficients of the perfect compatible Leibniz algebra. 2. We conjecture that the category CLeib does not satisfy the UCE condition, established in [3]. Namely, the composition of central extensions (the middle term in one of them must be perfect) is also a central extension. The fact described in the second point does not allow to obtain classical characterizations of universal central extensions. On the other hand, it is necessary to find an example that validates our conjecture.

2 Preliminaries on Leibniz Algebras A Leibniz algebra [11, 13] is a K-vector space (hereinafter, K is a field) g equipped with a bracket operation [−, −] : g ⊗ g → g satisfying the Leibniz identity [x, [y, z]] = [[x, y], z] − [[x, z], y], for all x, y, z ∈ g. There is a canonical inclusion functor that considers a Lie algebra as a Leibniz algebra, which is right adjoint to the Liezation functor Lie2 that assigns

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to a Leibniz algebra g the Lie algebra gLie = ggann , where gann = {[x, x] | x ∈ g}. A homomorphism of Leibniz algebras is a K-linear map which preserves the bracket. We denote by Leib the category of Leibniz algebras and homomorphisms between them. A subalgebra n of a Leibniz algebra g is called a two-sided ideal if [x, y], [y, x] ∈ n for all x ∈ n and y ∈ g. The center of a Leibniz algebra g is the two-sided ideal Z(g) = {c ∈ g | [c, x] = 0 = [x, c] for all x ∈ g}. An abelian Leibniz algebra is a Leibniz algebra with the trivial bracket. It is clear that a Leibniz algebra g is abelian if and only if Z(g) = g. An associative dialgebra [12] is a K-vector space D equipped with two operations , : D ⊗ D → D, called respectively left and right products, satisfying the following axioms for all x, y, z ∈ D, (x y) z = x (y z), (x y) z = x (y z), (x y) z = x (y z), (x y) z = x (y z), (x y) z = x (y z). A homomorphism of dialgebras is a K-linear map preserving both left and right products. We denote by Dias the category of dialgebras and homomorphisms between them. Any associative algebra becomes a dialgebra with x y = x · y = x y, so we have an inclusion functor Ass → Dias. This functor has a left adjoint functor Ass : Dias → Ass which assigns to a dialgebra D the quotient of D by the ideal generated by all elements x y − x y, x, y ∈ D. Furthermore, we have a functor Lb : Dias → Leib, that assigns to a dialgebra   (D, , ) the Leibniz algebra D, [−, −] , where [x, y] = x y − y x. The functor Lb admits as left adjoint the functor Ud : Leib → Dias, that assigns to a Leibniz algebra g its universal enveloping dialgebra, defined as the following quotient of the free dialgebra over the underlying K-vector space of g Ud (g) = T (g) ⊗ g ⊗ T (g) / {[x, y] − x y + y x | x, y ∈ g}. Therefore, we have the following diagram of categories and functors (Loday’s wonderful squares):

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3 Compatible Leibniz Algebras Definition 1 Let (D, , ) and (D,  ,  ) be two associative dialgebras. It is said that (D, , ) and (D,  ,  ) are compatible if the bilinear operations ⫣ = λ· +μ·  ⊩ = λ· +μ·  define an associative dialgebra structure on D, for any λ, μ ∈ K. In this case, we call (D, , ,  ,  ) a compatible associative dialgebra. Proposition 1 Let (D, , ) and (D,  ,  ) be two associative dialgebras. Then (D, , ,  ,  ) is a compatible associative dialgebra if and only if the following conditions hold for any x, y, z ∈ D: (i) (ii) (iii) (iv) (v)

(x (x (x (x (x

 

 



y) z + (x y) z + (x y) z + (x y) z + (x y) z + (x





y)  y)  y)  y)  y) 

z z z z z

= = = = =

x x x x x

(y (y

(y

(y

(y

  





z) + x z) + x z) + x z) + x z) + x

 







(y (y (y (y (y

z), z), z),

z),

z).

A homomorphism of compatible associative dialgebras is a K-linear map preserving the four operations. We denote by CDias the category of dialgebras and homomorphisms between them. Example 1 1. A compatible associative algebra [15] is a triple (A, ·, ◦) where · and ◦ are associative products over the K-vector space A such that (A, λ · +μ ◦) has structure of associative algebra. This property is equivalent to the following identity holds for all x, y, z ∈ A: (x ◦ y) · z + (x · y) ◦ z = x · (y ◦ z) + x ◦ (y · z). If we take = = · and  =  = ◦, then (A, , ,  ,  ) is a compatible associative dialgebra. This assignment defines an inclusion functor CAss → CDias, where CAss denotes the category of compatible associative algebras. By the way, it is easy to check that a compatible associative algebra (A, ·, ◦) gives rise to a compatible Lie algebra by making [x, y] = x · y − y · x and (x, y) = x ◦ y − y ◦ x. This assignment defines a functor Lie1 : CAss → CLie, where CLie denotes the category of compatible Lie algebras. 2. A totally compatible (associative) dialgebra [19] is a K-vector space A endowed with two associative operations ∗, : A ⊗ A → A satisfying the following identities for all a, b, c ∈ A, (a ∗ b) c = a ∗ (b c) = (a b) ∗ c = a (b ∗ c).

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Then A ⊗ A endowed with the operations (a ⊗ b) (a  ⊗ b ) = a ⊗ (b ∗ a  ∗ b ), (a ⊗ b) (a  ⊗ b ) = (a ∗ b ∗ a  ) ⊗ b , (a ⊗ b)  (a  ⊗ b ) = a ⊗ (b a  b ), (a ⊗ b)  (a  ⊗ b ) = (a b a  ) ⊗ b , is a compatible associative dialgebra. Definition 2 Let (g, [−, −]) and (g, (−, −)) be two Leibniz algebras. It is said that (g, [−, −]) and (g, (−, −)) are compatible if the bilinear operation {−, −} = λ · [−, −] + μ · (−, −) defines a Leibniz algebra structure on g, for any λ, μ ∈ K. In this case, we will say that (g, [−, −], (−, −)) is a compatible Leibniz algebra. Proposition 2 Let (g, [−, −]) and (g, (−, −)) be two Leibniz algebras. Then (g, [−, −], (−, −)) is a compatible Leibniz algebra if and only if the following condition holds for any x, y, z ∈ g: [x, (y, z)] − [(x, y), z] + [(x, z), y] + (x, [y, z]) − ([x, y], z) + ([x, z], y) = 0. (2) Proof Direct checking.



A homomorphism of compatible Leibniz algebras is a K-linear map f : (g , [−, −] , (−, −) ) → (g, [−, −], (−, −)) such that f [x, y] = [ f (x), f (y)], f (x, y) = ( f (x), f (y)), for all x, y ∈ g . Thus we have defined the semi-abelian category CLeib, whose objects are the compatible Leibniz algebras and whose morphisms are the homomorphisms of compatible Leibniz algebras. Example 2 1. Let (D, , ,  ,  ) be a compatible associative dialgebra, then (D, [−, −], (−, −)), where [x, y] = x y − y x, (x, y) = x  y − y  x, x, y ∈ D, is a compatible Leibniz algebra. This fact defines a functor CDias → CLeib. 2. Every compatible Lie algebra is also a compatible Leibniz algebra. This assignment defines an inclusion functor CLie → CLeib, where CLie denotes the category of compatible Lie algebras. Conversely,associatedwithacompatibleLeibnizalgebra(g, [−, −], (−, −))there is the compatible Lie algebra (gLie , [−, −], (−, −)). The homomorphism g  gLie is universal for all homomorphism of compatible Lie algebras f : g → h,

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which means that the above assignment defines a functor Lie2 : CLeib → CLie which is left adjoint to the inclusion functor CLie → CLeib. 3. A Lie-Leibniz algebra [1] is a K-vector space L, equipped with two bilinear maps [−, −], (−, −) : L × L → L, satisfying the following identities for all x, y ∈ L: (i) [x, x] = 0, (ii) (x, [y, z]) = ([x, y], z) − ([x, z], y), (iii) [x, (y, z)] = [(x, y), z] − [(x, z), y]. Obviously, Lie-Leibniz algebras are compatible Leibniz algebras. Conversely, associated with a compatible Leibniz algebra (g, [−, −], (−, −)) there is the LieLL = g/gann , [−, −], (−, −)), where gann = {[x, x], x ∈ g}. Leibniz algebra (gLie LL is universal for all homomorphism of Lie-Leibniz The homomorphism g  gLie algebras f : g → h. Clearly, if we divide a Lie-Leibniz algebra (g, [−, −], (−, −)) by the smallest two-sided ideal spanned by the elements (x, x), x ∈ g, then we obtain a compatible Lie algebra. We have the functors Lie1 : CAss −→ CLie (A, ·, ◦) → (A, [−, −], (−, −)) with [x, y] = x · y − y · x,

(x, y) = x ◦ y − y ◦ x, and

Lie2 : CLeib → CLie g → gLie = g/ { [x, x], (x, x) | x ∈ g} , with the two induced brackets. Now it is an easy task to show the commutativity of the following diagram using the multiplication changing functor.

Furthermore, there is the functor U2 : CLie → Ass is given by

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 U2 (g) = As X ∪ X | x y − yx + [x, y] x  y  − y  x  + (x, y) x y  − y  x + x  y − y  x + [x, y] + (x, y)



and where – X is a linear basis of g, – X  is a set such that X ∩ X  = ∅, and – the map  : X → X  , x → x  , is bijective. Let (g, [−, −], (−, −)) be a compatible Leibniz algebra. A subalgebra h of g is a K-vector subspace which is closed under both bracket operations, i.e. [h, h] ⊆ h and (h, h) ⊆ h. A subalgebra h of g is said to be a two-sided ideal if [h, g] ⊆ h, [g, h] ⊆ h, (h, g) ⊆ h, (g, h) ⊆ h. If h is a two-sided ideal of g, then g/h is endowed with a natural compatible Leibniz algebra structure induced by the bracket operations in g. The commutator ideal of two two-sided ideals h and j of g is the two-sided ideal [(h, j)] = {[h, j], [ j, h], (h, j), ( j, h), h ∈ h, j ∈ j}. The center of a compatible Leibniz algebra (g, [−, −], (−, −)) is the two-sided ideal Z(g) = {g ∈ g | [g, g  ] = [g  , g] = (g, g  ) = (g  , g) = 0, for all g  ∈ g}. The compatible Leibniz algebra (g, [−, −], (−, −)) is said to be abelian if both bracket operations are trivial, that is [g, g] = 0 = (g, g). Obviously, (g, [−, −], (−, −)) is an abelian compatible Leibniz algebra if and only if [(g, g)] = 0 if and only if Z(g) = g. An extension of compatible Leibniz algebras is a short exact sequence of compatible Leibniz algebras i

π

→ (g , [−, −] , (−, −) ) − → (g, [−, −], (−, −)) → 0, 0 → (h, [−, −] , (−, −) ) − i.e. π is a surjective homomorphism of compatible Leibniz algebras, i is injective and Ker(π ) = Im(i). Extensions of compatible Lie algebras are studied in [10].

4 Homology with Trivial Coefficients Let (g, [−, −], (−, −)) be a compatible Leibniz algebra and consider the double complex [18] (C∗,∗ , d∗ , δ∗ ), where C p,q = g⊗ p+q+1 , di+ p+1 : Ci, p → Ci−1, p is given by di+ p+1 (x1 ⊗ · · · ⊗ xi+ p+1 )  (−1)k+1 x1 ⊗ · · · ⊗ x j−1 ⊗ [x j , xk ] ⊗ · · · ⊗ xk ⊗ · · · ⊗ xi+ p+1 , = 1≤ j 0,

j=1

from ω(K [Yd ])ω(K [Z d ]) ⊂ K [Yd , Z d ]. The multiplication in Rd is defined by: xi x j = yi z j , β

β

xi (Ydα Z d ) = yi Ydα Z d , β

β

(Ydα Z d )x j = Ydα Z d z j , β

γ

α+γ

(Ydα Z d )(Yd Z dδ ) = Yd

β+δ

Zd

.

Here Rd2 is a vector subspace of ω(K [Yd , Z d ]). Clearly, the algebra Rd is generated by the set X d , the multiplication in Rd2 is as in the polynomial algebra K [Yd , Z d ] and Rd2 has the structure of a K [X d ]-bimodule. Proposition 1 ([15]) The algebra Fd (B) is isomorphic to the algebra Rd both as an algebra and as a multigraded vector space with an isomorphism defined by xi → xi , i = 1, . . . , d. Till the end of the paper we shall work in Rd instead of in Fd (B) but shall state the results for Fd (B). We assume that the group G L d (K ) acts on K Yd and K Z d in the same way as on K X d .

3 The Finite Generation Problem The famous Endlichkeitssatz of Emmy Noether [27] states the following. Theorem 1 For any finite subgroup G of G L d (K ) the algebra of invariants K [X d ]G is finitely generated. In the special case of finite groups the following result for Fd (B)G may be considered as an analogue of the theorem of Emmy Noether. Theorem 2 Let G be a subgroup of G L d (K ). (i) If every system of generators of the algebra of invariants K [X d ]G contains polynomials which are of degree ≥ 2, then the algebra of invariants Fd (B)G is not finitely generated. (ii) If the algebra K [Yd , Z d ]G is finitely generated, then Fd2 (B)G is a finitely generated K [Yd , Z d ]G -module.

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Proof (i) Changing the basis of the vector space K X d we may assume that the vector space (K X d )G of linear G-invariants has a basis X m = {x1 , . . . , xm } and m < d because by assumption K [X d ]G contains polynomials which do not belong to K [X m ] ⊂ K [X d ]G . Therefore K [X d ]G is generated by x1 , . . . , xm and by homogeneous polynomials h(X d ) which depend essentially on some of the variables xm+1 , . . . , xd . Let h 0 (X d ) ∈ K [X d ]G be such a polynomial and let for example it depends also on xd . As we already stated, we shall work in the algebra Rd instead of in Fd (B). Let us assume that RdG is finitely generated. Hence it has a system of generators consisting of x1 , . . . , xm and f 1 (Yd , Z d ), . . . , f n (Yd , Z d ) ∈ (Rd2 )G . We may assume that all f i (Yd , Z d ) are homogeneous with respect to Yd and to Z d . Since h 0 (Yd )h 0 (Z d ) belongs to RdG and depends on the variables yd and z d , it does not belong to the subalgebra of Rd generated by X m . Hence RdG has at least one generator f i (Yd , Z d ) ∈ (Rd2 )G . Let kmax = max{deg Z d ( f i (Yd , Z d )) | i = 1, . . . , n}. Since degYd ( f (Yd , Z d )) ≥ 1 for all nonzero f (Yd , Z d ) ∈ Rd2 and the same holds for deg Z d ( f i (Yd , Z d )), we have kmin = min{degYd ( f (Yd , Z d )) | 0 = f (Yd , Z d ) ∈ (Rd2 )G } ≥ 1. Let degYd ( f 0 (Yd , Z d )) = kmin for some f 0 (Yd , Z d ) ∈ (Rd2 )G . Then all polynomials p

v p (Yd , Z d ) = f 0 (Yd , Z d )h 0 (Z d ), p = 0, 1, 2, . . . , are G-invariants with the property degYd (v p (Yd , Z d )) = kmin . Hence they can be expressed as linear combinations of u (α,β,γ) (Yd , Z d ) =

m 

yiαi

i=1

n  j=1

γ

f j j (Yd , Z d )

m 

β

zi i

i=1

such that degYd (u (α,β,γ) (Yd , Z d )) =

m  i=1

αi +

n 

γ j degYd ( f j (Yd , Z d )) = kmin .

j=1

All u (α,β,γ) (Yd , Z d ) are homogeneous both in Yd and Z d . If γ1 + · · · + γn > 1, then degYd (u (α,β,γ) (Yd , Z d )) ≥ 2kmin > kmin which is impossible. Hence either γ1 + · · · + γn = 1 and α1 = · · · = αm = 0 or γ1 = · · · = γn = 0 and α1 + · · · + αm = kmin . In both cases degzd (u (α,β,γ) (Yd , Z d )) ≤ max{deg Z d ( f i (Yd , Z d )) | i = 1, . . . , n} = kmax .

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Since h 0 (X d ) essentially depends on xd we obtain that degzd (v p (Yd , Z d )) ≥ p which is impossible for p > kmax and this completes the proof of (i). (ii) Let the algebra K [Yd , Z d ]G be finitely generated. The multiplication in K [Yd , Z d ] defines a natural action on Fd2 (B): yi ∗ f (X d ) = xi f (X d ), z i ∗ f (X d ) = f (X d )xi , f (X d ) ∈ Fd2 (B), i = 1, . . . , d. Translating in terms of Rd it is sufficient to show that (Rd2 )G is a finitely generated K [Yd , Z d ]G -module. Since Rd2 is an ideal of K [Yd , Z d ], we obtain that (Rd2 )G is an ideal of the finitely generated algebra K [Yd , Z d ]G . By the Basissatz [22] of Hilbert (Rd2 )G is a finitely generated ideal, i.e. a finitely generated K [Yd , Z d ]G -module.  In order to specify Theorem 2 in the case of finite groups we need the following easy lemma. Lemma 1 Let G be a finite subgroup of G L d (K ). Then the algebra K [Yd , Z d ] is integral over the subalgebra K [Yd ]G K [Z d ]G . Proof Let G = {g1 , . . . , gn }. Then yi satisfies the equation n  j=1

(x − g j (yi )) = x n +

n  (−1)k ek (g1 (yi ), . . . , gn (yi ))x n−k = 0, k=1

where ek (X n ) = ek (x1 , . . . , xn ) =



xi1 · · · xik , k = 1, . . . , n,

i 1 d. Working in Rd instead of in Fd (B) and changing linearly the generators of the subalgebra we may assume that f1 ∈ Rd2 . Since f 22 also belongs to Rd2 and Rd2 is a commutative and associative algebra we obtain that f 1 ( f 2 f 2 ) = ( f 2 f 2 ) f 1 . On the other hand, x1 (x2 x2 ) = y1 y2 z 2 = y2 z 1 z 2 = (x2 x2 )x1 in Rd . Therefore f 1 and f 2 cannot be free generators of a free subalgebra of Fd (B). 

