Classical Hopf Algebras and Their Applications (Algebra and Applications, 29) [1st ed. 2021] 3030778444, 9783030778446

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Table of contents :
Preface
Conventions
Symbols
Contents
1 Introduction
1.1 Linearization
1.2 Coalgebras
1.3 Gebras
1.4 Natural Endomorphisms
1.5 Applications
1.6 Structure of the Book
Part I General Theory
2 Coalgebras, Duality
2.1 Preliminaries on Vector Spaces and Algebras
2.2 Coalgebras: Definition and First Properties
2.3 Primitive and Group-Like Elements
2.4 Tensors
2.5 Endomorphisms
2.6 The Structure of Coalgebras
2.7 Representative Functions
2.8 Comodules
2.9 Representations and Comodules
2.10 Algebra Endomorphisms and Pseudo-coproducts
2.11 Coalgebra Endomorphisms and Quasi-coproducts
2.12 Duals of Algebras and Convolution
2.13 Graded and Conilpotent Coalgebras
2.14 Bibliographical Indications
References
3 Hopf Algebras and Groups
3.1 Bialgebras, Hopf Algebras
3.2 Modules and Comodules
3.3 Characteristic Endomorphisms and the Dynkin Operator
3.4 Hopf Algebras and Groups
3.5 Algebraic Groups
3.6 Unipotent and Pro-unipotent Groups
3.7 Enveloping Algebras, Groups, Tangent Spaces
3.8 Filtered and Complete Hopf Algebras
3.9 Signed Hopf Algebras
3.10 Module Algebras and Coalgebras
3.11 Bibliographical Indications
References
4 Structure Theorems
4.1 Dilations, Unipotent Bialgebras, and Weight Decompositions
4.2 Enveloping Algebras
4.3 Cocommutative Unipotent Hopf Algebras
4.4 Commutative Unipotent Hopf Algebras
4.5 Cocommutative Hopf Algebras
4.6 Complete Cocommutative Hopf Algebras
4.7 Remarks and Complements
4.8 Bibliographical Indications
References
5 Graded Hopf Algebras and the Descent Gebra
5.1 Descent Gebras of Graded Bialgebras
5.2 Lie Idempotents
5.3 Logarithmic Derivatives
5.4 The Descent Gebra
5.5 Combinatorial Descents
5.6 Bibliographical Indications
References
6 Pre-lie Algebras
6.1 The Basic Concept
6.2 Symmetric Brace Algebras
6.3 Free Pre-Lie Algebras and Gebras of Trees
6.4 Left-Linear Groups and Faà di Bruno
6.5 Exponentials and Logarithms
6.6 The Agrachev–Gamkrelidze Group Law
6.7 Other Examples
6.8 Brace Algebras
6.9 Right-Handed Tensor Hopf Algebras
6.10 Commutative Shuffles and Quasi-shuffles
6.11 Bibliographical Indications
References
Part II Applications
7 Group Theory
7.1 Compact Lie Groups are Algebraic
7.2 Algebraic Envelopes
7.3 Free Groups and Free Lie Algebras
7.4 Tannaka Duality
7.5 Bibliographical Indications
References
8 Algebraic Topology
8.1 Homology of Groups and H-Spaces
8.2 Hopf Algebras with Divided Powers
8.3 Eilenberg–MacLane Spaces and the Bar Construction
8.4 The Steenrod Hopf Algebra and Its Dual
8.5 Bibliographical Indications
References
9 Combinatorial Hopf Algebras, Twisted Structures, and Species
9.1 Vector Species and S–Modules
9.2 Hopf Species
9.3 The Hopf Species of Decorated Forests
9.4 Twisted Hopf Algebras
9.5 The Tensor Gebra as a Twisted Hopf Algebra
9.6 From Twisted to Classical Hopf Algebras
9.7 The Gebra of Permutations
9.8 The Structure of Twisted Hopf Algebras
9.9 Bibliographical Indications
References
10 Renormalization
10.1 Wick Products
10.2 Diagrammatics
10.3 The Hopf Algebra of Feynman Graphs
10.4 Exponential Renormalization
10.5 Bibliographical Indications
References
Appendix A Categories
A.1 The Language of Categories
A.2 Equalizers, Generators and Products
A.3 Monoidal Categories
A.4 Abelian Categories
Appendix B Operads
B.1 Operads
B.2 Polynomial Functors
B.3 Symmetric Operads
B.4 Algebras over Operads
Index
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Algebra and Applications