6 Symmetric Polynomials in Free Bicommutative Algebras As usually, we assume that the symmetric group Sd acts on K X d as a group of permutation matrices. We shall need the following description of the symmetric polynomials K [Yd , Z d ] Sd in two sets of variables Yd and Z d . The general case of symmetric polynomials in any number of sets of variables is given by Schläfli [28], see also the books by MacMahon [25] and Weyl [31]. Theorem 7 The algebra K [Yd , Z d ] Sd is generated by the elementary symmetric polynomials ek (Yd ), ek (Z d ), k = 1, . . . , d, and their polarizations e p,q (Yd , Z d ) =



yi1 · · · yi p z j1 · · · z jq ,

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where p, q ≥ 1, p + q ≤ d, and the summation is on all ( p, q)-shuffles, that is, tuples (i 1 , . . . , i p , j1 , . . . , jq ) of pairwise different integers such that 1 ≤ i 1 < · · · < i p ≤ d, 1 ≤ j1 < · · · < jq ≤ d. The following result describes the symmetric polynomials in Fd (B). Theorem 8 The vector space Fd (B) Sd decomposes as Fd (B) Sd = K e1 (X d ) ⊕ Fd2 (B) Sd . There exist positive integers n p,q , where p, q are like in Theorem 7, such that, identifying the powers Fd2 (B) and Rd2 , the K [Yd ] Sn K [Z d ] Sd -module (Rd2 ) Sd is generated by a finite number of products 

k p,q

e p,q (Yd , Z d ), k p,q ≤ n p,q ,



k p,q ≥ 1, and e p (Yd )eq (Z d ), p, q = 1, . . . , d.

p+q≤d

Proof Since e1 (X d ) is the only symmetric linear polynomial in Fd (B) it is sufficient to prove the statement for (Rd2 ) Sd . By Theorem 7 (Rd2 ) Sd is spanned by the products d 

l e pp (Yd )

p=1



k p,q e p,q (Yd ,

p+q≤d

Zd )

d 

m

eq q (Z d )

q=1

which depend both on Yd and Z d . Hence (Rd2 ) Sd is generated as a K [Yd ] Sd K [Z d ] Sd module by the products 

k

p,q e p,q (Yd , Z d ) and e p (Yd )eq (Z d ).

p+q≤d

Now the proof follows from Corollary 1 (ii).



Example 1 Applying Theorem 7 for d = 2 we obtain that the algebra K [Y2 , Z 2 ] S2 is generated by e1 (Y2 ), e2 (Y2 ), e1 (Z 2 ), e2 (Z 2 ), e1,1 (Y2 , Z 2 ) = y1 z 2 + y2 z 1 . Direct computations show that 2 e1,1 (Y2 , Z 2 ) = e1 (Y2 )e1 (Z 2 )e1,1 (Y2 , Z 2 ) − e12 (Y2 )e2 (Z 1 ) − e2 (Y2 )e12 (Z 2 ) + 4e2 (Y2 )e2 (Z 2 ).

Hence as a K [Y2 ] S2 K [Z 2 ] S2 -module K [Y2 , Z 2 ] S2 is generated by 1 and e1,1 (Y2 , Z 2 ) and by Theorem 8 (R22 ) S2 is generated as a K [Y2 ] S2 K [Z 2 ] S2 -module by e1,1 (Y2 , Z 2 ) and the products e p (Y2 )eq (Z 2 ), p, q = 1, 2. Acknowledgements The author is very grateful to the anonymous referees for the careful reading and the valuable suggestions for improving the exposition.

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An Approach to the Classification of Finite Semifields by Quantum Computing J. M. Hernández Cáceres and I. F. Rúa

Abstract Finite semifields are finite nonassociative rings with an identity element, such that the set of nonzero elements is a loop under the product. Their number of elements is a prime power, known as order. They were considered first by Dickson [2], and studied by Albert [1] and Knuth [3]. Finite semifields of order 16 have been classified by Kleinfeld in [7], of order 32 by Knuth in [3], and by Walker in [8]. The classification of finite semifields can be rephrased as a problem of finding certain sets of matrices, which can be solved by computer search. So, classification of semifields with high order such as 64, was achieved by Rúa, Combarro, Ranilla in [4], or of order 243 by Rúa, Combarro, Ranilla in [5], and of order 81 by Dempwolff [9]. Based on this approach, classification of finite semifields of any order via a quantum procedure is possible. We present quantum techniques for the classification of semifields with 8 and 16 elements with their respective simulations, based on Grover’s quantum search algorithm. Keywords Finite semifields · Grover algorithm · Classification

1 Introduction It is well known that some quantum algorithms outperform their classical counterparts. For instance, Deutsch-Jozsa [10] was the first example of a quantum algorithm that performs better than the best classical algorithm. Integer factoring can be solved in polynomial time by Shor’s algorithm [11], while no classical algorithm with such complexity is known. A marked element in a list of N unordered ones (unstructured √ search) can be found by Grover’s algorithm [12] in expected time O( N ) (representing a quadratic advantage over classical algorithms). Additionally, it has also J. M. Hernández Cáceres · I. F. Rúa (B) University of Oviedo, Department of Mathematics, C/Calvo Sotelo s/n 33007, Oviedo, Spain e-mail: [email protected] J. M. Hernández Cáceres e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_16

245

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been shown in [13], that Grover’s algorithm is √ optimal in the sense that no quantum Turing machine can do this in less than O( N ) operations. In the computational (or effective) study of algebraic structures, Grover’s algorithm has been successfully applied for testing the commutativity of a finite dimensional algebra. In this direction, as was pointed out in [6] and [5], since the classification of all finite semifields of size 128 is completely out of reach with current classical computing technology, and since Grover’s algorithm has a quadratic speedup for finding marked items in long lists, the application of quantum computing to classify finite semifields seems promising. In this work, we introduce a quantum procedure for classifying finite semifields with 8 and 16 elements, based on Grover’s quantum search algorithm. We provide simulations supporting our approach, and we also discuss the scalability of the method to higher orders. The outline of the paper is as follows. In Sect. 2, we present notions on finite semifields and how the classification of finite semifields can be rephrased as the problem of finding certain sets of matrices. In Sect. 3, we present notions on the quantum circuit model and Grover’s algorithm. In Sect. 4, we present the main results and test them using a simulator for quantum circuits, finding the multiplication tables for the finite semifield F8 (which is the only finite semifield of order 8), and for finite commutative semifields of order 16. Finally, in Sect. 5, we draw some conclusions from this new approach to the classification of finite semifields.

2 Preliminaries In this section we collect definitions and facts on finite semifields. Proofs can be found, for instance, in [3]. Definition 1 A finite nonassociative ring D is called a presemifield if the set of nonzero elements D ∗ is closed under the product. If D has an identity element, then it is called a (finite) semifield. If D is a finite semifield, then D ∗ is a multiplicative loop. That is, there exists an element e ∈ D ∗ (the identity of D) such that ex = xe = x, for all x ∈ D, and for all a, b ∈ D ∗ , the equation ax = b (respectively xa = b) has a unique solution. The term finite semifield is due to Knuth. Any finite field is an associative finite semifield. The term proper finite semifield will mean a finite semifield in which the multiplication is not associative. Finite semifields are also known as finite nonassociative division algebras. Proposition 1 The characteristic of a finite semifield D is a prime number p. The associative-commutative center Z (D) of D is the finite field Fq with q = p c elements (c ∈ N). D is a finite-dimensional algebra over Z (D) of dimension d where |D| = q d . Knuth proved that a proper semifield of least order has 16 elements by showing that the only finite semifield of order 23 is the finite field F8 .

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247

Now, let S be a non-associative finite dimensional algebra over a field K . Let us fix a K -basis β = {x1 , . . . , xd } of S, so there exists a unique set of constants {Mi jk }i,d j,k=1 ⊆ K such that xi · x j =

d 

Mi jk xk for all i, j ∈ {1, . . . , n}.

k=1

The map L x : S → S given by L x (a) = x · a is a K -linear homomorphism. The column coordinate matrix of L xi with respect to β is the following: ⎡

Mi11 Mi21 Mi31 ⎢ Mi12 Mi22 Mi32 ⎢ Ai = ⎢ . .. .. ⎣ .. . . Mi1d Mi2d Mi3d

⎤ . . . Mid1 . . . Mid2 ⎥ ⎥ .. ⎥ . .. . . ⎦ . . . Midd

Consider the set M := {A1 , . . . , Ad }. Then, if x = d

αi L xi , and so we have that A x =

i=1

d

d

αi xi with αi ∈ K then L x =

i=1

αi Ai .

i=1

Note that x = 0 is not a zero divisor if and only if L x is a K -linear isomorphism if and only if A x is invertible. Moreover, S is a division algebra (in particular, a finite semifield) if and only if any non zero linear combination of M is invertible. The study of finite semifields can be rephrased as a problem of matrices, as the following proposition shows. Proposition 2 ([4]) Any finite semifield D of order q d can be described by a set of d matrices {A1 , . . . , Ad }, known as standard basis, such that 1. A1 is the identity matrix. d αi Ai is invertible for all nonzero tuples (α1 , . . . , αd ) ∈ Fqd ; 2. i=1

↓

3. The first column of the matrix Ai is the column vector ei with a 1 in the ith position, and 0 everywhere else. The semifield D can be identified with the algebra (Fqd , +, ·) where the multiplid xi Ai y. cation is given by x · y = i=1

As a consequence of this proposition we have: Corollary 1 Any non-scalar linear combination of a standard basis has a characteristic polynomial without linear factors.

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3 Quantum Preliminaries and Grover’s Quantum Search The quantum circuit model is one of the most popular models for quantum computing. In it, qubits store data, operations are performed with quantum gates, and results are obtained via measurements. All of these are ruled by the laws of quantum mechanics. So, let us recollect some basic notions on qubits and quantum gates that will be useful for the next section. Details can be found for instance in [15]. Let us begin with some notation. A ket is a term of the form |v . Mathematically it denotes a vector v ∈ Cn . The bra of a vector v ∈ Cn , denoted by v| , is defined as v| = (|v )† , where † is the conjugate transpose. In contrast to normal bits, which can be in t 1, a quantum bit t state 0 or in state or qubit for short, can be in state |0 = 1 0 or |1 = 0 1 (where t means the transpose of a matrix), or it can be in a superposition α |0 + β |1 where α, β ∈ C and |α|2 + |β|2 = 1, which are called the amplitudes of the state. All of these possible values for the state of a single qubit are normalized states of vectors in C2 . In fact, {|0 , |1 } , is an orthonormal basis of C2 , which is called the computational basis. n The computational basis for C2 would be {|0 ⊗ |0 ⊗ · · · ⊗ |0 , |0 ⊗ |0 ⊗ · · · ⊗ |1 , . . . , |1 ⊗ |1 · · · ⊗ |1 }

= {|0 , |1 , . . . , 2n − 1 } so, a generic state of a multi-qubit system is |ψ =

n 2 −1

i=0

αi |i where

n 2 −1

|αi |2 = 1

i=0

and αi ∈ C for 0 ≤ i ≤ 2n − 1. Now, in order to perform computations, we need to manipulate the state of qubits. In the quantum circuit model, operations are a special type of linear transformations applied to the vectors representing the states of the qubits. These linear transformations are represented by unitary matrices and they are called quantum gates. Recall that a matrix U is unitary if U † U = UU † = In , where In is the identity matrix of size n × n. Some important one-qubit quantum gates are  H=

√1 2

       1 0 1 1 10 01 π , I2 = , X= , T = 1 −1 01 10 0 ei 4

where the X gate is the quantum version of the NOT gate, H is the Hadamard gate (which achieves superposition), and T is known as the π/8 half-phase gate. Since the tensor product between two unitary operators is unitary, two one-qubit gates acting on each qubit independently form a two-qubit gate whose action is given by U |ψ1 ⊗ |ψ2 = (U1 ⊗ U2 )(|ψ1 ⊗ |ψ2 ) = U1 |ψ1 ⊗ U2 |ψ2 where |ψ1 , |ψ2 are two states of qubits and U1 and U2 are one-qubit gates acting respectively on states |ψ1 and |ψ2 . In general, we can construct a n-qubit gate U as U = U1 ⊗ U2 where U1 is a n 1 -qubit gate, U2 is a n 2 -qubit gate and n = n 1 + n 2 . For example, if we take the gate H and make the tensor product of itself n times we

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249

have a n-qubit gate H ⊗n = H ⊗ H ⊗ · · · ⊗ H. However, there are unitary matrices that cannot be written as the tensor product of smaller unitary matrices. Indeed, an example of this is one of the most important two-qubit gates, the CNOT or controlledNOT, given by ⎡

1 ⎢0 C N O T = |0 0| ⊗ I2 + |1 1| ⊗ N O T = ⎢ ⎣0 0

0 1 0 0

0 0 0 1

⎤ 0 0⎥ ⎥. 1⎦ 0

The CNOT gate in a quantum circuit is represented as •

(1)

X The horizontal lines are called wires and they represent the qubits that we are working with, and the circuit is read from left to right. An example of a three-qubit gate is the CCNOT or Toffoli gate. The CCNOT is a doubly-controlled NOT gate. Its matrix is given by ⎡

1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 CC N O T = |0 0| ⊗ I4 + |1 1| ⊗ C N O T = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎥ ⎥ 1⎦ 0

The Toffoli gate is a universal reversible logic gate, in fact it can be used to make the AND gate. For instance, a ∧ b (where ∧ is AND, a, b are Boolean variables) can be seen as • (2) • X Also, (a ∧ b) ⊕ (d ∧ e) (where ⊕ stands for XOR and a, b, c, d are Boolean variables) could be described with 4 Toffoli gates and 3 CNOT gates

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J. M. Hernández Cáceres and I. F. Rúa

• • X

•

• • X

X

• • X

• •

X

(3)

• • X

X The CNOT, Hadamard and π/8 half-phase gates are a universal set of gates [15, Sect. 4.5], which means that we can approximate any quantum transformation on an arbitrary number of qubits using only these gates. These gates are fundamental for quantum computing theory, and by virtue of their universality they can represent any computation done by quantum systems up to any desired accuracy. For example, the following circuit shows a possible decomposition (or transpilation, which is the process of rewriting a given input circuit to map the topology of a specific quantum device) of a Toffoli gate, in the base [H, T, T † , C N O T ]: •

•

• H

X

T†

X

T

•

T

X

T†

X

•

T

•

X

T†

X

T

H

(4)

Now let us talk about oracles, which are a special kind of unitary transformations. An oracle is a black box that has the ability to recognize solutions to the search problem, and so it is a unitary operator defined by its action on the computational basis. For example, for any state |x in the computational basis,  Uω |x =

− |x if x = ω |x if x = ω

is an oracle that adds a negative phase to the solution state ω, where ω is the element we wish to find. This oracle will be a diagonal matrix, where the entry that corresponds to the marked item will have a negative phase. A particular instance of the search problem can conveniently be represented by a function f that takes a proposed solution x, and its output would be  0 if x is not a solution (x = ω) f (x) = . 1 if x is a solution (x = ω) Then the oracle can be described as U f |x = (−1) f (x) |x and the oracle coordinate matrix will be

An Approach to the Classification of Finite Semifields by Quantum Computing

⎛ (−1) f (0) 0 f (1) ⎜ 0 (−1) ⎜ Uω = ⎜ .. .. ⎝ . . 0 0

... ... .. .

0 0 .. .

. . . (−1) f (2

251

⎞ ⎟ ⎟ ⎟. ⎠ n

−1)

There is an additional element: the measurement operator. This is not unitary or reversible. This operation is usually done at the end of a computation, when we want to measure qubits. It is denoted by the gauge symbol: (5) For example, if we want to estimate the state of a qubit |ψ = α |0 + β |1 with α, β ∈ C and |α|2 + |β|2 = 1, we measure and find that the probability of getting 0 is |α|2 and the probability of getting 1 is |β|2 . After the measurement is done the qubit has collapsed into that state. There is a procedure to measure; in effect, to find the probability of measuring a state of |ψ in state |x we can compute | x| ψ |2 . Finally, a quantum algorithm consists in an initial state which is transformed by a series of quantum gates (included oracle ones), and eventually measured. For instance, Grover’s algorithm starts with the uniform superposition |s , which is constructed from H ⊗n |0 ⊗n , then applies the oracle reflection U f to the state |s , and after that applies an additional reflection Us . This transformation is, thus, Us U f , which corresponds to a rotation (it actually rotates the initial state |s closer towards |ω ). The procedure can be seen in the following circuit: |0 ⊗n

H⊗

n

Uf 

2 |s s| − In  

(6)

√ Repeat O( N /k) times

where U f is known as the oracle, Us = 2 |s s| − In as the diffusion operator, |s = H ⊗n |0 ⊗n , N is the number of elements in the list, and k is the number or solutions in the search space. What makes Grover’s algorithm so powerful is a quadratic speedup for finding marked items in long lists.

4 Quantum Computational Search of Finite Semifields 4.1 General Procedure In this subsection we show a general procedure for the classification of finite semifields using Grover’s algorithm.