Pierre Cartier Frédéric Patras

Classical Hopf Algebras and Their Applications

Algebra and Applications Volume 29

Series Editors Michel Broué, Université Paris Diderot, Paris, France Alice Fialowski, Eötvös Loránd University, Budapest, Hungary Eric Friedlander, University of Southern California, Los Angeles, CA, USA Iain Gordon, University of Edinburgh, Edinburgh, UK John Greenlees, Warwick Mathematics Institute, University of Warwick, Coventry, UK Gerhard Hiß, Aachen University, Aachen, Germany Ieke Moerdijk, Utrecht University, Nijmegen, Utrecht, The Netherlands Christoph Schweigert, Hamburg University, Hamburg, Germany Mina Teicher, Bar-Ilan University, Ramat-Gan, Israel

Algebra and Applications aims to publish well-written and carefully refereed monographs with up-to-date expositions of research in all fields of algebra, including its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, and further applications in related domains, such as number theory, homotopy and (co)homology theory through to discrete mathematics and mathematical physics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications within mathematics and beyond. Books dedicated to computational aspects of these topics will also be welcome. Announcement (30 November 2020) Alain Verschoren (1954-2020), Professor of Mathematics and Honorary Rector of the University of Antwerp, became an editor of the Algebra and Applications series in 2000. His contribution to the development of the series over two decades was pivotal. We, the Springer mathematics editorial staff and the editors of the series, mourn his passing and bear him in fond and grateful remembrance.

More information about this series at http://www.springer.com/series/6253

Pierre Cartier Frédéric Patras •

Classical Hopf Algebras and Their Applications

123

Pierre Cartier Institut des Hautes Études Scientifiques Bures-sur-Yvette, France

Frédéric Patras Laboratoire J.A.Dieudonné Université Côte d’Azur Nice, France

ISSN 1572-5553 ISSN 2192-2950 (electronic) Algebra and Applications ISBN 978-3-030-77844-6 ISBN 978-3-030-77845-3 (eBook) https://doi.org/10.1007/978-3-030-77845-3 Mathematics Subject Classification: 16T05, 16S30, 16T15, 16T30, 16W10 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved 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

Preface

The present volume is dedicated to classical Hopf algebras and their applications. By classical Hopf algebras, we mean Hopf algebras as they first appeared in the works of Borel, Cartier, Hopf, and others in the 1940s and 50s: commutative or cocommutative Hopf algebras. The purpose of the book is twofold. It first of all offers a modern and systematic treatment of the structure theory of Hopf algebras, using the approach of natural operations. According to it, the best way to understand the structure of Hopf algebras is by means of their endomorphisms and their combinatorics. We therefore put weight on notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras, or Lie idempotents, to quote a few. We have included in this treatment the case of enveloping algebras of pre-Lie algebras, extremely important in the recent literature, many interesting Lie algebras actually being the Lie algebras obtained by antisymmetrization of a pre-Lie product. Second, the book surveys important application fields, explaining how Hopf algebras arise there, what problems they allow to address, and presenting the corresponding fundamental results. Each application field would require a textbook on its own, we have therefore limited our exposition to introducing the main ideas and accounting for the most fundamental results on which the use of Hopf algebras in the field are grounded. The book should thus be useful as a general introduction and reference on classical Hopf algebras, their structure and endomorphisms; as a textbook for Master 2 or doctoral-level programs; and mostly and ultimately to scholars in algebra and the main application fields of Hopf algebras. As for the book itself, it is the result of a long-lasting project. It originates ultimately in 1989, when one of the authors initiated a Ph.D. under the direction of the other. One of the ideas that emerged then was that the combinatorics of dilations underlying the theory of finite integration as appearing in Hilbert’s third problem (on equidecomposability of polytopes under pasting and gluing operations) had far-reaching applications and generalizations. It extends, for example, to properties of the direct sum of the symmetric group algebras or the study of power maps on H-spaces. This lead to a purely combinatorial proof of structure theorems