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J. M. Hernández Cáceres and I. F. Rúa

The multiplication table of a finite semifield D with q d elements is related to d matrices satisfying the properties of Proposition 2 (namely, take β = {x1 = 1, x2 , . . . , xd } a Fq −basis of D, so that the coordinate matrices of the maps L xi for i = 1, . . . , d with respect to such a basis satisfy those conditions). So, we focus our attention on finding the standard basis {A1 , A2 , . . . , Ad }. In order to do that, we build up a Boolean function using the determinant of each linear combination of the standard basis in terms of the operations XOR, AND, NOT. That Boolean function is used to construct an oracle with Toffoli and CNOT gates, to be used in an implementation of Grover’s algorithm. However, it is important to notice that we must know how many iterations are needed, and also to verify that we are not leaving any solution out (with high probability). For that, we follow the methodology from [19], and so we apply Grover’s algorithm, which gives as tuples all the solutions for the standard basis (with a probability as high as desired). Finally, following [4] and [5], semifields can be naturally classified up to isomorphism, isotopy and S3 -action. In summary, the previous procedure can be seen in the following flowchart: Theory reduction (using Proposition 2, Corollary 1)

Standard basis {A1 , A2 , . . . , Ad } where d stands for the dimension of the finite semifield

Boolean function Build the oracle

Applying [19]’s methodology

Grover’s algorithm

All solutions: Tuples of the form (a1 , a2 , . . . , ad(d−1)2 ) where ai ∈ F2 for i = 1, . . . , d(d − 1)2

[4],[5]

Classification by isomorphism, isotopy and S3 -action

An Approach to the Classification of Finite Semifields by Quantum Computing

253

4.2 Semifields of Order 8 In this subsection we will apply Grover’s algorithm to the problem of the classification of finite semifields. We consider the small cases of order 8 and 16 (commutative). For these cases, we explicitly construct a Boolean function that would become the oracle to be used in Grover’s algorithm for the classification of the finite semifields. Consider the following standard basis: ⎛ ⎞ ⎛ ⎛ ⎞ ⎞  100 0 a1 a4 0 a7 a10  A1 = ⎝0 1 0⎠ , A2 = ⎝1 a2 a5 ⎠ , A3 = ⎝0 a8 a11 ⎠ . 001 0 a3 a6 1 a9 a12 If we want to classify the finite semifields of order 8 by Proposition 1, we need to find a1 , . . . , a12 ∈ F2 such that each linear combination except the trivial one yields a matrix with determinant different from zero, i.e., det (α1 A1 + α2 A2 + α3 A3 ) = 1 for each nonzero tuple (α1 , α2 , α3 ) ∈ F32 . Now, in one hand the determinant of a matrix is based on multiplications and additions. On the other hand, the product mod 2 is an AND, and the addition or subtraction mod 2 is an exclusive OR (XOR). Hence we rewrite each determinant in terms of AND, XOR and NOT, and create a Boolean function from them, f : {0, 1}12 −→ {0, 1} (a1 , . . . , a12 ) −→ f (a1 , . . . , a12 ) such that f (a1 , . . . , a12 ) = 1 if and only if the corresponding set of matrices satisfies the above-mentioned conditions. Specifically, f (a1 , . . . , a12 ) = ((a10 ∧ a8 ) ⊕ (a11 ∧ a7 )) ∧ ((a1 ∧ a6 ) ⊕ (a3 ∧ a4 )) ∧ ((a1 ∧ a11 ) ⊕(a1 ∧ a12 ) ⊕ (a1 ∧ a5 ) ⊕ (a1 ∧ a6 ) ⊕ (a10 ∧ a2 ) ⊕ (a10 ∧ a3 ) ⊕(a10 ∧ a8 ) ⊕ (a10 ∧ a9 ) ⊕ (a11 ∧ a7 ) ⊕ (a12 ∧ a7 ) ⊕ (a2 ∧ a4 ) ⊕(a3 ∧ a4 ) ⊕ (a4 ∧ a8 ) ⊕ (a4 ∧ a9 ) ⊕ (a5 ∧ a7 ) ⊕ (a6 ∧ a7 )) ∧((a10 ∧ a8 ) ⊕ a10 ⊕ (a11 ∧ a7 ) ⊕ (a11 ∧ a9 ) ⊕ (a12 ∧ a8 ) ⊕ a12 ⊕(∼ a8 )) ∧ ((a1 ∧ a6 ) ⊕ a1 ⊕ (a2 ∧ a6 ) ⊕ a2 ⊕ (a3 ∧ a4 ) ⊕(a3 ∧ a5 ) ⊕ (∼ a6 )) ∧ ((a1 ∧ a11 ) ⊕ (a1 ∧ a12 ) ⊕ (a1 ∧ a5 ) ⊕(a1 ∧ a6 ) ⊕ a1 ⊕ (a10 ∧ a2 ) ⊕ (a10 ∧ a3 ) ⊕ (a10 ∧ a8 ) ⊕(a10 ∧ a9 ) ⊕ a10 ⊕ (a11 ∧ a3 ) ⊕ (a11 ∧ a7 ) ⊕ (a11 ∧ a9 ) ⊕(a12 ∧ a2 ) ⊕ (a12 ∧ a7 ) ⊕ (a12 ∧ a8 ) ⊕ a12 ⊕ (a2 ∧ a4 ) ⊕(a2 ∧ a6 ) ⊕ a2 ⊕ (a3 ∧ a4 ) ⊕ (a3 ∧ a5 ) ⊕ (a4 ∧ a8 ) ⊕ (a4 ∧ a9 ) ⊕a4 ⊕ (a5 ∧ a7 ) ⊕ (a5 ∧ a9 ) ⊕ (a6 ∧ a7 ) ⊕ (a6 ∧ a8 ) ⊕ a6 ⊕ a7 ⊕(∼ a8 ))

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where ⊕ stands for XOR, ∧ for AND, ∼ for NOT. Hence, this expression for f would have 57 gates AND, 60 XOR gates, and 3 NOT gates. This function is to be used in an oracle in order to perform Grover’s algorithm, as stated in the previous section. Because of its length, we give the circuit for ((a10 ∧ a8 ) ⊕ (a11 ∧ a7 )) ∧ ((a1 ∧ a6 ) ⊕ (a3 ∧ a4 )) (the rest are analogous): • • (7) • • • •

•

•

•

• •

X X

X

• •

X

• •

•

•

• • X

X •

X •

X X

X

• • X

X

X X • X

The solutions (i.e., the tuples with f (x) = 1) found by independent simulations of Grover’s algorithm are:

An Approach to the Classification of Finite Semifields by Quantum Computing Result 010011001101 001111001110 110011101100 111001110111 011011011110 101101110111 000111011100 100111101101

255

Shots 129 150 128 129 125 108 129 126

This means that from 1024 shots, 150 correspond to 001111001110. In terms of our variables, this must be interpreted in the following way: read it from right to left and the solution to each ai corresponds to the order in which the variables were written in the Boolean expression, i.e., a10 a8 a11 a7 a1 a6 a3 a4 a12 a5 a2 a9 . Thus, a10 = 0, a8 = 1, a11 = 1, a7 = 1, a1 = 0, a6 = 0, a3 = 1, a4 = 1, a12 = 1, a5 = 1, a2 = 0, a9 = 0. The same applies to each result. Note that all the results satisfy the condition that, for all i, j, the ith column of A j is the jth column of Ai . This means that the corresponding finite semifield is commutative. Actually, all of them provide standard bases of the only finite semifield of order 8: the Galois field F8 . In practice, all these solutions can be effectively found using the methodology of [19].

4.3 Description of Semifields of Order 16 Now let us move to the case of semifields of order 16, where the corresponding standard bases contain binary matrices of dimension 4 × 4. Consider the following set: ⎧ ⎪ ⎪ ⎨

⎛ 0 ⎜1 A 1 = I4 , A 2 = ⎜ ⎝0 ⎪ ⎪ ⎩ 0

a1 a2 a3 a4

a5 a6 a7 a8

⎛ ⎞ 0 a13 a9 ⎜0 a14 ⎟ a10 ⎟ ,A =⎜ a11 ⎠ 3 ⎝1 a15 a12 0 a16

a17 a18 a19 a20

⎛ ⎞ 0 a25 a21 ⎜0 a26 ⎟ a22 ⎟ ,A =⎜ a23 ⎠ 4 ⎝0 a27 a24 1 a28

a29 a30 a31 a32

⎞⎫ a33 ⎪ ⎪ ⎬ a34 ⎟ ⎟ . ⎠ a35 ⎪ ⎪ ⎭ a36

(8) If we follow the same procedure as before, we end up with an expression for a Boolean function f : {0, 1}36 → {0, 1} with 4189 AND, 2314 XOR, and 7 NOT gates, which make the writing of its circuit really expensive. As proof of concept, we restrict ourselves to the commutative case (which incidentally is interesting in general, because of what has been studied and mentioned in [14]). Thus, we consider that for all i, j the ith column of A j is the jth column of Ai . Now, note that x 4 + x 3 + x 2 + x + 1 = (x + 1)4 + (x + 1)3 + 1 in F2 [x]. So, by Corollary 1 and a suitable change of basis β, we can simplify the matrix A2 as

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J. M. Hernández Cáceres and I. F. Rúa

$ % $ % C x4 + x + 1 , C x4 + x3 + 1

(9)

% ' & $ 2 C x +x +1 $ 2 0 % 0 C x +x +1

(10)

or

where C( p(x)) is the companion matrix of the polynomial p(x) ∈ F2 [x]. Hence, we need to find binary values a1 , a2 , . . . , a12 such that each linear combination of the matrices in the standard basis, except for the trivial one, yields a matrix with determinant different from zero. The second matrix can be chosen among the following ones: ⎛ ⎞ 0001 ⎜1 0 0 1 ⎟ ⎟ 1. x 4 + x + 1, A2 = ⎜ ⎝0 1 0 0⎠ . 0010 In this case, the Boolean function has 122 AND, 133 XOR, and 4 NOT. Following the methodology of [19], after 1024 simulations of Grover’s algorithm, each with 50 iterations, we found: Result 100011101001

Shots 1024

Analogously as before, we should read the result from left to right. So, for this case, a12 = 1, a4 = 0, a3 = 0, a8 = 0, a2 = 1, a1 = 1, a6 = 1, a5 = 0, a11 = 1, a9 = 0, a9 = 0, a10 = 1. ⎛ ⎞ 0001 ⎜1 0 0 0 ⎟ ⎟ 2. x 4 + x 3 + 1, A2 = ⎜ ⎝0 1 0 0 ⎠ 0011 Here, the complexity of the Boolean function is 149 AND, 174 XOR, 9 NOT. Result 111011011110

Shots 1024

Following the methodology of [19], after 1024 simulations of Grover’s algorithm, each with 50 iterations, we found: So, a12 = 1, a9 = 1, a4 = 1, a3 = 0, a8 = 1, a5 = 1, a2 = 0, a1 =1, a6 = 1, a11 = 1, a10 = 1, a7 = 0.

An Approach to the Classification of Finite Semifields by Quantum Computing

⎛

0 ⎜1 2 2 3. (x + x + 1) , A2 = ⎜ ⎝0 0

1 1 0 0

0 0 0 1

257

⎞ 0 0⎟ ⎟. 1⎠ 1

Now, the Boolean function has 161 AND, 194 XOR, 9 NOT. Following the methodology of [19], after 1024 simulations of Grover’s algorithm, each with 20 iterations, we found:

Result

Shots

110011010111 011101101011 101110111100 101110101011 011101010111 110011111100

177 176 144 194 170 163

In all three cases, the only semifield found is the finite field, as this is the only commutative finite semifield of order 16 that exists.

4.4 Estimation of Costs for the General Case, in Terms of Quantum Gates In this subsection, for the sake of simplicity, we restrict to binary semifields. Let D be a finite semifield of order 2d , that we want to describe with a standard basis {A1 , A2 , . . . , Ad }. Consider all possible linear combinations $ from % of S = {A2 , . . . , Ad }, i.e., without involving the identity matrix. There are d−1 1 the form Ai , which have the first column full of zeros except for one position which is 1. Therefore, the determinant of Ai would have (after expansion) at most (d − 1)!(d − 2) products and (d − 1)! − 1 additions (since we are working modulo

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J. M. Hernández Cáceres and I. F. Rúa

2, subtractions can be seen as additions). Now, let us consider all linear combinations of the form Ai + A j . The first column has two ones and d − 2 zeros, and on the remaining columns each position ai j has one sum of 2 variables. So, the determinant of Ai + A j would be the sum of two determinants, each one of a matrix of dimension (d − 1) × (d − 1) in which each entry$ is% a sum of two different variables. So, the number of products would be at most d2 2d (d − 1)!(d − 2), and of additions, $d % d 2 ((d − 1)! − 1). Now if we take k matrices, and add them, together they would 2 have the following form: in the first column, k ones and d − k zeros, and in $ the% remaining columns each ai j position has k − 1 sums of k variables. There are d−1 k $ % matrices of this form, the number of products would be at most dk k d (d − 1)!(d − 2), $ % and of additions dk k d ((d − 1)! − 1). k Now, consider the matrices B + A1 , where B = Ai for k = 3, . . . , d. Then i=2

by [16], det (B + A1 ) = det (A1 ) + det (B) +

d−1 

$ % di det B/Ai1

i=1

= 1 + det (B) +

d−1 

% $ di det B/Ai1 .

i=1

% $ Here, for each k, di det B/Ai1 is defined as the sum of all the determinants in which i rows of B are substituted by the rows of A1 . Note that the % $ corresponding number of products and additions of det B/Ai1 is less or equal than det(B), hence, the number of products on det(B + A1 ) would be at most d times the products of det(B), and for additions, d times the sums of det(B) plus 1. There are 2d−1 − 1 nontrivial linear combinations on S, and of the form A1 + B the number is the same. So, joining all of them, we get 2d − 2 (removing the determinant of the identity matrix, and the null matrix) the same ones that if we consider all of A1 ∪ S. Therefore, the number of required products would be at most ' d−1 &  d −1 k=1

k

k d (d − 1)!(d − 2) +

' d−1 &  d −1 d d k (d − 1)!(d − 2) k k=1

' d−1 &  d −1 d k (d − 1)!(d − 2) = (d + 1) k k=1 and the number of sums

' ' d−1 & d−1 &   % d −1 d d −1 $ d k ((d − 1)! − 1) + dk ((d − 1)! − 1) + 1 k k k=1 k=1

An Approach to the Classification of Finite Semifields by Quantum Computing

259

' ' d−1 & d−1 &   d −1 d d −1 k ((d − 1)! − 1) (1 + d) + k k k=1 k=1 ' d−1 &  d −1 d k ((d − 1)! − 1) (1 + d) + 2d−1 − 1. = k k=1

=

Now, in order to construct the Boolean function we need d(d − 1)2 variables, so 2 f : {0, 1}d(d−1) → {0, 1}. In Sect. 3 we showed a possible decomposition of the Toffoli gate. We should mention that by the Solovay–Kitaev theorem [15, Appendix 3], the asymptotic growth is the same whatever decomposition we choose so, since as we saw, every Toffoli gate in a quantum circuit may execute up to six CNOT gates, this function would have a cost in terms of quantum gates of at least Number

Gate

6a + b 2a 3a 4a

C N OT H T† T

where ' d−1 &  d −1 d k (d − 1)!(d − 2) a = (d + 1) k k=1 ' d−1 &  d −1 d k ((d − 1)! − 1) (1 + d) + 2d−1 − 1. b= k k=1

5 Conclusions and Future Work In order to classify finite semifields of order 2d , we show that at least d(d − 1)2 qubits are required. Also, we give an estimate on the number of quantum gates needed to build up the quantum circuit, showing that this approach is way far than expensive. As a future work, we would look for another alternatives of quantum computing techniques to classify finite semifields, as for instance Quantum Annealing, a form of computation that efficiently samples the low-energy configurations of a quantum system [17, 18]. Acknowledgements This work has been partially supported by the Spanish Government, under Projects MTM-2017-83506-C2-2-P and PID2021-123461NB-C22.

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References 1. Albert, A.A.: Finite division algebras and finite planes. Proc. Symp. Appl. Math 10, 53–70 (1960) 2. Dickson, L.E.: Linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc 7, 370–390 (1906) 3. Knuth, D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965) 4. Rúa, I.F., Combarro, E.F., Ranilla, J.: Classification of semifields of order 64. J. Algebra 322(11), 4011–4029 (2009) 5. Rúa, I.F., Combarro, E.F., Ranilla, J.: Determination of division algebras with 243 elements. Finite Fields Appl. 18(6), 1148–1155 (2012) 6. Combarro, E.F., Ranilla, J., Rúa, I.F.: A quantum algorithm for the commutativity of finite dimensional algebras. IEEE Access 7, 45554–45562 (2019) 7. Kleinfield, E.: Techniques for enumerating Veblen-Wedderburn systems. J. ACM 7, 330–337 (1960) 8. Walker, R.J.: Determination of division algebras with 32 elements. Proc. Symp. Appl. Math. AMS 15, 83–85 (1962) 9. Dempwolff, U.: Semifield planes of order 81. J. Geome. 89, 1–16 (2008) 10. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 439(1907), 553–558 (1992) 11. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). https://doi.org/10.1137/ S0097539795293172 12. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, ACM, New York, NY, USA (1996), pp. 212–219. https://doi.org/10.1145/237814.237866 13. Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997) 14. Lavrauw, M., Seekey, J.: Symplectic 4-Dimensional semifields or order 84 and 94 (2022). https://arxiv.org/abs/2205.08995 15. Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, Cambridge. https://doi.org/10.1017/ CBO9780511976667 16. Xu, S.J., Darouach, M., Schaefers, J.: Expansion of det (A + B) and robustness analysis of uncertain state space systems. IEEE Access Trans. Autom. Control 38(11) (1993) 17. Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355–5363 (1998). https://doi.org/10.1103/PhysRevE.58.5355 18. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. Chem. Phys. Lett. 219(5–6), 343–348 (1994) 19. Fernández-Combarro, E., Rúa, I.F., Orts, F., Ortega, G., Puertas, A.M., Garzón, E.M.: Quantum algorithms to compute the neighbour list of N -body simulations (2021, submitted)

On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras Abdennour Kitouni, Stephen Mboya, Elvice Ongong’a, and Sergei Silvestrov

Abstract The aim of this work is to explore some properties of n-ary skewsymmetric Hom-algebras and n-Hom-Lie algebras related to their ideals, derived series and central descending series. We extend the notions of derived series and central descending series to n-ary skew-symmetric Hom-algebras and provide various general conditions for their members to be Hom-subalgebras, weak ideals or Hom-ideals in the algebra or relatively to each other. In particular we study the invariance under the twisting maps of the derived series and central descending series and their subalgebra and ideal properties for a class of 3-dimensional Hom-Lie algebras and some 4-dimensional 3-Hom-Lie algebras. We also introduce a type of generalized ideals in n-ary Hom-algebras and present a few basic properties. Keywords Hom-algebra · n-Hom-Lie algebra · Derived series · Central descending series · Ideals 2020 Mathematics Subject Classification: 17B61, 17D30, 17A40, 17A42, 17B30

A. Kitouni · S. Silvestrov (B) Division of Mathematics and Physics, Mälardalen University, Box 833, 72123 Västerås, Sweden e-mail: [email protected] A. Kitouni e-mail: [email protected] S. Mboya · E. Ongong’a Department of Mathematics, University of Nairobi, Box 30197, Nairobi, Kenya e-mail: [email protected] E. Ongong’a e-mail: [email protected] S. Mboya Division of Mathematics and Physics, Mälardalen University, Box 833, 72123 Västerås, Sweden E. Ongong’a Strathmore Institute of Mathematical Sciences, Strathmore University, Box 59857, Nairobi, Kenya © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_17