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Preface

for graded connected cocommutative Hopf algebras around which the content of the central chapters of the first part of the book is organized. At the time, the interest of the mathematical community for classical Hopf algebras was limited. A certain number of classical tools and structure results were available and were for the most part enough for the needs of applications, for example, in rational homotopy—the subdomain of algebraic topology where torsion phenomena are ignored. The situation evolved progressively, leading to the writing of the present book that brings together classical results, some of which go back to the 1950s, and recent advances under the unifying point of view of combinatorial structure results and techniques. Many developments have contributed to the renewal of interest for classical Hopf algebras. In algebraic combinatorics, the works of Ch. Reutenauer, J.-Y. Thibon, and others generated again interest for the combinatorial theory of free Lie algebras, Lie idempotents, (noncommutative) representation theory of symmetric groups, and related objects. From the mid-1990s onward, the theme of combinatorial Hopf algebras, whose first idea can be traced back to Rota, gained momentum and grew steadily up to becoming one of the leading arguments of contemporary algebraic combinatorics. Another line of development has several independent origins: deformation theory, differential calculus and differential geometry, numerical analysis and control, theoretical physics... It relates to the notion of pre-Lie algebras and to Hopf algebras of trees, forests, and diagrams. The notion of pre-Lie algebra dates back from the early 1960s (Gerstenhaber, Vinberg) and can even be found earlier in the work of Lazard. From the group and Lie theoretic point of view, which is also one of the Hopf algebras, a key step in the development of the theory of pre-Lie algebras is due to Agrachev and Gamkrelidze in the beginning of the 1980s. In hindsight, their work started to develop the extension to pre-Lie algebras of the combinatorial theory of Lie algebras and their enveloping algebras. However, the systematic development of the theory is much more recent. The work of Connes and Kreimer on Hopf algebras in perturbative quantum field theory around 2000 played here a particularly important role. They featured the role of pre-Lie algebras of trees and Feynman diagrams and their enveloping algebras in renormalization. Brouder rapidly connected their insights with methods and results in numerical analysis. Pre-Lie algebras and their enveloping (Hopf) algebras came to the forefront of researches on Hopf algebras and their applications. The recent surge of Hopf algebra techniques in stochastics (with rough paths, regularity structures) connects to this line of development. In algebraic topology, homological algebra and related areas, where the very notion of Hopf algebra was born, the use of Hopf algebra techniques was classical since the 1940s. Besides in the study of topological groups, they appear, for example, in the study of loop spaces, algebras of operations such as Steenrod’s or homology of Eilenberg–MacLane spaces. Pre-Lie algebras first appear in this context with the work of Gerstenhaber. They relate to the more general idea of brace operations that was introduced in the mid-1990s by Getzler, Gerstenhaber, and Voronov in the context of cochain complexes and the theory of operads. Here,

Preface

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the Hopf algebras at play have a particular structure: they are free or cofree as (co) associative (co)algebras (free or cofree (co)commutative when arising from pre-Lie algebras). This idea of Hopf algebras with extra structures proved also important, as those structures carry with themselves the existence of additional properties and operations. Algebraic combinatorics and combinatorial Hopf algebras; algebraic topology, homological algebra, and operadic structures; pre-Lie algebras together with their many applications: we can give only very fragmentary indications about the many developments that occured during the last 30 years and have deeply reshaped the subject of classical Hopf algebras. We mention specifically these three lines of thought since they motivated various choices made in the writing of this book. We also wanted to point out with these examples the high level of activity surrounding the subject of Hopf algebras, which appears over and over as a central topic in contemporary mathematics. Overall, the subject is too vast to be covered by a single textbook, we therefore had to make choices. The book is structured into two parts: general theory and applications. In the first part, we give a systematic account of the structure theory of commutative or cocommutative Hopf algebras with emphasis put on enveloping algebras of graded or complete Lie algebras and the dual polynomial Hopf algebras, mostly over a field of characteristic 0. The second part is dedicated to several key applications of the theory, classical, and recent. These application chapters can be read separately, but we advise the reader seriously interested in using Hopf algebras to read all of them as they offer complementary insights. Many techniques and intuitions can actually be carried over from an application field to another. It is impossible to acknowledge here all those who contributed along the years by discussion, collaborations, and joint works to the building of the picture of Hopf algebras and their applications addressed hereafter. Frédéric Patras would like to thank especially Kurusch Ebrahimi-Fard together with those others with whom he developed long-lasting research projects on the topics addressed in this book; many of these projects have run over the last 20 years and are still ongoing: Christian Brouder, Patrick Cassam-Chenaï, Loïc Foissy, Joachim Kock, Claudia Malvenuto, Simon Malham, Dominique Manchon, Frédéric Menous, Christophe Reutenauer, Nikolas Tapia, Jean-Yves Thibon, Anke Wiese, and Lorenzo Zambotti. A special thought to a late friend, Manfred Schocker: we had started together a vast program on Hopf algebras in combinatorics that was interrupted by his premature death, the chapter dedicated to combinatorial Hopf algebras is a tribute to his memory. Limours, France Nice, France