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1 Introduction Hom-Lie algebras and more general quasi-Hom-Lie algebras were introduced first by Hartwig, Larsson and Silvestrov in [43], where the general quasi-deformations and discretizations of Lie algebras of vector fields using more general σ -derivations (twisted derivations) and a general method for construction of deformations of Witt and Virasoro type algebras based on twisted derivations have been developed, initially motivated by the q-deformed Jacobi identities observed for the q-deformed algebras in physics, q-deformed versions of homological algebra and discrete modifications of differential calculi [7, 32–38, 44, 45, 60–62]. The general abstract quasi-Lie algebras and the subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras as well as their general colored (graded) counterparts have been introduced in [43, 54–56, 73]. Subsequently, various classes of Hom-Lie admissible algebras have been considered in [64]. In particular, in [64], the Hom-associative algebras have been introduced and shown to be Hom-Lie admissible, that is leading to Hom-Lie algebras using commutator map as new product, and in this sense constituting a natural generalization of associative algebras, as Lie admissible algebras leading to Lie algebras via commutator map as new product. In [64], moreover, several other interesting classes of Hom-Lie admissible algebras generalizing some classes of non-associative algebras, as well as examples of finite-dimensional Hom-Lie algebras have been described. Hom-algebras structures are very useful since Hom-algebra structures of a given type include their classical counterparts and open more possibilities for deformations, extensions of cohomological structures and representations. Since these pioneering works [43, 54–57, 64], Hom-algebra structures have developed in a popular broad area with increasing number of publications in various directions (see for example [8, 28, 41, 53, 54, 58, 65–67, 69, 71, 72, 78, 79] and references therein). Ternary Lie algebras appeared first in the generalization of Hamiltonian mechanics by Nambu [68]. Besides Nambu mechanics, n-Lie algebras revealed to have many applications in physics. The mathematical algebraic foundations of Nambu mechanics have been developed by Takhtajan in [74]. Filippov, in [42] independently introduced and studied structure of n-Lie algebras and Kasymov [46] investigated their properties. Properties of n-ary algebras, including solvability and nilpotency, were studied in [30, 46, 76]. Kasymov [46] pointed out that n-ary multiplication allows for several different definitions of solvability and nilpotency in n-Lie algebras, and studied their properties. Further properties, classification, and connections of n-ary algebras to other structures such as bialgebras, Yang-Baxter equation and Manin triples for 3-Lie algebras were studied in [15–24, 46]. The structure of 3-Lie superalgebras induced by Lie superalgebras, classification of 3-Lie superalgebras and application to constructions of B.R.S. algebras have been considered in [2–4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation are considered in [5]. In [31], Leibniz nalgebras have been studied. The general cohomology theory for n-Lie algebras and Leibniz n-algebras was established in [39, 70, 75]. The structure and classification of finite-dimensional n-Lie algebras were considered in [21, 42, 59] and many other

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authors. For more details of the theory and applications of n-Lie algebras, see [40] and references therein. Hom-type generalization of n-ary algebras, such as n-Hom-Lie algebras and other n-ary Hom algebras of Lie type and associative type, were introduced in [13], by twisting the defining identities by a set of linear maps. The particular case, where all these maps are equal and are algebra morphisms has been considered and a way to generate examples of n-ary Hom-algebras from n-ary algebras of the same type have been described. Further properties, construction methods, examples, representations, cohomology and central extensions of n-ary Hom-algebras have been considered in [9, 11, 12, 48, 77, 80]. These generalizations include n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. In [50], constructions of n-ary generalizations of BiHom-Lie algebras and BiHomassociative algebras have been considered. Generalized derivations of n-BiHom-Lie algebras have been studied in [27]. Generalized derivations of multiplicative n-ary Hom- color algebras have been studied in [29]. Cohomology of Hom-Leibniz and n-ary Hom-Nambu-Lie superalgebras has been considered in [1] Generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu superalgebras have been considered in [63]. A construction of 3-Hom-Lie algebras based on σ -derivation and involution has been studied in [6]. Multiplicative n-Hom-Lie color algebras have been considered in [25]. In [14], Awata, Li, Minic and Yoneya introduced a construction of (n + 1)-Lie algebras induced by n-Lie algebras using combination of bracket multiplication with a trace in their work on quantization of the Nambu brackets. Further properties of this construction, including solvability and nilpotency, were studied in [10, 17, 47]. In [11, 12], this construction was generalized using the brackets of general Hom-Lie algebra or n-Hom-Lie and trace-like linear forms satisfying conditions depending on the twisting linear maps defining the Hom-Lie or n-Hom-Lie algebras. In [26], a method was demonstrated of how to construct n-ary multiplications from the binary multiplication of a Hom-Lie algebra and a (n − 2)-linear function satisfying certain compatibility conditions. Solvability and nilpotency for n-Hom-Lie algebras and (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras have been considered in [49]. In [51], properties and classification of n-Hom-Lie algebras in dimension n + 1 were considered, and 4-dimensional 3-Hom-Lie algebras for various special cases of the twisting map have been computed in terms of structure constants as parameters and listed in classes in the way emphasizing the number of free parameters in each class. n-Hom-Lie algebras are fundamentally different from n-Lie algebras, especially when the twisting maps are not invertible or not diagonalizable. When the twisting maps are not invertible, the Hom-Nambu-Filippov identity becomes less restrictive since when elements of the kernel of the twisting maps are used, several terms or even the whole identity might vanish. Isomorphisms of Hom-algebras are also different from isomorphisms of algebras since they need to intertwine not only the multiplications but also the twisting maps.

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The aim of this work is to explore some properties of n-ary skew-symmetric Homalgebras and n-Hom-Lie algebras related to their ideals, derived series and central descending series. We extend the notions of derived series and central descending series to n-ary skew-symmetric Hom-algebras and provide various general conditions for their members to be Hom-subalgebras, weak ideals or Hom-ideals in the algebra or relatively to each other. In particular we study the invariance under the twisting maps of the derived series and central descending series and their subalgebra and ideal properties for a class of 3-dimensional Hom-Lie algebras and some 4-dimensional 3-Hom-Lie algebras. We also introduce a type of generalized ideals in n-ary Homalgebras and present a few basic properties. In Sect. 2, we present the basic definitions and properties of Hom-Lie algebras, n-ary skew-symmetric Hom-algebras and nHom-Lie algebras needed for this study. In Sect. 3, we introduce the notions of -ideals and k-ideals generalizing the notion of ideal and subalgebra in n-ary Homalgebras, which include n-Hom-Lie algebras. We study also some of their basic properties. In Sect. 4, we generalize and improve the fundamental properties of kderived series and k-Central descending series. We extend part of them to the case of n-ary skew-symmetric Hom-algebras, we consider different types of twisting maps in addition to weak morphisms and prove stronger properties than the results given in [49]. In Sect. 5, we study the conditions of the results in [49] on a class of 4dimensional 3-Hom-Lie algebras and present examples where the conditions do not hold but we get some properties. In Sect. 6, we present a similar but more extensive study of a class of Hom-Lie algebras while considering all the possible twisting maps. For each case inside this class, the derived series and central descending series of the whole algebra are computed, and it is determined whether they are weak ideals, Hom-subalgebras or Hom-ideals and the set of twisting maps realizing these properties are given when relevant. Furthermore all twisting maps corresponding to multiplicative Hom-Lie algebras are computed.

2 Definitions and Properties of n-Hom-Lie Algebras In this section, we present the basic definitions and properties of n-Hom-Lie algebras needed for our study. Throughout this article, N denotes the set of natural numbers including 0, it is assumed that all vector spaces are over a field K of characteristic 0, and for any subset S of a vector space, S denotes the linear span of S. HomLie algebras are a generalization of Lie algebras introduced in [43] while studying σ -derivations. The n-ary case was introduced in [13]. Definition 1 ([43, 64]) A Hom-Lie algebra (A, [·, ·], α) is a vector space A together with a bilinear map [·, ·] : A × A → A and a linear map α : A → A satisfying, for all x, y, z ∈ A, [x, y] = −[y, x] [α(x), [y, z]] = [[x, y], α(z)] + [α(y), [x, z]]

Skew-symmetry Hom-Jacobi identity

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In Hom-Lie algebras, by skew-symmetry, the Hom-Jacobi identity is equivalent to 

[α(x), [y, z]] = [α(x), [y, z]] + [α(y), [z, x]] + [α(z), [x, y]] = 0.

(x,y,z)

Hom-Jacobi identity in cyclic form

Definition 2 ([43, 54]) Let A = (A, [·, ·]A , α) and B = (B, [·, ·]B , β) be HomLie algebras. Hom-Lie algebra morphisms are linear maps f : A → B satisfying, for all x, y ∈ A, f ([x, y]A ) = [ f (x), f (y)]B , f ◦ α = β ◦ f.

(1) (2)

Linear maps f : A → B satisfying only condition (1) are called weak morphisms of Hom-Lie algebras. Definition 3 ([28, 64]) A Hom-Lie algebra (A, [·, ·], α) is said to be multiplicative if α is an algebra morphism, and it is said to be regular if α is an isomorphism. Definition 4 An n-ary Hom-algebra (A, [·, . . . , ·], {αi }1≤i≤s ) is a vector space A together with an n-ary operation, that is an n-linear map [·, . . . , ·] : An → A and linear maps αi : A → A, 1 ≤ i ≤ s. An n-ary Hom-algebra is said to be skew-symmetric if its n-ary operation is skew-symmetric, that is satisfying for all x1 , . . . , xn ∈ A, [xσ (1) , . . . , xσ (n) ] = sgn(σ )[x1 , . . . , xn ],

Skew-symmetry

(3)

The n-ary Hom-algebras (A, [·, . . . , ·], {αi }1≤i≤s ) with α1 = · · · = αs = α will be denoted by (A, [·, . . . , ·] , α). Definition 5 An n-ary Hom-algebra (A, [·, . . . , ·] , α) is called multiplicative if α is an algebra morphism, and regular if α is an algebra isomorphism. The following definition is a specialization of the standard definition of a subalgebra in general algebraic structures to the case of n-ary Hom-algebras considered in this paper. Definition 6 A Hom-subalgebra B = (B, [·, . . . , ·]B , β1 , . . . , βs ) of an n-ary Homalgebra A = (A, [·, . . . , ·]A , α1 , . . . , αs ), is an n-ary Hom-algebra consisting of a subspace B of A satisfying, for all x1 , . . . , xn ∈ B and for all 1 ≤ i ≤ s, 1) αi (B) ⊆ B 2) [x1 , . . . , xn ]A ∈ B, with the multiplication [·, . . . , ·]B = [·, . . . , ·]A and linear maps βi = αi , 1 ≤ i ≤ s restricted from A to B. If only condition 2 is satisfied then B is said to be a weak subalgebra.

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The following definition is a direct extension of the corresponding definition in [28, 64, 80] to arbitrary n-ary skew-symmetric Hom-algebras. Definition 7 In an n-ary skew-symmetric Hom-algebra (A, [·, . . . , ·], α1 , . . . , αs ), a subspace I of A is a Hom-ideal if for all x1 , . . . , xn−1 ∈ A and y ∈ I , 1) αi (I ) ⊆ I for all 1 ≤ i ≤ s. 2) [x1 , . . . , xn−1 , y] ∈ I (or equivalently [y, x1 , . . . , xn−1 ] ∈ I by skew-symmetry). A subspace I ⊆ A only satisfying the condition 2 is called a weak ideal. Definition 8 ([13, 80]) n-ary Hom-algebra morphisms of n-ary Hom algebras A = (A, [·, . . . , ·]A , {αi }1≤i≤s ) and B = (B, [·, . . . , ·]B , {βi }1≤i≤s ) are such linear maps f : A → B that, for all x1 , . . . , xn ∈ A, f ([x1 , . . . , xn ]A ) = [ f (x1 ), . . . , f (xn )]B , f ◦ αi = βi ◦ f, for all 1 ≤ i ≤ s.

(4) (5)

Linear maps satisfying only condition (4) are called weak morphisms of n-ary Homalgebras. Definition 9 ([13]) An n-Hom-Lie algebra (A, [·, . . . , ·], {αi }1≤i≤n−1 ) is a n-ary skew-symmetric Hom-algebra satisfying, for all x1 , . . . , xn−1 , y1 , . . . , yn ∈ A Hom-Nambu-Filippov identity   α1 (x1 ), . . . , αn−1 (xn−1 ), [y1 , . . . , yn ] = n  [α1 (y1 ), . . . , αi−1 (yi−1 ), [x1 , . . . , xn−1 , yi ], αi (yi+1 ), . . . , αn−1 (yn )].

(6)

i=1

Remark 1 If αi = I d A for all 1 ≤ i ≤ n − 1, then one gets an n-Lie algebra ([42]). Therefore, the class of n-Lie algebras is included in the class of n-Hom-Lie algebras. For any vector space A and any linear maps α1 , . . . , αn−1 , if [x1 , . . . , xn ]0 = 0 for all x1 , .., xn ∈ A, then (A, [·, . . . , ·]0 , α1 , . . . , αn−1 ) is an n-Hom-Lie algebra. The following proposition, providing a way to construct an n-Hom-Lie algebra from an n-Lie algebra and an algebra morphism, was first introduced in the case of Lie algebras and then generalized to the n-ary case in [13]. A more general version of this theorem, given in [80], states that the category of n-Hom-Lie algebras is closed under twisting by weak morphisms. Proposition 1 ([13, 80]) Let β : A →  A be a weak morphism of n-Hom-Lie algebra A = A, [·, . . . , ·] , {αi }1≤i≤n−1 , and define [·,. . . , ·]β by [x1 , . . . , xn ]β = β ([x1 , . . . , xn ]) . Then, A, [·, . . . , ·]β , {β ◦ αi }1≤i≤n−1 is an n-Hom-Lie algebra.  (A, [·, . . . , ·] , α) is multiplicative and β ◦ α = α ◦ β, then Moreover, if the algebra A, [·, . . . , ·]β , β ◦ α is multiplicative. The following particular case of Proposition 1 is obtained if α = I d A .

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Corollary 1 Let (A, [·, . . . , ·]) be an n-Lie algebra, β : A → A an algebra morphism, and [·, . . . , ·]β is defined by [x1 , . . . , xn ]β = β ([x1 , . . . , xn ]). Then,  A, [·, . . . , ·]β , β is a multiplicative n-Hom-Lie algebra. The following notions are a direct extension of the corresponding notions in [49] to arbitrary n-ary skew-symmetric Hom-algebras. Definition 10 Let (A, [·, . . . , ·] , α1 , . . . , αn−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra, and let I be a weak ideal of A. For 2 ≤ k ≤ n, we define the k-derived series of the ideal I by: p+1

Dk0 (I ) = I and Dk

p

p

(I ) = [Dk (I ), . . . , Dk (I ), A, . . . , A].  

 

n−k

k

We define the k-central descending series of I by: p+1

Ck0 (I ) = I and Ck

p

(I ) = [Ck (I ), I, . . . , I , A, . . . , A].    

k−1

n−k

Definition 11 Let (A, [·, . . . , ·] , α1 , . . . , αn−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra, and let I be a weak ideal of A. For 2 ≤ k ≤ n, the ideal I is said to be k-solvable (resp. k-nilpotent) if there exists r ∈ N such that Dkr (I ) = {0} (resp. Ckr (I ) = {0}), and smallest r ∈ N satisfying this condition is called the class of k-solvability (resp. the class of k-nilpotency) of I . The following direct extension of the corresponding result in [49] to arbitrary n-ary skew-symmetric Hom-algebras is proved in the same way as in [49] since the proof does not involve the Hom-Nambu-Filippov identity. Lemma 1 Let A = (A, [·, . . . , ·] A , α1 , . . . , αn−1 ), B = (B, [·, . . . , ·] B , β1 , . . . , βn−1 ) be two n-ary skew-symmetric Hom-algebras, f : A → B be a surjective n-ary Hom-algebras morphism and I a weak ideal of A . Then, for all r ∈ N and 2 ≤ k ≤ n,     f Dkr (I ) = Dkr ( f (I )) and f Ckr (I ) = Ckr ( f (I )) . This lemma also implies that if two n-Hom-Lie algebras are isomorphic, they would also have isomorphic members of the derived series and central descending series, which also means that if two algebras have a significant difference in the derived series or the central descending series, for example different dimensions of given corresponding members, then these algebras cannot be isomorphic. Remark 2 ([51]). For the whole algebra A, all the k-central descending series, for all 2 ≤ k ≤ n, are equal. Therefore all the notions of k-nilpotency, for all 2 ≤ k ≤ n, p are equivalent, and we denote Ck (A) for any 2 ≤ k ≤ n by C p (A). Definition 12 Let (A, [·, . . . , ·] , α1 , . . . , αn−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra. Define Z (A), the center of A, by

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  Z (A) = {z ∈ A : x1 , . . . , xn−1 , z = 0, ∀x1 , . . . , xn−1 ∈ A}. Note that Z (A) is a weak ideal of A in the sense of Definition 7. Lemma 2 ([51]) Let (A, [·, . . . , ·] , α) be an n-Hom-Lie algebra. If A is k-nilpotent, for any 2 ≤ k ≤ n, then the center Z (A) of A is not trivial. Lemma 3 ([52]) When A = (A, [·, . . . , ·] , (αi )1≤i≤n−1 ) is an n-ary skew-symmetric Hom-algebra, it holds that 1) If A is nilpotent, then Z (A ) is not trivial. 2) If dim A = n + 1, then dim Z (A ) = 0 or dim Z (A ) = 1 or Z (A ) = A. The following direct extension of the corresponding result in [51] to arbitrary n-ary skew-symmetric Hom-algebras is proved in the same way as in [51] since the proof does not involve the Hom-Nambu-Filippov identity. Proposition 2 ([51, 52]) Let A = (A, [·, . . . , ·], {αi }1≤i≤n−1 ) be an (n + 1)-dimensional n-ary skew-symmetric algebra. The algebra A is nilpotent and non abelian if and only if dim Z (A ) = 1 and [A, . . . , A] = Z (A ). Proposition 3 ([51, 52]) Let A = (A, [·, . . . , ·], {αi }1≤i≤n−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra. A is nilpotent of class p if and only if {0}  C p−1 (A) ⊆ Z (A).

3 Generalized Ideals For non-skew-symmetric algebras, positioning of the ideal property in the n-ary multiplication becomes important. Throughout this section, n-ary multiplications will be denoted by m for convenience. Definition 13 Let A = (A, m, α1 , . . . , αn−1 ) be an n-ary Hom-algebra. A linear subspace I of A is called a Hom-ideal if for all x1 , . . . , xn−1 ∈ A, s ∈ I , 1 ≤ i ≤ n − 1, 1) αi (I ) ⊆ I , 2) m(s, x1 , . . . , xn−1 ), m(x1 , s, x2 , . . . , xn−1 ), . . . , m(x1 , . . . , xn−1 , s) ∈ I . The above mentioned definition corresponds to two-sided ideals in binary (noncommutative) rings and algebras, that is the one that fully defines a congruence and allows to define a factor ring or algebra. One would like to consider also generalizations of left ideals and right ideals in the case of n-ary algebras and n-ary Hom-algebras. Definition 14 Let A = (A, m, α1 , . . . , αn−1 ) be an n-ary Hom-algebra,  be a subset of {1, . . . , n}. A linear subspace I of A is called a -Hom-ideal if

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1) αi (I ) ⊆ I, ∀1 ≤ i ≤ n − 1, / . 2) m(x1 , . . . , xn ) ∈ I, ∀x1 , . . . , xn : x j ∈ I if j ∈ , x j ∈ A if j ∈ If a linear subspace I of A only satisfies condition 2, then I is called a weak -ideal. We denote by I (A ) (resp. Iw (A )) the set of all -Hom-ideals of A (resp. the set of all weak -ideals of A ). Definition 15 Let A = (A, m, α1 , . . . , αs ) be an n-ary Hom-algebra, k ∈ {1, . . . , n} and I be a linear subspace of A. We say that I is a k-Hom-ideal (resp. weak kideal) if it is a -Hom-ideal (resp. weak k-ideal) for all  ⊆ {1, . . . , n} such that Car d() = k. We denote by Ik (A ) (resp. Ikw (A )) the set of all k-Hom-ideals of A (resp. all weak k-ideals of A ). In other words Ik (A ) =



I (A ),

⊆{1,...,n}:Car d=k

Ikw (A ) =



Iw (A ).

⊆{1,...,n}:Car d=k

In these general definitions, the usual notion of Hom-ideal corresponds to that of 1-Hom-ideal (k-Hom-ideal for k = 1), while {1, . . . , n}-Hom-ideals or equivalently n-Hom-ideals are Hom-subalgebras. Similarly, weak 1-ideals are weak ideals and weak n-ideals are weak subalgebras. Proposition 4 Let A = (A, m, α1 , . . . , αs ) be an n-ary Hom-algebra, 1 , 2 be subsets of {1, . . . , n}. If 1 ⊆ 2 then I1 (A ) ⊆ I2 (A ) (resp. Iw1 (A ) ⊆ Iw2 (A )). Proof Suppose that 1 ⊆ 2 ⊆ {1, . . . , n} and let I ∈ I1 (A ) (resp. I ∈ Iw1 (A )), / then m(x1 , . . . , xn ) ∈ I for all x1 , . . . , xn where x j ∈ I if j ∈ 1 and x j ∈ A if j ∈ 1 , which means that m(x1 , . . . , xn ) ∈ I for all x1 , . . . , xn where x j ∈ I if j ∈ 2 and x j ∈ A if j ∈ / 2 since 1 ⊆ 2 , that is I ∈ I2 (A ) (resp. I ∈ Iw2 (A )).  Proposition 5 Let A = (A, m, α1 , . . . , αs ) be an n-ary Hom-algebra and 1 ≤ k ≤ l ≤ n. Then Ik (A ) ⊆ Il (A ) (resp. Ikw (A ) ⊆ Ilw (A )). Proof For 1 ≤ k ≤ l ≤ n, I ∈ Ik (A ) ⇐⇒ I ∈ I (A ), ∀ ⊆ {1, . . . , n} : Car d = k Prop 4

=⇒ I ∈ I (A ), ∀ ⊆ {1, . . . , n} : Car d = l and ∃ ⊆ {1, . . . , n} : Car d = k and  ⊆  =⇒ I ∈ I (A ), ∀ ⊆ {1, . . . , n} : Car d = l =⇒ I ∈ Il (A ).

Therefore Ik (A ) ⊆ Il (A ). The same proof holds if Ik (A ), I (A ) are replaced  by Ikw (A ), Iw (A ).

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4 Ideal Properties of Derived Series and Central Descending Series The following results (Lemma 4, Propositions 6 and 7) extending the corresponding results from [49] give the fundamental properties of derived series and central descending series. Lemma 4 Let A = (A, [·, . . . , ·] , α1 , . . . , αs ) be an n-ary skew-symmetric Homalgebra, and let I be a weak ideal of A . Then, 1) for 2 ≤ k ≤ n − 1 and r ∈ N, r r Dk+1 (I ) ⊆ Dkr (I ) and Ck+1 (I ) ⊆ Ckr (I );

2) for 2 ≤ k ≤ n and r ∈ N, Ckr +1 (I ) ⊆ Ckr (I ) and Dkr +1 (I ) ⊆ Dkr (I ). Proposition 6 Let A = (A, [·, . . . , ·] , α1 , . . . , αs ) be an n-ary skew-symmetric Hom-algebra, and let I be a weak ideal of A . For all 2 ≤ k ≤ n and all r ∈ N, we have that Dkr (I ) and Ckr (I ) are weak subalgebras of A . Proposition 7 Let A = (A, [·, . . . , ·] , α1 , . . . , αs ) be an n-ary skew-symmetric Hom-algebra, and let I be an Hom-ideal of A . If all the linear maps αi , 1 ≤ i ≤ s are weak morphisms, then the subspaces Dkr (I ) and Ckr (I ) are Hom-subalgebras of A , for all 2 ≤ k ≤ n and all r ∈ N. If, in addition, these maps are surjective, and A is an n-Hom-Lie algebra then Dkr (I ) and Ckr (I ) are Hom-ideals of A . These results can be improved by either imposing weaker conditions or getting more detailed conclusions. Proposition 8 If A = (A, [·, . . . , ·] , (αi )1≤i≤s ) is an n-ary skew-symmetric Homp+1 algebra and I is a weak ideal of A , then for all p ∈ N, 2 ≤ k ≤ n, Dk (I ) is a p p+1 p weak ideal of Dk (I ), and Ck (I ) is a weak ideal of Ck (I ). In particular, Dk1 (A) 1 and Ck (A) are weak ideals of A . Proof For any p ∈ N, and 2 ≤ k ≤ n we have 

p+1

Dk

   p p p p p+1 (I ), Dk (I ), . . . , Dk (I ) ⊆ Dk (I ), . . . , Dk (I ), A, . . . , A = Dk (I ),  

 

n−k

k

 p+1    p p p p+1 Ck (I ), Ck (I ), . . . , Ck (I ) ⊆ Ck (I ), I, . . . , I , A, . . . , A = Ck (I ).    

k−1 p+1

That is Dk

p

p+1

(I ) is a weak ideal of Dk (I ) and Ck

n−k p

(I ) is a weak ideal of Ck (I ). 

On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras

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Proposition 9 Let A = (A, [·, . . . , ·] , (αi )1≤i≤s ) be an n-ary skew-symmetric Homalgebra and let I be a Hom-ideal of A . If for all 1 ≤ i ≤ s, αi satisfies one of the following conditions 1) ∀x1 , . . . , xn ∈ A, αi ([x1 , . . . , xn ]) = [αi (x1 ), . . . , αi (xn )] (αi is a weak morphism),   2) ∀x1 , . . . , xn ∈ A, αi ([x1 , . . . , xn ]) = x1 , . . . , xn−1 , αi (xn ) , n   x1 , . . . , xk−1 , αi (xk ), xk+1 , . . . , xn 3) ∀x1 , . . . , xn ∈ A, αi ([x1 , . . . , xn ]) = k=1

(αi is a derivation), then, for all p ∈ N and 2 ≤ k ≤ n,

p Dk (I )

and

p Ck (I )

p

are Hom-subalgebras of A . p

Proof By Proposition 6, we have that Dk (I ) and Ck (I ) are weak subalgebras of A , it only remains to show that they are invariant under αi for all 1 ≤ i ≤ s. We proceed by induction over p ∈ N, for all 2 ≤ k ≤ n. For p = 0, Ck0 (I ) = Dk0 (I ) = I p is invariant under all αi by hypothesis. Suppose now that for p ∈ N, Dk (I ) and p p+1 Ck (I ) are invariant under all αi . In order to show that αi (a) ∈ Dk (I ) and αi (b) ∈ p+1 p+1 p+1 Ck (I ) for elements of the  a ∈ Dk (I ) and b ∈ C k (I ), it is enough to consider p form a = d1 , . . . , dk , dk+1 , . . . , dn , where d1 , . . . , dk ∈ Dk (I ) and dk+1 , . . . , dn ∈ p A and b = [c1 , . . . , cn ] where c1 ∈ Ck (I ), c2 , . . . , ck ∈ I and ck+1 , . . . , cn ∈ A. If αi satisfies condition 1, then αi (a) = αi

    d1 , . . . , dk , dk+1 , . . . , dn = αi (d1 ), . . . , αi (dk ), αi (dk+1 ), . . . , αi (dn ) , p+1

which is an element of Dk hypothesis, and αi (b) = αi

p

(I ) since αi (d1 ), . . . , αi (dk ) ∈ Dk (I ) by inductive

    c1 , . . . , ck , ck+1 , . . . , cn = αi (c1 ), . . . , αi (ck ), αi (ck+1 ), . . . , αi (cn ) , p+1

p

which is an element of Ck (I ) since αi (c1 ) ∈ Ck (I ) and αi (c2 ), . . . , αi (ck ) ∈ I by inductive hypothesis. If αi satisfies condition 2, then αi (a) = αi

    d1 , . . . , dk , dk+1 , . . . , dn = d1 , . . . , dk , dk+1 , . . . , αi (dn ) , p+1

which is an element of Dk esis, and αi (b) = αi

p

(I ) since αi (dn ) ∈ Dk (I ) if k = n by inductive hypoth-

    c1 , . . . , ck , ck+1 , . . . , cn = c1 , . . . , ck , ck+1 , . . . , αi (cn ) , p+1

which is an element of Ck (I ) since αi (cn ) ∈ I if k = n by hypothesis. If αi satisfies condition 3, then

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αi (a) = αi

n      d1 , . . . , dk , dk+1 , . . . , dn = d1 , . . . , αi (dq ), . . . , dn , q=1 p+1

which is an element of Dk hypothesis, and αi (b) = αi

p

(I ) since αi (d1 ), . . . , αi (dk ) ∈ Dk (I ) by inductive

n      c1 , . . . , ck , ck+1 , . . . , cn = c1 , . . . , αi (cq ), . . . , cn , q=1 p+1

p

which is an element of Ck (I ) because αi (c1 ) ∈ Ck (I ) by inductive hypothesis and αi (c2 ), . . . , αi (ck ) ∈ I by hypothesis. p+1 p+1 Therefore, in all three cases, Dk (I ) and Ck (I ) are invariant under αi , which completes the proof.  By combining Propositions 8 and 9, we get the following corollary. Corollary 2 Let A = (A, [·, . . . , ·] , (αi )1≤i≤s ) be an n-ary skew-symmetric Homalgebra. If I is a Hom-ideal of A and each αi , 1 ≤ i ≤ s satisfies either conditions p+1 p p+1 1, 2 or 3 of Proposition 9, then Dk (I ) is a Hom-ideal of Dk (I ) and Ck (I ) is a p Hom-ideal of Ck (I ). Corollary 3 Let A = (A, [·, . . . , ·] , α1 , . . . , αn−1 ) be an n-Hom-Lie algebra and let I be a Hom-ideal of A . If all the linear maps αi , 1 ≤ i ≤ n − 1 are surjective, and each of them satisfies either condition 1, 2 or 3 of Proposition 9, then Dkr (I ) and Ckr (I ) are Hom-ideals of A , for all 2 ≤ k ≤ n and all r ∈ N. The proof follows the same idea as the last point in Proposition 7 where the weak morphism property of the twisting maps is replaced by conditions 1, 2 or 3 of Proposition 9. The proof uses the Hom-Nambu-Filippov identity and the invariance of Dkr (I ) and Ckr (I ) under the twisting maps. From the proof of Proposition 9 (condition 2), one gets also Proposition 10 Let A = (A, [·, . . . , ·] , (αi )1≤i≤s ) be an n-ary Hom-algebra and let I be a weak ideal of A . If for all 1 ≤ i ≤ s, αi satisfies   ∀x1 , . . . , xn ∈ A, αi ([x1 , . . . , xn ]) = x1 , . . . , xn−1 , αi (xn ) , p

p

then, for all p ∈ N \ {0} and 2 ≤ k ≤ n − 1, Dk (I ) and Ck (I ) are Hom-subalgebras of A . Proposition 11 Let A = (A, [·, . . . , ·] , (αi )1≤i≤s ) be an n-ary skew symmetric r   u i,1,k , . . . , u i,n−1,k , · + λI d, where Hom-algebra. If for all 1 ≤ i ≤ s, αi = k=1

λ ∈ K and u i,1,k , . . . , u i,n−1,k ∈ A, then every weak ideal I of A is a Hom-ideal.

On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras

273

Proof Suppose that for all 1 ≤ i ≤ n − 1, αi is of the assumed form. Let I be a weak ideal of A , and let x ∈ I . Then, αi (x) =

r    u i,1,k , . . . , u i,n−1,k , x + λx ∈ I, k=1

  since all the u i,1,k , . . . , u i,n−1,k , x are elements of I because I is a weak ideal. 

5 Some Examples in the Case of n-Hom-Lie Algebras We consider the class of 4-dimensional 3-Hom-Lie algebras, over a field K of characteristic 0, studied in [52] with the bracket defined on the basis (ei )1≤i≤4 by [e1 , e2 , e3 ] = 0 [e1 , e2 , e4 ] = c(1, 2, 4, 1)e1 + c(1, 2, 4, 2)e2 + c(1, 2, 4, 3)e3 + c(1, 2, 4, 4)e4 [e1 , e3 , e4 ] = c(1, 3, 4, 1)e1 + c(1, 3, 4, 2)e2 + c(1, 3, 4, 3)e3 + c(1, 3, 4, 4)e4 [e2 , e3 , e4 ] = 0, where c(i, j, k, r ) ∈ K, and the twisting map is defined by ⎛ 0 ⎜0 ⎜ [α] = ⎝ 0 0

0 0 0 0

0 1 0 0

⎞ 0 0⎟ ⎟. 1⎠ 0

The first member in the derived series and the central descending series of these algebras are given by D31 (A) = D21 (A) = C31 (A) = {c(1, 2, 4, 1)e1 + c(1, 2, 4, 2)e2 + c(1, 2, 4, 3)e3 + c(1, 2, 4, 4)e4 , c(1, 3, 4, 1)e1 + c(1, 3, 4, 2)e2 + c(1, 3, 4, 3)e3 + c(1, 3, 4, 4)e4 } We consider the two following subclasses 1) [e1 , e2 , e3 ] = 0 [e1 , e2 , e4 ] = e4 [e1 , e3 , e4 ] = c(1, 3, 4, 1)e1 + c(1, 3, 4, 3)e3 + c(1, 3, 4, 4)e4 [e2 , e3 , e4 ] = 0, c(1, 3, 4, 1) = 0,

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2) [e1 , e2 , e3 ] = 0 [e1 , e2 , e4 ] = e4 [e1 , e3 , e4 ] = c(1, 3, 4, 2)e2 + c(1, 3, 4, 3)e3 + c(1, 3, 4, 4)e4 [e2 , e3 , e4 ] = 0, (c(1, 3, 4, 2), c(1, 3, 4, 3)) = (0, 0). For the cases 1 and 2, D31 (A) is not invariant under α, that is, it is not a Hom-ideal. In the case 1, D22 (A) = D21 (A). In the case 2, D22 (A) = {c(1, 3, 4, 3)ω2 + c(1, 3, 4, 2)ω3 } = {0}, where ω2 = [e1 , e3 , e4 ] and ω3 = [e1 , e2 , e4 ]. So, dim D31 (A) = 2 and dim D22 (A) = 1. Denote by v the generator of D22 (A): v = c(1, 3, 4, 3)c(1, 3, 4, 2)e2 + c(1, 3, 4, 3)2 e3 + (c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2))e4 . In general, D22 (A) is a weak subalgebra of A . We study whether D22 (A) can be a Hom-subalgebra in this class. To this end, we calculate the image by α of D22 (A): α(v) = c(1, 3, 4, 3)2 e2 + (c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2))e3 . If c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2) = 0, then α(c(1, 3, 4, 3)ω2 + c(1, 3, 4, 2)ω3 ), c(1, 3, 4, 3)ω2 + c(1, 3, 4, 2)ω3 are linearly independent, which means that D22 (A) is not invariant under α and thus D22 (A) is a weak subalgebra but not a Hom-subalgebra of A . If c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2) = 0, then c(1, 3, 4, 2) = −c(1, 3, 4, 3)c(1, 3, 4, 4), α(v) = c(1, 3, 4, 3)2 e2 , v = c(1, 3, 4, 3)2 (e3 − c(1, 3, 4, 4)e2 ). If c(1, 3, 4, 4) = 0, then c(1, 3, 4, 3) = 0 since otherwise c(1, 3, 4, 2) = 0 too, which contradicts to the assumptions of case 2. If c(1, 3, 4, 4) = 0 then c(1, 3, 4, 2) = 0, and thus c(1, 3, 4, 3) = 0 by the hypothesis of case 2. Thus these vectors are linearly independent since e2 and e3 are linearly independent. Thus in case 2, D22 (A ) cannot be invariant under α and hence D22 (A ) is a weak subalgebra but not a Homsubalgebra of A . Since D22 (A ) is not a Hom-subalgebra of A , it is not a Hom-ideal either. Let us study now whether D22 (A ) is a weak ideal of A . We have: [e1 , e2 , v] = (c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2))e4 .

On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras

275

If (c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2)) = 0, then if c(1, 3, 4, 3) = 0, then [e1 , e2 , v] and v are linearly independent. Thus D22 (A ) is a weak subalgebra but not a weak ideal of A . If c(1, 3, 4, 3) = 0, then c(1, 3, 4, 2) = 0 by the assumptions of case 2, and hence [e1 , e3 , v] = (c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2)) (c(1, 3, 4, 2)e2 + c(1, 3, 4, 4)e4 ). This vector is linearly independent from v, which means that it is not an element of D22 (A ). Thus D22 (A ) is not a weak ideal of A . If (c(1, 3, 4, 3)c(1, 3, 4, 4) + c(1, 3, 4, 2)) = 0, then v = c(1, 3, 4, 3)2 (e3 − c(1, 3, 4, 4)e2 ).   In this case, e j , ek , v = 0 if and only if ( j, k) = (1, 4) or ( j, k) = (4, 1). Therefore, we compute only [e1 , e4 , v] and get [e1 , e4 , v] = c(1, 3, 4, 3)v. Thus, D22 (A ) is a weak ideal of A . In this case the bracket of A is given by [e1 , e2 , e3 ] = 0 [e1 , e2 , e4 ] = e4 [e1 , e3 , e4 ] = −c(1, 3, 4, 3)c(1, 3, 4, 4)e2 + c(1, 3, 4, 3)e3 + c(1, 3, 4, 4)e4 [e2 , e3 , e4 ] = 0, where c(1, 3, 4, 3) = 0. Example 1 If we take K = C, c(1, 3, 4, 4) = ±i and c(1, 3, 4, 3) = −2 then we get the following two examples where D22 (A ) is a weak ideal of A : 1) [e1 , e2 , e3 ] = 0 [e1 , e2 , e4 ] = e4 [e1 , e3 , e4 ] = 2ie2 − 2e3 + ie4 [e2 , e3 , e4 ] = 0, 2) [e1 , e2 , e3 ] = 0 [e1 , e2 , e4 ] = e4 [e1 , e3 , e4 ] = −2ie2 − 2e3 − ie4 [e2 , e3 , e4 ] = 0.