Pierre Cartier Frédéric Patras

Conventions

All linear structures are defined over a ground field k, excepted otherwise specified. Categories are written with bold symbols, for example, Alg stands for the category of algebras with a unit over k. Algebras are algebras with unit, coalgebras are coalgebras with counit, excepted otherwise specified. Ideals are two-sided, that is, simultaneously left and right ideals, excepted otherwise specified. We tend to abbreviate notations. For example, we will often write A for an algebra, instead of the triple ðA; mA ; gA Þ, where mA and gA stand for the product and the unit.

Symbols When a symbol (e.g., Cogk ) is followed by “resp., . . .” (e.g., resp., Cog), this means that the first symbol is the complete symbol associated to a notion, whereas the following ones stand for abbreviations used to alleviate the notation when no confusion can arise. A þ : augmentation ideal of an augmented algebra A. Abe: category of abelian groups. Algk (resp., Alg): category of associative unital algebras. AutC ðXÞ (resp., AutðXÞ): automorphisms of X in the category C. Algck (resp., Algc ): category of complete augmented algebras. cð1Þ  . . .  cðnÞ : (abbreviated) Sweedler notation for Dn ðcÞ. C: complex numbers. CðA; BÞ: set of morphisms in the category C from A to B (also denoted HomC ðA; BÞ). CðMÞ: subcoalgebra of a coalgebra C associated to a C-comodule M. Cmd (resp., CmdC ): category of comodules over a coalgebra C. Cogk (resp., Cog): category of coassociative counital coalgebras.

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Comk (resp., Com): category of commutative unital algebras. dij : Kronecker delta function (dij ¼ 1 if i ¼ j and 0 else). dM (resp., d): coproduct for a comodule M. dV : identity of V viewed as an element of V  V  . dx : delta function (dx ðyÞ :¼ dyx ). DC , DH (resp., D): coproduct of the coalgebra C, the Hopf algebra H... Dd : deconcatenation coproduct. Du : unshuffle coproduct. Dl : l  1-fold iteration from C to C l of a coassociative coproduct D : C ! C  C.  reduced coproduct associated to a coproduct D. D: : dual (V  is the dual of V). gA (resp., g): unit map k ! A of an algebra, coaugmentation gC : k ! C of a coalgebra. EncðVÞ ¼ V  V  ¼ End _ ðV  Þ: vector space of linear endomorphisms of a finite-dimensional vector space V viewed as a coalgebra. End _ ðVÞ ¼ V   V: dual coalgebra of the algebra EndðVÞ ¼ V  V  of linear endomorphisms of a finite-dimensional vector space V. End C ðXÞ (resp., EndðXÞ): set of endomorphisms of X in C. eC , eA (resp., e): counit map C ! k of a coalgebra, augmentation eA : A ! k of an algebra. f jX : restriction of a map f to X, a subset of the domain of f . f jA : when a vector space decomposes as W ¼ A  B and f is a linear map from V to W, it denotes the composition of the projection from W to A along B with f . We call f jA the corestriction of f to A. Fin: category whose objects are the finite (possibly empty) subsets of N and whose morphisms are the bijections. CðBÞ: set (resp., group) of group-like elements of a coalgebra (resp., Hopf algebra) B. GLðn; kÞ: n-th general linear group over k. Grp: category of groups. HomC ðA; BÞ: set of morphisms from A to B in the category C. Hopk (resp., Hop): category of Hopf algebras. Hopck (resp., Hopc ): category of complete Hopf algebras. Id C (resp., Id): identity map of an object C in a given category. k: ground field. kG: group algebra of the group G. kG : k-valued functions on G. k½V: space of polynomials over a vector space V. Lx f : left translate of f , Lx f ðyÞ ¼ f ðxyÞ. Liek (resp., Lie): category of Lie algebras. Link (resp., Lin): category of vector spaces.