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6 Study of Derived Series and Central Descending Series of a Class of 3-dimensional Hom-Lie Algebras Consider a 3-dimensional K-linear space V with the basis elements {X 1 , X 2 , X 3 } and the bilinear skew-symmetric multiplication defined as follows: [X 1 , X 2 ] = λ1 X 2 , [X 1 , X 3 ] = λ2 X 3 , λ1 , λ2 , λ3 ∈ K [X 2 , X 3 ] = λ3 X 1 . The goal of this section is to present all twisting maps making these skewsymmetric algebras into Hom-Lie algebras and multiplicative Hom-Lie algebras, and to study their solvability, nilpotency and the properties of their derived series and central descending series in light of the results presented in earlier sections. The results are summarized in Table 1 and further discussion and analysis thereafter. We consider different dimensions of [V , V ] and for each value of the dimension, we give all the cases for λ1 , λ2 , λ3 realizing it. We exclude the cases dim [V , V ] = 3 and dim [V , V ] = 0 because their derived series and central descending series stabilize respectively at V and {0} from r = 1 and thus are not interesting for our study. In column 1, we list those different cases for λ1 , λ2 , λ3 . In column 2, for each case of λ1 , λ2 , λ3 , we provide the linear space of all linear maps α such that (V , [·, ·] , α) is a Hom-Lie algebra. In column 3, we state whether, in the corresponding case, the considered algebra is solvable (resp. nilpotent) or not, and we give all the different non-zero members of the derived series and central descending series of V starting from the r = 1 and indicate whether or not they are weak ideals. Each cell of column 3 corresponds to a case of column 1. These cells are further divided, when relevant, to separate different members of derived series and central descending series. In column 4, for each member of the derived series and central descending series, we give the subspace of the linear space of linear maps α given in column 2 such that this member of the derived series or central descending series is α-invariant. They are given together with the condition characterizing them as subspaces of the corresponding space of linear maps in column 2. In the cases where the cells of column 3 are divided, the corresponding cells in column 4 are divided accordingly. In column 5, for each case, we give the subset of the space of linear maps given in column 2 consisting of algebra morphisms, or, in other words, linear maps α such that (V , [·, ·] , α) is a multiplicative Hom-Lie algebra. We know by Proposition 7 that if (V , [·, ·] , α) is multiplicative, then all Dr (V ) and C r (V ) are invariant under α. The aim is to see for which non-multiplicative linear maps α this property is still satisfied. For example, in the first row, we consider the case where λ1 = 0 and λ2 , λ3 = 0, indicated in column 1. In column 2, a matrix, with some free parameters, representing the linear space of all linear maps α such that (V , [·, ·] , α), with λ1 = 0 and λ2 , λ3 = 0, is a Hom-Lie algebra. In column 3, the cell is divided into three parts. It is stated in the first part that the considered algebra with the chosen values of λ1 , λ2 , λ3 is class

⎞ a11 a12 a13 ⎟ ⎜ λ1 ⎝ λ3 a13 a22 a23 ⎠ a31 a32 0

λ2 = 0 λ1 , λ3  = 0 dim[V , V ] = 2

⎛

⎞ a11 a12 a13 ⎜ ⎟ 0 a23 ⎠ ⎝ a21 λ2 − λ3 a12 a32 a33

⎛

[α] such that (V , [· , ·] , α) is a Hom-Lie algebra

λ1 = 0 λ2 , λ3  = 0 dim[V , V ] = 2

Structure constants

D 2 (V ) = {X 2 } Not weak ideal

Not nilpotent Class-3-solvable For r ≥ 1 C r (V ) = {X 1 , X 2 } = D 1 (V ) Weak ideal

  D 2 (V ) = {X 3 } Not weak ideal

Not nilpotent Class-3-solvable For all r ≥ 1 C r (V ) = {X 1 , X 3 } = D 1 (V ) Weak ideal

Derived series, central descending series of V

=0 ⎞ a12 0 ⎟ 0 0 ⎠ a32 a33

a13 = a23 ⎛ a11 ⎜ ⎝ a21 − λλ23 a12

a31 = a32 = 0 ⎛ ⎞ a11 a12 a13 ⎜ λ1 ⎟ ⎝ λ3 a13 a22 a23 ⎠ 0 0 0 a12 = a32 = 0 ⎛ ⎞ a11 0 a13 ⎜ λ1 ⎟ ⎝ λ3 a13 a22 a23 ⎠ a31 0 0

=0 ⎞ a12 a13 ⎟ 0 0 ⎠ a32 a33

a21 = a23 ⎛ a11 ⎜ ⎝ 0 − λλ23 a12

[α] such that C r (V ), Dr (V ) are α−invariant

⎞ 00 0 ⎜ ⎟ ⎝0 0 a23 ⎠ 00 0

⎛

⎞ 0 0 0 ⎜ ⎟ ⎝0 0 0 ⎠ 0 a32 0

⎛

(continued)

[α] such that (V , [· , ·] , α) is multiplicative

Table 1 All twisting maps making the considered skew-symmetric algebras into Hom-Lie algebras and multiplicative Hom-Lie algebras. Derived series and central descending series and their properties. Solvability and nilpotency of these algebras

On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras 277

λ3 = 0 λ1 , λ2  = 0 dim[V , V ] = 2

Structure constants

Table 1 (continued)

⎛ ⎞ a11 0 0 ⎜ ⎟ ⎝a21 a22 a23 ⎠ a31 a32 a33

[α] such that (V , [· , ·] , α) is a Hom-Lie algebra

For r ≥ 1 C r (V ) = {X 2 , X 3 } D 1 (V ) = {X 2 , X 3 } Weak ideal

Not nilpotent Class-2-solvable

Derived series, central descending series of V

⎛ ⎞ a11 0 0 ⎜ ⎟ ⎝a21 a22 a23 ⎠ a31 a32 a33

[α] such that C r (V ), Dr (V ) are α−invariant

λ1 = λ2 ⎛ 1 0 ⎜ ⎝a21 a22 a31 a32 ⎛ 1 0 ⎜ ⎝a21 a22 a31 a32

⎞ 0 ⎟ a23 ⎠ a33 ⎞ 0 ⎟ 0 ⎠ a33

λ1 = ±λ2 ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎝a21 a22 0 ⎠ a 0 a33 ⎛ 31 ⎞ a11 0 0 ⎜ ⎟ ⎝a21 0 0⎠ a31 0 0 ⎞ ⎛λ 1 λ2 0 0 ⎟ ⎜ ⎝a21 0 0⎠ a31 a32 0 ⎛λ ⎞ 2 λ1 0 0 ⎜ ⎟ ⎝a21 0 a23 ⎠ a31 0 0

(continued)

[α] such that (V , [· , ·] , α) is multiplicative

278 A. Kitouni et al.

⎛ ⎞ a11 a12 a13 ⎜ ⎟ ⎝a21 a22 a23 ⎠ a31 a32 a33

⎛ ⎞ a11 0 a13 ⎜ ⎟ ⎝a21 a22 a23 ⎠ a31 a32 a33

λ1 , λ3 = 0 λ2  = 0 dim[V , V ] = 1

[α] such that (V , [· , ·] , α) is a Hom-Lie algebra

λ1 , λ2 = 0 λ3  = 0 dim[V , V ] = 1

Structure constants

Table 1 (continued)

For r ≥ 1 C r (V ) = {X 3 } D 1 (V ) = {X 3 } Weak ideal

Not nilpotent Class-2-solvable

C 1 (V ) = {X 1 } D 1 (V ) = {X 1 } Weak ideal

Class-2-nilpotent Class-2-solvable

Derived series, central descending series of V

a13 = a23 = 0 ⎛ ⎞ a11 0 0 ⎜ ⎟ ⎝a21 a22 0 ⎠ a31 a32 a33

a = a31 = 0 ⎞ ⎛21 a11 a12 a13 ⎟ ⎜ ⎝ 0 a22 a23 ⎠ 0 a32 a33

[α] such that C r (V ), Dr (V ) are α−invariant

0 ⎜ ⎝a21 a31 ⎛ 1 ⎜ ⎝a21 a31

⎛

⎞ 0 0 ⎟ a22 0 ⎠ a32 0 ⎞ 0 0 ⎟ a22 0 ⎠ 0 a33 (continued)

⎛ ⎞ a22 a33 − a23 a32 a12 a13 ⎜ ⎟ 0 a22 a23 ⎠ ⎝ 0 a32 a33

[α] such that (V , [· , ·] , α) is multiplicative ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎝a21 a22 a23 ⎠ a 0 a33 ⎞ ⎛ 31 1 0 0 ⎟ ⎜ ⎝a21 0 a23 ⎠ a31 a32 0 λ1 = −λ2 ⎛ ⎞ −1 0 0 ⎜ ⎟ ⎝a21 0 a23 ⎠ a31 a32 0

On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras 279

λ2 , λ3 = 0 λ1  = 0 dim[V , V ] = 1

Structure constants

Table 1 (continued)

⎛ ⎞ a11 a12 0 ⎜ ⎟ ⎝a21 a22 a23 ⎠ a31 a32 a33

[α] such that (V , [· , ·] , α) is a Hom-Lie algebra

C 1 (V ) = {X 2 } D 1 (V ) = {X 2 } Weak ideal

Not nilpotent Class-2-solvable

Derived series, central descending series of V

a12 = a32 = 0 ⎞ ⎛ a11 0 0 ⎟ ⎜ ⎝a21 a22 a23 ⎠ a31 0 a33

[α] such that C r (V ), Dr (V ) are α−invariant

0 ⎜ ⎝a21 a31 ⎛ 1 ⎜ a ⎝ 21 a31 ⎛ a11 ⎜ ⎝a21 a31

⎛

⎞ 0 0 ⎟ 0 a23 ⎠ 0 a33 ⎞ 0 0 ⎟ a22 0 ⎠ 0 a33 ⎞ 0 0 ⎟ 0 0 ⎠ 0 a33

[α] such that (V , [· , ·] , α) is multiplicative ⎛ ⎞ a11 0 0 ⎜ ⎟ ⎝a21 a22 0⎠ a31 0 0

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3 solvable and not nilpotent. In the second part, D 1 (V ) which is equal, in this case, to C r (V ) for all r ≥ 1 is given and it is stated that it is a weak ideal of V . In the third part, D 2 (V ) is given and it is stated that it is not a weak ideal of V . In column 4, the cell is divided into two parts. The first part contains a matrix (with some free parameters) representing the subspace of the space of linear maps given in column 2, consisting of linear maps α such that D 1 (V ) = C r (V ) is α-invariant together with the system of linear equations characterizing it. In the second part, in the same way, the subspace of linear maps α such that D 2 (V ) is α-invariant is given. In column 5, we give a matrix, with free parameters (only one in this case), representing the subset of the linear space given in column 2 consisting of algebra morphisms, or, in other words, linear maps α such that (V , [·, ·] , α), with λ1 = 0 and λ2 , λ3 = 0, is a multiplicative Hom-Lie algebra. Unless otherwise stated, the scalars present in the table are arbitrary elements of K. In all the cases presented in this table, C 1 (V ) and D 1 (V ) are weak ideals of V as stated by Proposition 8. In the first and second cases, D 2 (V ) are not weak ideals, this is an example where the conditions of Proposition 7 are not satisfied and its conclusion does not hold. In these cases, there is no surjective α such that (V , [·, ·] , α) is a multiplicative Hom-Lie algebra as shown in the fifth column of the table. In these cases, the linear maps described in column 4 are not surjective and not necessarily weak morphisms, however C r (V ) for r ≥ 1 and D 1 (V ) are Homideals of V . This indicates that it may be possible to consider weaker conditions in Proposition 7, see for example Proposition 9. In these cases, the set of linear maps α such that (V , [·, ·] , α) is a Hom-Lie algebra, does not contain the identity map, which means that the algebras described in theses cases are Hom-Lie algebras but are not Lie algebras. In all other cases, the identity map is among the linear maps α such that (V , [·, ·] , α) is a Hom-Lie algebra, and thus these are also Lie algebras. Proposition 12 Let A = (A, [·, ·]) be a 3-dimensional skew-symmetric algebra and let H om A be the subspace of α ∈ End(A) such that (A, [·, ·] , α) is a Hom-Lie algebra. If A is nilpotent, then dim H om A = 9. Proof Suppose that A is nilpotent and 3-dimensional, then C 1 (A) = Z (A) by Proposition 2, and thus ∀x, y, z ∈ A, [x, [y, z]] = 0, which implies that ∀α ∈ End(A), ∀x, y, z ∈ A, [α(x), [y, z]] + [α(y), [z, x]] + [α(z), [x, y]] = 0. Therefore dim H om A = dim End(A) = 9.



The case 4 in the table is an example where Proposition 12 applies, the algebra is nilpotent and 3-dimensional. This implies that the set of α such that (V , [·, ·] , α) is a Hom-Lie algebra is the whole End(V ). In this case C 1 (V ) = Z (V ), which can be seen from the definition of the bracket [X 1 , X 2 ] = 0 , [X 1 , X 3 ] = 0 , [X 2 , X 3 ] = λ3 X 1 .

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The first two relations say that X 1 ∈ Z (V ), and the last one says that X 2 , X 3 ∈ / Z (V ) and C 1 (V ) = {X 1 }. This fact can be also obtained by Proposition 2. If λ1 , λ2 , λ3 = 0 then the set of α such that (V , [·, ·] , α) is a Hom-Lie algebra is ⎛ ⎞ a11 a12 a13 λ given by ⎝ λ13 a13 a22 a23 ⎠. This algebra is not nilpotent, not solvable and for − λλ23 a12 a32 − λλ21 a22 all r ≥ 1, C r (V ) = Dr (V ) = V is invariant under all the linear maps α. The linear maps α such that (V , [·, ·] , α) is multiplicative are given by ⎛ ⎞ 1 0 0 2 0 ⎠ where a22 = − λλ21 (a) when λ1 = ±λ2 , ⎝0 a22 λ2 0 0 − λ1 a22 (b) when λ1 = λ2 , ⎛ ⎞ ⎞ ⎛ 1 0 0 1 0 0 2 2 22 ⎠ for a (ii) ⎝0 a22 0 ⎠ where a22 = −1 (i) ⎝0 a22 − 1+a 32  = 0 a32 0 0 −a 22 0 a −a 32

22

(c) when λ1 = −λ2 , ⎛ a11 ⎜ λ (i) ⎝ λ13 a13

(−1+a11 )(1+a11 )λ3 2a13 λ2 − 21 (1 + a11 ) (1−a11 )(1+a11 ) (−1+a11 )2 (1+a11 )λ3 2a13 4a13 λ2

⎞ 1 a12 0 −1 0 ⎠ (ii) ⎝ 0 a2 λ λ2 − λ3 a12 − 12λ3 2 −1 ⎛ ⎞ 100 (iv) ⎝0 1 0⎠. 001 ⎛

a13

⎞

⎟ a13 λ2 − (1+a ⎠, for a13 = 0, a11 = −1 11 )λ3 1 − 2 (1 + a11 ) ⎛ ⎞ −1 0 0 (iii) ⎝ 0 0 a132 ⎠ for a32 = 0 0 a32 0

In this case, when λ1 = −λ2 the set of linear maps α such that (V , [·, ·] , α) is a Hom-Lie algebra does not contain the identity map, which means that the algebras described in theses cases are Hom-Lie algebras but are not Lie algebras. If λ1 = −λ2 , the identity map is among the linear maps α such that (V , [·, ·] , α) is a Hom-Lie algebra, and thus these are also Lie algebras. Note that this case does not satisfy the hypothesis of Proposition 12, and the dimension of the subspace of α such that (V , [·, ·] , α) is a Hom-Lie algebra dropped to 6. Acknowledgements Elvice Ongong’a and Stephen Mboya are grateful to the International Science Program (ISP), Uppsala University for the support in the framework of the Eastern Africa Universities Mathematics Programme (EAUMP). Elvice Ongong’a, Stephen Mboya and Abdennour Kitouni also thank the research environment in Mathematics and Applied Mathematics (MAM), the Division of Mathematics and Physics of the School of Education, Culture and Communication at Mälardalen University for hospitality and creating excellent conditions for research and research education. The authors are grateful to anonymous referees for helpful suggestions.

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Okubo Algebras with Isotropic Norm Alberto Elduque

Abstract Okubo algebras form an important class of nonunital composition algebras of dimension 8. Contrary to what happens for unital composition algebras, they are not determined by their multiplicative norms. Okubo algebras with isotropic norm are characterized here by the existence of a special grading. In the split case, and under some restrictions on the ground field, the automorphism group of the most symmetric of these gradings is the projective unitary group PU(3, 22 ), whose structure is showcased by the grading. Keywords Composition algebras · Okubo algebras · Special grading · Finite field Unital composition algebras (also termed Hurwitz algebras) over a field are the analogues of the classical algebras or real and complex numbers, quaternions, and octonions. In particular, their dimension is restricted to 1, 2, 4 or 8. The reader may consult [19, Chap. 2], [13, Chap. VIII], or [18]. In recent years, a new class of composition algebras, the so called symmetric composition algebras, better suited for the study of the triality phenomenon, have made their appearance. These split in two disjoint families: para-Hurwitz algebras and Okubo algebras. (See e.g. [13, Chap. VIII] or [6]). Two Hurwitz (or para-Hurwitz) algebras over a field are isomorphic if and only if their norms are isometric. Since the norm of any composition algebra is a Pfister form, this implies in particular that, up to isomorphism, in each dimension 2, 4, or 8, there is a unique Hurwitz (or para-Hurwitz) algebra with isotropic norm. These are the split Hurwitz (or para-Hurwitz) algebras. Over a field F, the split Hurwitz algebras are, up to isomorphism, F × F, Mat2 (F), and the split Cayley algebra Cs (F). Supported by grant MTM2017-83506-C2-1-P (AEI/FEDER, UE), by grant PID2021-123461NBC21, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, and by grants E22_17R and E22_20R (Gobierno de Aragón, Grupo de investigación “Álgebra y Geometría”). A. Elduque (B) Departamento de Matemáticas e Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0_18

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The situation for Okubo algebras is not that neat. Over many fields (see Corollaries 2 and 3), the norm of any Okubo algebra is isotropic, but this does not imply that the Okubo algebra is the split one. A characteristic feature of the Okubo algebras  with isotropic norm is that they are endowed with a grading by (Z/3)2 : O = 0=a∈(Z/3)2 Oa , where the dimension of all (nonzero) homogeneous components is 1. The paper is structured as follows. The first section will provide the basic definitions about composition algebras. The split Okubo algebra will be defined in a very precise way. Okubo algebras will then be defined as the twisted forms of the split one. Sect. 2 will review the main known results on Okubo algebras. The short Sect. 3 will be devoted to showing that over any finite field the only Okubo algebra is, up to isomorphism, the split one. This will be instrumental in Sect. 4, where the Okubo algebras with isotropic norm will be characterized in terms of the existence of some gradings on them. The automorphisms groups of these gradings will be computed. The largest group that appears is the projective unitary group PU(3, 22 ). The classical group PSU(3, 22 ) is not simple, and its structure will become clear by means of the only Okubo algebra over the field of four elements. All the algebras considered in this paper will be defined over a ground field F.

1 Composition Algebras A quadratic form n : V → F on a vector space V over a field F is said to be nondegenerate if so is its polar form: n(x, y) := n(x + y) − n(x) − n(y), that is, if its radical V ⊥ := {v ∈ V : n(v, V ) = 0} is trivial. Moreover, n is said to be nonsingular if either it is nondegenerate or it satisfies that the dimension of V ⊥ is 1 and n(V ⊥ ) = 0. The last possibility only occurs over fields of characteristic 2. Definition 1 A composition algebra over a field F is a triple (C, ·, n) where • (C, ·) is a nonassociative algebra, and • n : C → F is a nonsingular quadratic form which is multiplicative, that is, n(x · y) = n(x)n(y)

(1)

for any x, y ∈ C. The unital composition algebras are called Hurwitz algebras. These have the property that any element satisfies the degree 2 equation: x ·2 − n(x, 1)x + n(x)1 = 0.

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The map x → x = n(x, 1)1 − x is both an involution and an isometry, and it satisfies n(x · y, z) = n(y, x · z) = n(x, z · y) for all x, y, z. For simplicity, we will refer to the composition algebra C, instead of (C, ·, n). The well-known Generalized Hurwitz Theorem (see e.g. [13, (33.17)]) asserts that the Hurwitz algebras are, up to isomorphism, either F, quadratic étale F-algebras, quaternion algebras over F, or Cayley algebras over F. In particular, the dimension of a Hurwitz algebra is restricted to 1, 2, 4, or 8. Moreover, two Hurwtiz algebras are isomorphic if and only if their norms are isometric. It turns out that the dimension of any finite-dimensional not necessarily unital composition algebra is restricted too to 1, 2, 4, or 8, but there are examples of nonunital composition algebras of arbitrary infinite dimension [10]. Definition 2 A composition algebra (S, ∗, n) is said to be a symmetric composition algebra if n(x ∗ y, z) = n(x, y ∗ z) for all x, y, z ∈ S. The condition on the definition above is equivalent to (x ∗ y) ∗ x = x ∗ (y ∗ x) = n(x)y

(2)

for all x, y ∈ S. The dimension of any symmetric composition algebra is finite, and hence restricted to 1, 2, 4, or 8. The first examples of symmetric composition algebras are given as follows. Let (C, ·, n) be a Hurwitz algebra and consider the composition algebra (C, •, n) with the new product given by x • y = x · y. Then n(x • y, z) = n(x · y, z) = n(x, z · y) = n(x, z · y) = n(x, y • z), for any x, y, z, so that (C, •, n) is a symmetric composition algebra. (Note that 1 • x = x • 1 = x = n(x, 1)1 − x for any x: 1 is a para-unit of (C, •, n)). These symmetric composition algebras are called para-Hurwitz. It is easy to prove that two para-Hurwitz algebras are isomorphic if and only if so are the corresponding Hurwitz algebras. But apart from para-Hurwitz algebras, there is a new class of eight-dimensional symmetric composition algebras with different properties. These are the Okubo algebras. Let Mat 3 (F) be the associative algebra of 3 × 3-matrices and let sl(3, F) be the corresponding special Lie algebra, consisting of the zero trace matrices. Any element a ∈ Mat 3 (F) is a root of its characteristic polynomial: det(X 1 − a) = X 3 − tr(a)X 2 + sr(a)X − det(a) where tr(a) is the trace, det(a) the determinant, and sr(a) is a quadratic form on the coordinates of a which, if the characteristic is not 2, equals 21 (tr(a)2 − tr(a 2 )).

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(The same happens for any central simple associative algebra of degree 3. In this case tr is the generic trace, sr a suitable quadratic form, and det its generic norm). The polar form of the quadratic form sr : sr(a, b) := sr(a + b) − sr(a) − sr(b), satisfies (in any characteristic) sr(a, b) = tr(a)tr(b) − tr(ab) for any a, b ∈ Mat 3 (F). If the characteristic of F is not 3, then the restriction of the quadratic form sr to sl(3, F) = 1⊥ is nondegenerate. Assume for a while that the characteristic of our ground field F is not 3, and that F contains a primitive cubic root ω of 1. On sl(3, F), define a new multiplication and a quadratic form as follows: x ∗ y = ωx y − ω2 yx −

ω − ω2 tr(x y), 3

(3)

n(x) = sr(x). A straightforward computation (see [9]) gives n(x ∗ y) = n(x)n(y), (x ∗ y) ∗ x = x ∗ (y ∗ x) = n(x)y  for all x, y ∈ sl(3, F) and hence sl(3, F), ∗, n) is an eight-dimensional symmetric composition algebra. In particular, take F = C, the field of complex numbers, and consider the Pauli matrices: ⎛ ⎞ ⎛ ⎞ 10 0 010 x = ⎝0 ω 0 ⎠ , y = ⎝0 0 1⎠ , (4) 0 0 ω2 100 in Mat3 (C), which satisfy x 3 = y 3 = 1,

yx = ωx y.

(5)

For i, j ∈ Z/3, (i, j) = (0, 0), define z i, j :=

ω−i j i j x y . ω − ω2

(6)

Then {z i, j : (i, j) = (0, 0)} is a basis of sl(3, C). Using (5) we obtain:

z i, j ∗ z i  , j 

⎧ ⎪ if  = 2 or i + i  = 0 = j + j  , ⎨0 = −z i+i  , j+ j  if  = 1, ⎪ ⎩ otherwise, z i+i  , j+ j 

(7)

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where  = det ii jj . Moreover, we get n(z i, j ) = 0 for any (i, j) = (0, 0), and n(z i, j , z i  , j  ) is 1 for i + i  = 0 = j + j  and 0 otherwise. Thus the Z-span   OZ = Z−span z i, j | i, j ∈ Z/3, (i, j) = (0, 0) is closed under ∗, and n restricts to a nonsingular multiplicative quadratic form on OZ . This allows us to define Okubo algebras over arbitrary fields: Definition 3 The algebra OF := OZ ⊗Z F, with the induced multiplication and nonsingular quadratic form, is called the split Okubo algebra over the field F. The twisted forms of (OF , ∗, n) are called Okubo algebras. Remark 1 This is not the original definition of these algebras given by Okubo [14] and Okubo and Osborn [15], but it is equivalent to it. Let F be an algebraic closure of the field F. Take 0 = α, β ∈ F, and consider cubic roots α 1/3 and β 1/3 in F. Then the elements in O ⊗Z F given by z 1,0 ⊗ α 1/3 , z 0,1 ⊗ β 1/3 generate, by multiplication and linear combinations with coefficients in F, a twisted form of the split Okubo algebra, with a basis (over F!) consisting of the elements z˜ i, j = z i, j ⊗ α i/3 β j/3 , for 0 ≤ i, j ≤ 2, (i, j) = (0, 0). This Okubo algebra will be denoted by Oα,β , with O1,1 being the split Okubo algebra. The multiplication table in this basis is given in Fig. 1.

2 Classification of Okubo Algebras The classification of Okubo algebras has a different flavor depending on the characteristic of the ground field being = 3 or 3. ∗

z˜1,0

z˜2,0

z˜0,1

z˜0,2

z˜1,1

z˜2,2

z˜1,2

z˜2,1

z˜1,0

z˜2,0

0

−˜ z1,1

0

−˜ z2,1

0

0

−α˜ z0,1

z˜2,0

0

α˜ z1,0

0

−˜ z2,2

0

−α˜ z1,2

−α˜ z0,2

0

z˜0,1

0

−˜ z2,1

z˜0,2

0

0

−β z˜2,0

0

−˜ z2,2

z˜0,2

−˜ z1,2

0

0

β z˜0,1

−β z˜1,0

0

−β z˜1,1

0

z˜1,1

0

−α˜ z0,1

−˜ z1,2

0

z˜2,2

0

−β z˜2,0

0

z˜2,2

−α˜ z0,2

0

0

−β z˜2,1

0

αβ z˜1,1

0

−αβ z˜1,0

z˜1,2

−˜ z2,2

0

−β z˜1,0

0

0

−αβ z˜0,1

β z˜2,1

0

z˜2,1

0

−α˜ z1,1

0

−β z˜2,0

−α˜ z0,2

0

0

α˜ z1,2

Fig. 1 Multiplication table of Oα,β

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2.1 Characteristic not 3 If A is a finite-dimensional associative algebra, A0 will denote the subspace of generic trace 0 elements of A. Theorem 1 ([2, 8, 9]) Let F be a field of characteristic not 3 and let ω be a primitive cubic root of 1 in an algebraic closure of F. (i) If ω ∈ F, then the Okubo algebras over F are, up to isomorphism, exactly the algebras (A0 , ∗, n), where A is a central simple associative algebra over F of degree 3 and ∗ and n are given by (3). Two Okubo algebras are isomorphic if and only if so are the corresponding central simple associative algebras. (ii) If ω ∈ / F, and K = F[ω],  then the Okuboalgebras over F are, up to isomorphism, exactly the algebras Skew(A0 , τ ), ∗, n , where A is a central simple associative algebra over K of degree 3, endowed with a K/F-involution τ of the second kind, Skew(A0 , τ ) is the subspace of generic trace 0 elements x with τ (x) = −x, and ∗ and n are given by (3). Two Okubo algebras are isomorphic if and only if so are the corresponding pairs (A, τ ), as K-algebras with involution. Corollary 1 ([1, Corollary 8.9]) Let F be a field of characteristic not 3 and let ω be a primitive cubic root of 1 in an algebraic closure of F. (i) If ω ∈ F, then the group of automorphisms of the Okubo algebra (A0 , ∗, n), as in Theorem 1.(i) is isomorphic to Aut(A). (ii) If / F, and K = F[ω], then the group of automorphisms of the Okubo algebra   ω∈ Skew(A0 , τ ), ∗, n as in Theorem 1. (ii) is isomorphic to Aut(A, τ ) (the group of automorphisms of A commuting with τ ). In both cases, any automorphism of A (commuting with τ in (ii)) acts by restriction to A0 or Skew(A0 ), respectively. The fact that any central simple division algebra over a field containing a primitive cubic root of unity is cyclic (see e.g. [16, p. 286]) has the following consequence: Corollary 2 ([9, Proposition 7.3]) Let F be a field of characteristic not 3 containing a primitive cubic root of 1. Then the norm of any Okubo algebra over F is isotropic. Remark 2 Actually, if our field F contains a primitive cubic root ω of 1, any central simple  algebra A of degree 3 is a symbol algebra, that is, it is isomorphic to the algebra α, β F,ω for some nonzero scalars α, β ∈ F, defined as the algebra generated by two elements x, y subject to the relations (compare with (5)) x 3 = α,

y 3 = β,

yx = ωx y.

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It is clear that then A0 , with multiplication and norm as in (3), is isomorphic to the Okubo algebra Oα,β . If our field F has characteristic = 3 but does not contain a primitive cubic root of 1, × consider the quadratic  field  extension K = F[ω] as in Theorem 1. For 0 = α, β ∈ F , the symbol algebra α, β K,ω is endowed with a second kind involution τ fixing the generators x and y. Then  τ

ω−i j i j x y ω − ω2

 =

ωi j ω−i j i j j i y x = − x y , ω2 − ω ω − ω2

and it turns out that the Okubo algebra Skew(A0 , τ ), with multiplication and norm as in (3), is isomorphic to the Okubo algebra Oα,β . Among the Okubo algebras with isotropic norm, the split one is characterized as follows: Theorem 2 ([5, Theorem 5.9]) An Okubo algebra over a field of characteristic not 3 is split if and only if its norm is isotropic and it contains a nonzero idempotent (e ∗ e = e).

2.2 Characteristic 3 In characteristic 3, the classification does not follow the same lines: Theorem 3 ([2]) Up to isomorphism, the Okubo algebras over a field F of characteristic 3 are precisely the algebras Oα,β for α, β ∈ F× . Corollary 3 The norm of any Okubo algebra over a field of characteristic 3 is isotropic. In [2], precise conditions for two Okubo algebras, over a field F of characteristic 3, to be isomorphic are given. In particular, if F is perfect, the only Okubo algebra, up to isomorphism, is the split one. Over fields of characteristic 3 there exist three kinds of (nonzero) idempotents in Okubo algebras (see [5]): Quaternionic if the restriction of the norm to the centralizer of the idempotent has rank 4. Quadratic if the restriction of the norm to the centralizer of the idempotent has rank 2. Singular if the restriction of the norm to the centralizer of the idempotent has rank 1. The split Okubo algebra is characterized as follows:

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Theorem 4 ([5, Corollary 6.5]) Let O be an Okubo algebra over a field F of characteristic 3 with norm n. The following conditions are equivalent: (i) O is split. (ii) n is isotropic and O contains a singular idempotent. (iii) n is isotropic and O contains a quaternionic idempotent.

3 Okubo Algebras Over Finite Fields are Split By Wedderburn’s Little Theorem, there are no quaternion division algebras over a finite field, and hence the only four-dimensional Hurwitz algebra is, up to isomorphism, the algebra of 2 × 2-matrices. In particular, its norm is isotropic. As any Cayley algebra contains quaternion subalgebras, it follows that its norm is isotropic too, and again the only Hurwitz (or para-Hurwitz) algebras of dimension 4 or 8 over a finite field are the split ones. However, the isometry class of the norm does not determine the isomorphism class of Okubo algebras. Even so, the only Okubo algebra over a finite field is the split one. To prove this, we need to recall the following well-known result: Lemma 1 Let F be a finite field, K/F a quadratic field extension ([K : F] = 2), τ the nontrivial F-automorphism of K, U a finite-dimensional K-vector space, and h : U × U → K a nondegenerate hermitian form: h is F-bilinear, h(αu, v) = αh(u, v),   and h(v, u) = τ h(u, v) for any u, v ∈ U and α ∈ K. Then there is a K-basis {u 1 , . . . , u n } in U with h(u i , u i ) = 1 and h(u i , u j ) = 0 for 1 ≤ i = j ≤ n. Proof This is well known. If F is the field of q elements, then τ (α) = α q for any α. Let μ be a generator of the cyclic group K× . Hence F× is generated by μq+1 , so for any α ∈ F there is an element β ∈ K such that α = β q+1 . It cannot be the case that h(u, u) equals 0 for all u because this would imply h(u, v) = −h(v, u) = −τ (h(u, v)), and this would give τ (α) = −α for any α ∈ K, a contradiction, so let us take an element u ∈ U with h(u, u) = α = 0. Pick β ∈ K with β q+1 = α, then:  q   h β −1 u, β −1 u = β −1 β −1 h(u, u) = α −1 h(u, u) = 1. Now the argument can be repeated with the restriction h|(Ku)⊥ .



Theorem 5 Let O be an Okubo algebra over a finite field F. Then O is split. Proof If the characteristic is 3, this follows from the isomorphism conditions in [2], which shows in particular that the only Okubo algebra over a perfect field is the split one. If the characteristic of F is not 3 and F contains a primitive cubic root of 1, then by Wedderburn’s Little Theorem the only central simple associative algebra over F

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is Mat 3 (F), and hence Theorem 1 shows that there is, up to isomorphism, a unique Okubo algebra over F. If the characteristic of F is not 3 and F does not contain a primitive cubic root of 1, let K = F[ω] be the field obtained by adjoining one such root to F. Again, the only central simple associative K-algebra is Mat 3 (K), and any K/F-involution of the second kind in Mat 3 (K) is given by the adjoint relative to a nondegenerate K/F-hermitian form on K3 . Since there is only one such form, up to isometry, by Lemma 1, the result follows. 

4 Special Gradings on Okubo Algebras with Isotropic Norm Let us recall the definition of equivalent and isomorphic gradings (see [7]).   Definition 4 Let Γ : A = g∈G Ag and Γ  : B = h∈H Bh be two gradings by the groups G and H on the algebras A and B. 1. Γ and Γ  are said to be equivalent if there is an algebra isomorphism ϕ : A → B such that for any g ∈ G in the support SuppΓ (this is the set of elements g ∈ G such that Ag = 0), there is an element h ∈ H such that ϕ(Ag ) = Bh . In this case we will write (A, Γ ) eq (B, Γ  ). 2. Γ and Γ  are said to be isomorphic if G = H and there is an algebra isomorphism ϕ : A → B such that ϕ(Ag ) = Bg for all g ∈ G. 3. The group Aut(Γ ) is the group of autoequivalences of Γ (i.e. automorphisms ϕ of A that permute the homogeneous components of Γ ). Its subgroup Stab(Γ ) is the group of isomorphisms of Γ (i.e., autoequivalences that fix each homogeneous component). The quotient W (Γ ) = Aut(Γ )/Stab(Γ ) is called the Weyl group of Γ . It must be remarked that, in dealing with gradings by groups on symmetric composition algebras, it is enough to consider gradings by abelian groups (see [7, Proposition 4.49]). Following the notation  used by Hesselink for Lie algebras [12], a grading by an abelian group Γ : A = g∈G Ag on an algebra A will be called special if the homogeneous component corresponding to the neutral element is trivial: Ae = 0. Okubo algebras with isotropic norm are characterized by the existence of special gradings on them. Theorem 6 Let (O, ∗, n) be an Okubo algebra over a field F. The following conditions are equivalent: (i) The norm n is isotropic. (ii) There are scalars α, β ∈ F× such that O is isomorphic to Oα,β . (iii) O admits a special grading by an abelian group.

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Proof The equivalence of (i) and (ii) follows from [3, Theorem 4] and Remark 2 in case the characteristic of F is not 3, and from Theorem 3 in characteristic 3. It is clear that Oα,β is (Z/3)2 -graded, with deg(˜z i, j ) = (i, j) (modulo 3 ).