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Linck (resp., Linc ): category of complete filtered vector spaces. Linkf (resp., Linf ): category of filtered vector spaces. Lingk (resp., Ling ): category of graded vector spaces. mA , mH (resp., m): algebra product of the algebra A, the Hopf algebra H... ml : ðl  1Þ-fold iterated product of an algebra, from Al to A. M n : n-th tensor power M  . . .  M of M. M n ðkÞ: square matrices of size n  n over k. ModA : category of left modules over an algebra A. Mon: category of monoids. ½n :¼ f1; . . .; ng. N: nonnegative integers. N : positive integers. mH (resp., m): mH :¼ gH  eH , unit map of the convolution algebra End Lin ðHÞ of linear endomorphisms of a Hopf algebra. ½0 :¼ ;. ObðCÞ: class of objects of the category C. Oðn; kÞ: n  n orthogonal group. N Q ai : in an associative algebra, and denotes the ordered product a1 . . . aN . i¼1

PrimðCÞ: set of primitive elements of a coaugmented coalgebra C. Q: rational numbers. Ry f : right translate of f , Ry f ðxÞ ¼ f ðxyÞ. RðGÞ: representative functions on a monoid or a group. R: real numbers. : shuffle product. Set: category of sets. Sn : n-th symmetric group. Spe: category of vector species. T: switch map (Tðx  yÞ :¼ y  x), also written T C;D when mapping C  D to D  C. TðVÞ: space of tensors a V n over a vector space V. n2N

TSðVÞ: space of symmetric tensors over a vector space V. Z: integers.

Contents

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Coalgebras, Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries on Vector Spaces and Algebras . . . 2.2 Coalgebras: Definition and First Properties . . . . . 2.3 Primitive and Group-Like Elements . . . . . . . . . . 2.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Structure of Coalgebras . . . . . . . . . . . . . . . 2.7 Representative Functions . . . . . . . . . . . . . . . . . . 2.8 Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Representations and Comodules . . . . . . . . . . . . . 2.10 Algebra Endomorphisms and Pseudo-coproducts 2.11 Coalgebra Endomorphisms and Quasi-coproducts 2.12 Duals of Algebras and Convolution . . . . . . . . . . 2.13 Graded and Conilpotent Coalgebras . . . . . . . . . . 2.14 Bibliographical Indications . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hopf Algebras and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Bialgebras, Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modules and Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Introduction . . . . . . . . . . . . . 1.1 Linearization . . . . . . . . . 1.2 Coalgebras . . . . . . . . . . 1.3 Gebras . . . . . . . . . . . . . 1.4 Natural Endomorphisms 1.5 Applications . . . . . . . . . 1.6 Structure of the Book . .

Part I

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General Theory

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3.3 Characteristic Endomorphisms and the Dynkin Operator 3.4 Hopf Algebras and Groups . . . . . . . . . . . . . . . . . . . . . 3.5 Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Unipotent and Pro-unipotent Groups . . . . . . . . . . . . . . 3.7 Enveloping Algebras, Groups, Tangent Spaces . . . . . . . 3.8 Filtered and Complete Hopf Algebras . . . . . . . . . . . . . 3.9 Signed Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Module Algebras and Coalgebras . . . . . . . . . . . . . . . . . 3.11 Bibliographical Indications . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Structure Theorems . . . . . . . . . . . . . . . . . . . . . . 4.1 Dilations, Unipotent Bialgebras, and Weight Decompositions . . . . . . . . . . . . . . . . . . . . . 4.2 Enveloping Algebras . . . . . . . . . . . . . . . . . . 4.3 Cocommutative Unipotent Hopf Algebras . . 4.4 Commutative Unipotent Hopf Algebras . . . . 4.5 Cocommutative Hopf Algebras . . . . . . . . . . 4.6 Complete Cocommutative Hopf Algebras . . . 4.7 Remarks and Complements . . . . . . . . . . . . . 4.8 Bibliographical Indications . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Graded Hopf Algebras and the Descent Gebra . 5.1 Descent Gebras of Graded Bialgebras . . . . 5.2 Lie Idempotents . . . . . . . . . . . . . . . . . . . . 5.3 Logarithmic Derivatives . . . . . . . . . . . . . . 5.4 The Descent Gebra . . . . . . . . . . . . . . . . . . 5.5 Combinatorial Descents . . . . . . . . . . . . . . . 5.6 Bibliographical Indications . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Pre-lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Basic Concept . . . . . . . . . . . . . . . . . . 6.2 Symmetric Brace Algebras . . . . . . . . . . . . 6.3 Free Pre-Lie Algebras and Gebras of Trees 6.4 Left-Linear Groups and Faà di Bruno . . . . 6.5 Exponentials and Logarithms . . . . . . . . . . . 6.6 The Agrachev–Gamkrelidze Group Law . . . 6.7 Other Examples . . . . . . . . . . . . . . . . . . . . 6.8 Brace Algebras . . . . . . . . . . . . . . . . . . . . . 6.9 Right-Handed Tensor Hopf Algebras . . . . . 6.10 Commutative Shuffles and Quasi-shuffles . . 6.11 Bibliographical Indications . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 113 117 124 129 131 134 135 142 144 148 151 153