(8)

This is a special grading, so (ii) implies (iii). Conversely, according to [4, Theorem 4.4], the only special gradings on Okubo algebras are, up to equivalence, the above (Z/3)2 -gradings on the Okubo algebras  Oα,β . We will denote by Γα,β the (Z/3)2 -grading on Oα,β defined in (8). Corollary 4 Any special grading Γ on an Okubo algebra O is equivalent to the canonical grading Γα,β on Oα,β for some 0 = α, β ∈ F:     O, Γ eq Oα,β , Γα,β . The proof of the above Theorem is based on [4, Theorem 4.4], which relies on the next technical result, that has its own independent interest. Lemma 2 ([4, Theorem 3.12 and Corollary 3.17]) Let (S, ∗, n) be an eightdimensional symmetric composition algebra over a field F, and let x, y ∈ S be elements such that n(x) = n(y) = 0, n(x, x ∗ x) = α = 0 = β = n(y, y ∗ y), and n(Fx + Fx ∗ x, Fy + Fy ∗ y) = 0. Then (S, ∗, n) is an Okubo algebra and either x ∗ y = 0 or y ∗ x = 0, but not both. Moreover, if y ∗ x = 0, then (S, ∗, n) is isomorphic to Oα,β under an isomorphism that takes x to z˜ 1,0 and y to z˜ 0,1 . In the above Lemma, if x ∗ y = 0, interchanging the roles of x and y, it follows that (S, ∗, n) is isomorphic to Oβ,α under an isomorphism that takes y to z˜ 1,0 and x to z˜ 0,1 .  Let Γ : O = g∈G Og be a special grading on an Okubo algebra. Denote by H the subgroup of G generated by the support SuppΓ = {g ∈ G | Og = 0}. Consider the following map: ΦΓ : H −→ F× /(F× )3  [1] if h = e, h → [n(u, u ∗ u)] for 0 = u ∈ Oh , otherwise. (Here [α] := (αF× )3 for all α ∈ F× ).

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Lemma 3 The above map ΦΓ is a well defined group homomorphism. Proof By [4, Theorem 4.4], H = {e} ∪ SuppΓ (disjoint union) is isomorphic to (Z/3)2 . Besides, for any e = h ∈ H and 0 = u ∈ Oh , 0 = u ∗ u ∈ Oh 2 and n(u, u ∗ u) = 0 because n(Og , Oh ) = 0 unless gh = e (and n(Og ) = 0 unless g 2 = e, [4, §4]). Therefore, ΦΓ is well defined. For 0 = u ∈ Oh , (2) gives   (u ∗ u) ∗ (u ∗ u) = − (u ∗ u) ∗ u ∗ u + n(u, u ∗ u)u = −n(u)u + n(u, u ∗ u)u = n(u, u ∗ u)u, as n(Oh ) = 0. Hence we get   n u ∗ u, (u ∗ u) ∗ (u ∗ u) = n(u, u ∗ u)2 so that ΦΓ (h 2 ) = ΦΓ (h)2 . Take now e = h, g ∈ H with g = h, h 2 , and pick nonzero elements x ∈ Oh and y ∈ Og . By Lemma 2, either x ∗ y = 0 or y ∗ x = 0, but not both. If x ∗ y = 0 (the other case is similar), then using repeatedly (2) and the fact that two homogeneous spaces are orthogonal relative to n unless the product of their degrees is e, we get (y ∗ y) ∗ x = −(x ∗ y) ∗ y = 0 and (y ∗ x) ∗ (y ∗ x) = −x ∗ (y ∗ (y ∗ x)) = x ∗ (x ∗ (y ∗ y)) = −(y ∗ y) ∗ (x ∗ x), so that     n y ∗ x, (y ∗ x) ∗ (y ∗ x) = −n y ∗ x, (y ∗ y) ∗ (x ∗ x)   = −n (y ∗ x) ∗ (y ∗ y), x ∗ x = −n(y, y ∗ y)n(x, x ∗ x). This shows ΦΓ (gh) = ΦΓ (g)ΦΓ (h).

(9) 

Note that the image of the homomorphism ΦΓ is a 3-elementary abelian group of rank ≤ 2. Denote by μ3 (F) the group of cubic roots of 1 in the field F. This is either trivial or cyclic of order 3 (isomorphic to Z/3). Among the finite classical groups it is well known that the projective special linear groups PSL(n, p m ) (for a prime number p) are all simple with the exceptions of PSL(2, 2) and PSL(2, 3). Their action on the projective line show easily that PSL(2, 2) is, up to isomorphism, the symmetric group of degree 3 and PSL(2, 3) is the alternating group of degree 4. Besides, the projective special unitary groups PSU(n, p 2m ) are simple with the following exceptions: PSU(2, 22 ), PSU(2, 32 ), and PSU(3, 22 ). Moreover, PSU(2, 22 ) is the symmetric group of degree 3 and PSU(2, 32 ) is the alternating group of order 4. (See e.g. [11, Chap. 11]). The structure of the remaining case, PSU(3, 22 ), turns out to be related to Okubo algebras and their special gradings.

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Theorem 7 Let Γ be a special grading on an Okubo algebra O over the field F. Then the short exact sequence 1 −→ Stab(Γ ) −→ Aut(Γ ) −→ W (Γ ) −→ 1

(10)

splits. Moreover, Stab(Γ ) = μ3 (F)2 , which is trivial if the characteristic of F is 3 or if char F = 3 but F does not contain a primitive cubic root of 1, and it is isomorphic to (Z/3)2 if char F = 3 and F contains a primitive cubic root of 1. The following possibilities appear: • The image of ΦΓ is trivial. In this case the algebra O is the split Okubo algebra, (O, Γ ) eq O1,1 , Γ1,1 , and W (Γ ) is isomorphic to the special linear group SL(2, 3).     • The image of ΦΓ is cyclic of order 3. In this case O, Γ eq O1,α , Γ1,α , where [α] is a generator of the image of ΦΓ , and the Weyl group is cyclic of order 3. • The image of ΦΓ is isomorphic to (Z/3)2 . In this case the Weyl group W (Γ ) is  [α] and  [β] of  the imageof ΦΓ are picked, then either trivial, and iftwo generators O, Γ eq Oα,β , Γα,β , or O, Γ eq Oβ,α , Γβ,α , but not both. Proof The subgroup H generated by the support of Γ is isomorphic to (Z/3)2 , and the proof of Theorem 6 shows that (O, Γ ) eq (Oα,β , Γα,β ) for some α, β ∈ F× . Then W (Γ ) can be identified with a subgroup of Aut(H ), and hence by a subgroup  2 of GL(2, 3) = Aut Z/3 . Equation (7) and Lemma 2 shows that W (Γ ) is, under the identification above, precisely W (Γ ) = { f ∈ SL(2, 3) | ΦΓ ( f.a) = ΦΓ (a) ∀a ∈ (Z/3)2 }.

(11)

If the image of ΦΓ is trivial, and a, b generate H , there are elements x ∈ Oa , y ∈ Ob with n(x, x ∗ x) = 1 = n(y, y ∗ y) and either x ∗ y = 0 or y ∗ x = 0, but not both. If x ∗ y = 0, Lemma 2 shows that there is an isomorphism O → O1,1 mapping x to z˜ 1,0 and y to z˜ 0,1 , while if x ∗ y = 0, then (2) shows that (y ∗ y) ∗ x = −(x ∗ y) ∗ y = 0. Lemma 2 gives x ∗ (y ∗ y) = 0. Replacing y by y ∗ y, which is homogeneous too, we get that there is an isomorphism   O → O1,1 mapping x to z˜ 1,0 and y ∗ y to z˜ 0,1 . This shows (O, Γ ) eq O1,1 , Γ1,1 . Moreover, W (Γ ) is here the whole SL(2, 3), and (7) shows that for any f ∈ SL(2, 3), the linear map that takes z˜ a to z˜ f (a) for any a ∈ (Z/3)2 is an automorphism. This shows that (10) splits. If the image of ΦΓ is cyclic of order 3, take a ∈ ker ΦΓ and b ∈ H \ ker ΦΓ with ΦΓ (b) = [α]. By Lemma 2 either Oa ∗ Ob = 0, or Ob ∗ Oa = 0. In the latter case, as 2 above, we have Oa 2 ∗ Ob = 0. Thus, changing  a by a if necessary, we may assume Oa ∗ Ob = 0. Lemma 2 shows (O, Γ ) eq O1,α , Γ1,α . Identify H with (Z/3)2 by means of a → (1, 0) and b → (0, 1). Because of (11), any f ∈ W (Γ ) satisfies f.a = a or f.a = a 2, and f.b is either b, ab, or a 2 b. In other words, the coordinate i matrix of f is ±1 0 1 for some i ∈ Z/3. But f is in SL(2, 3), so the first entry must be 1. Hence W (Γ ) is cyclic of order 3. Take x ∈ Oa and y ∈ Ob with n(x, x ∗ x) = 1

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  and n(y, y ∗ y) = α. As n x ∗ y, (x ∗ y) ∗ (x ∗ y) = −α (see (9)), Lemma 2 shows the existence of an automorphism ϕ of O with ϕ(x) = x and ϕ(y) = −x ∗ y. Using repeatedly (2), we get   ϕ 3 (y) = − x ∗ x ∗ (x ∗ y) = (x ∗ y) ∗ (x ∗ x) = n(x, x ∗ x)y = y, so we obtain ϕ 3 = id. This shows that (10) splits in this case, as ϕ ∈ Aut(Γ ). Finally, If the image of ΦΓ is isomorphic to (Z/3)2 , ΦΓ is one-to-one, and W (Γ ) is trivial because of (11). Let a, b be generators of H with ΦΓ (a) = [α] and ΦΓ (b) = [β], and take x ∈ Oa with n(x, x ∗ x) = α and y ∈ Ob with n(y,y ∗ y) = β. Lemma 2 shows that if Oa ∗ Ob = 0 (i.e., x ∗ y = 0), then (O, Γ ) eq Oα,β , Γα,β by means of an isomorphism that takes x to z˜ 1,0 and y to z˜ 0,1 , while if Oa ∗ Ob = 0,  then Ob ∗ Oa = 0 and then (O, Γ ) eq Oβ,α , Γβ,α . If F is a field with char F = 3, ω will denote a primitive cubic root of 1 in an algebraic closure of F. As a consequence of Theorem 7, Aut(Γ ) is W (Γ ) if char F = 3 or char F = 3 and ω∈ / F. Otherwise Aut(Γ ) is the semidirect product (Z/3)2  W (Γ ), where W (Γ ) is identified with a subgroup of Aut(Γ ) as shown in the proof above. In particular, for the canonical grading Γ1,1 on the split Okubo algebra O1,1 over a field F with char F = 3 and ω ∈ F, we obtain: Aut(Γ1,1 ) = (Z/3)2  SL(2, 3),

(12)

where the action of SL(2, 3) on (Z/3)2 is the natural one. Moreover, this is independent of the field F. The smallest such field is the field of four elements F4 = {0, 1, ω, ω2 = 1 + ω}. The (unique up to isomorphism  by Theorem 5) Okubo algebra O = O1,1 over F4 is the algebra sl(3, F4 ), ∗, n as in Theorem 1, whose group of automorphisms is PGL(3, 4) (the group of automorphisms of Mat 3 (F4 )). In other words, the automorphisms of O are obtained by conjugation by elements in GL(3, 4). Let τ denote the nontrivial automorphism of F4 : τ (α) = α 2 for all α ∈ F4 . Its fixed subfield is F2 = {0, 1}. Let V be the three-dimensional vector space F34 over F4 , and basis is endow V with the hermitian form h : V × V → F4 where the canonical  orthonormal: h(u i , u j ) = δi j . (Note that we have h(v, u) = τ h(v, u) for all u, v ∈ V ). The corresponding group of isometries is the unitary group U(3, 22 ), which is the subgroup of GL(3, 4) consisting of those matrices a with a † a = 1, where † denotes the second kind involution on Mat 3 (F4 ) determined by h: h(a.u, v) = h(u, a † .v) for all a ∈ Mat 3 (F4 ) and u, v ∈ V . Pauli matrices x, y in (4) are unitary (x † x = 1 = y † y) and their minimal polynomial is X 3 − 1. It follows that the matrices z i j as in (6) are all unitary too and, for (i, j) = (0, 0), their minimal polynomial is again X 3 − 1. (Note that in F4 , ω − ω2 = 1, so z i, j = ω−i j x i y j ).

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The set of (nonzero) homogeneous elements of Γ1,1 is H=

 (0,0)=(i, j)∈(Z/3)

F× 4 z i, j . 2

Lemma 4 The orbit of x under the action by conjugation of U(3, 22 ) is precisely H. Proof Since the z i, j ’s above are unitary and with the same minimal polynomial than x, it is clear that H is contained in the orbit of x. The stabilizer of x consists of the diagonal matrices in U(3, 22 ) (27 elements), while the size of U(3, 22 ) is 36 × 6 × 3 = 648. Hence the size of the orbit of x is 648/27 = 24, which coincides with the size of H.  Theorem 8 Let = O1,1 be the split Okubo algebra over a field F of characteristic not 3 and containing a primitive cubic root of 1. Then Aut(Γ1,1 ) is isomorphic to the projective unitary group PU(3, 22 ). Proof By Theorem 7, Aut(Γ1,1 ) is the semidirect product in (12), and this is independent of F (as long as char F = 3 and ω ∈ F). In particular, as the size of SL(2, 3) = 24, Aut(Γ1,1 ) has 9 × 24 = 216 elements. is 8×6 2 Lemma 4 shows that PU(3, 22 ) permutes the homogeneous components of the grading Γ1,1 over F4 , and hence PU(3, 22 ) is contained in Aut(Γ1,1 ). But PU(3, 22 ) has 648/3 = 216 elements, and the result follows.  Theorem 8 gives a surprising proof of the next result. Corollary 5 The projective unitary group PU(3, 22 ) is, up to isomorphism, the semidirect product (Z/3)2  SL(2, 3), and the projective special unitary group PSU(3, 22 ) is the semidirect product of (Z/3)2 and the quaternion group Q 8 (the Sylow 2-subgroup of SL(2, 3)). Proof The first part follows at once from Theorems 8 and 7. The classical group PSU(3, 22 ) is a normal subgroup of index 3 in PU(3, 22 ), and hence it contains its derived subgroup. It follows that, under the isomorphism PU(3, 22 ) ∼ = (Z/3)2  SL(2, 3), PSU(3, 22 ) is the semidirect product (Z/3)2  Q, for an order 8 normal subgroup of SL(2, 3). Then  1  the unique (normal)  Q is necessarily and 11 −1 are order 4 elements Sylow 2-subgroup of SL(2, 3). The matrices 01 −1 0 of SL(2, 3) that anticommute. Hence Q is generated by these two elements and it is  isomorphic to the quaternion group Q 8 . The result follows. According to Corollary 1, the group PU(3, 22 ) is (isomorphic to) the group of automorphisms of the Okubo algebra O over the field of two elements. The group Aut(O) was shown in [17, Propositions 7.10 and 7.13] to be naturally isomorphic to the group of automorphisms of the Okubo quasigroup OK(2) and its structure was determined, using GAP, to be an extension of SL(2, 3) by (Z/3)2 . By Corollary 1, Aut(O) is, up to isomorphism PU(3, 22 ) and Corollary 5 explains its structure.

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References 1. Chernousov, V., Elduque, A., Knus, M.-A., Tignol, J.-P.: Algebraic groups of type D4 , triality, and composition algebras. Doc. Math. 18, 413–468 (2013) 2. Elduque, A.: Symmetric composition algebras. J. Algebra 196(1), 282–300 (1997) 3. Elduque, A.: Okubo Algebras and Twisted Polynomials, Recent Progress in Algebra (Taejon/Seoul, 1997), pp. 101–109, Contemp. Math. 224, Amer. Math. Soc., Providence, RI (1999) 4. Elduque, A.: Gradings on symmetric composition algebras. J. Algebra 322(10), 3542–3579 (2009) 5. Elduque, A.: Order 3 elements in G 2 and idempotents in symmetric composition algebras. Canad. J. Math. 70(5), 1038–1075 (2018) 6. Elduque, A.: Composition algebras. In: Abdenacer, M. (ed.) Algebra and Applications I: Non-associative Algebras and Categories, Chapter 2, pp. 27–57. Sciences-Mathematics, ISTEWiley, London (2021) 7. Elduque, A., Kochetov, M.: Gradings on Simple Lie Algebras, Mathematical Surveys and Monographs, vol. 189. American Mathematical Society, Providence, RI (2013) 8. Elduque, A., Myung, H.C.: Flexible composition algebras and Okubo algebras. Comm. Algebra 19(4), 1197–1227 (1991) 9. Elduque, A., Myung, H.C.: On flexible composition algebras. Comm. Algebra 21(7), 2481– 2505 (1993) 10. Elduque, A., Pérez-Izquierdo, J.M.: Infinite-dimensional quadratic forms admitting composition. Proc. Amer. Math. Soc. 125(8), 2207–2216 (1997) 11. Grove, L.C.: Classical Groups and Geometric Algebra, Graduate Studies in Mathematics 39. American Mathematical Society, Providence, Rhode Island (2002) 12. Hesselink, W.H.: Special and pure gradings of Lie algebras. Math. Z. 179(1), 135–149 (1982) 13. Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The Book of Involutions, American Mathematical Society Colloquium Publications 44. American Mathematical Society, Providence, RI (1998) 14. Okubo, S.: Pseudo-quaternion and pseudo-octonion algebras. Hadronic J. 1(4), 1250–1278 (1978) 15. Okubo, S., Osborn, J.M.: Algebras with nondegenerate associative symmetric bilinear forms permitting composition II. Comm. Algebra 9(20), 2015–2073 (1981) 16. Pierce, R.S.: Associative Algebras, Graduate Texts in Mathematics 88. Springer-Verlag, New York (1982) 17. Smith, J.D.H., Vojtˇechovský, P.: Okubo quasigroups. Doc. Math. 27, 535–580 (2022) 18. Springer, T.A., Veldkamp, F.D.: Octonions, Jordan Algebras and Exceptional Groups. Springer Monographs in Mathematics, Springer-Verlag, Berlin (2000) 19. Zhevlakov, K.A., Slin’ko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are nearly associative. Academic Press, New York (1982)

Author Index

A Ayupov, S., 4

Kitouni, A., 262 Kudaybergenov, K., 4

B Benito, P., 17 Blasco, V., 149

L Ladra, M., 97 Laliena, J., 163

C Cabrera, Y., 213 Casas, J., 97, 113

D Daza-García, A., 149 Draper, C., 29 Drensky, V., 231

E Elduque, A., 287

G Gil, C., 213

M Mancini, M., 127 Martín, C., 213 Martín, D., 213 Martínez, C., 171 Mboya, S., 262 Mello, T., 185

O Ongong’a, E., 262

P Pacheco Rego, N., 113 Palomo, F., 29 Petrogradsky, V., 81

H Hernández, J., 245

K Kamiya, N., 65

R Rúa, I., 245 Rodrigues, D., 185 Roldán-López, J., 17

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Albuquerque et al. (eds.), Non-Associative Algebras and Related Topics, Springer Proceedings in Mathematics & Statistics 427, https://doi.org/10.1007/978-3-031-32707-0

303

304 S Silvestrov, S., 262 T Turdibaev, R., 137

Subject Index Z Zelmanov, E., 171