4

Contents

Part II

xv

Applications

7

Group Theory . . . . . . . . . . . . . . . . . . . . 7.1 Compact Lie Groups are Algebraic 7.2 Algebraic Envelopes . . . . . . . . . . . 7.3 Free Groups and Free Lie Algebras 7.4 Tannaka Duality . . . . . . . . . . . . . . 7.5 Bibliographical Indications . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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157 157 163 165 167 174 175

8

Algebraic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Homology of Groups and H-Spaces . . . . . . . . . . . . . . 8.2 Hopf Algebras with Divided Powers . . . . . . . . . . . . . 8.3 Eilenberg–MacLane Spaces and the Bar Construction . 8.4 The Steenrod Hopf Algebra and Its Dual . . . . . . . . . . 8.5 Bibliographical Indications . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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177 177 182 186 189 195 197

9

Combinatorial Hopf Algebras, Twisted Structures, and Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Vector Species and S–Modules . . . . . . . . . . . . 9.2 Hopf Species . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Hopf Species of Decorated Forests . . . . . . 9.4 Twisted Hopf Algebras . . . . . . . . . . . . . . . . . . 9.5 The Tensor Gebra as a Twisted Hopf Algebra . 9.6 From Twisted to Classical Hopf Algebras . . . . 9.7 The Gebra of Permutations . . . . . . . . . . . . . . . 9.8 The Structure of Twisted Hopf Algebras . . . . . 9.9 Bibliographical Indications . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Renormalization . . . . . . . . . . . . . . . . . . . . . 10.1 Wick Products . . . . . . . . . . . . . . . . . . 10.2 Diagrammatics . . . . . . . . . . . . . . . . . . 10.3 The Hopf Algebra of Feynman Graphs 10.4 Exponential Renormalization . . . . . . . . 10.5 Bibliographical Indications . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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223 224 228 234 237 244 245

Appendix A: Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Appendix B: Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Chapter 1

Introduction

The main purpose of this volume is to give a modern, up-to-date, presentation of the theory of Hopf algebras and their applications, classical and recent. The Hopf algebras we consider are the classical ones: the ones that appeared in the 40s and 50s in algebraic topology, the theory of algebraic groups and representation theory. That is, they will be most often commutative or cocommutative and associated to a group of characters or group-like elements. Concretely, we will consider typically enveloping algebras of graded or complete Lie algebras and their dual Hopf algebras, Hopf algebras of representative functions, of trees and graphs, and similar ones. The account will include certain Hopf algebras carrying extra algebraic structures, such as, for example, enveloping algebras of pre-Lie algebras. Applications developed will include duality phenomena in group theory; classical Hopf algebra structures in algebraic topology; combinatorial Hopf algebras; and Hopf algebraic renormalization. Let us start with some elementary and very general principles governing the theory of Hopf algebras (and variants thereof that will be studied in this book). We will illustrate them on the simple example of finite groups, although it will appear later on that their range of application is much larger. We refrain from making at this point all definitions rigorous, this introduction aiming at indicating the behavior of some objects and structures that will be defined thoroughly later.

1.1 Linearization One of the key ideas underlying the theory of Hopf algebras, closely related to the standard properties of the exponential and the logarithm, is linearization : that is, making (non-linear) group-theoretical problems into linear ones. The correspondence between groups and Lie algebras that can often be understood through a common © Springer Nature Switzerland AG 2021 P. Cartier and F. Patras, Classical Hopf Algebras and Their Applications, Algebra and Applications 29, https://doi.org/10.1007/978-3-030-77845-3_1

1

2

1 Introduction

embedding of both objects into a completion of the enveloping algebra of the Lie algebra is one of its most striking and successful illustrations. When Hopf algebras have extra structures, interesting new group-theoretical phenomena arise. A nice example is provided by graded pre-Lie algebras and their enveloping algebras: two exponentials relating the pre-Lie algebra to the associated group can be defined in this context, and the study of their interactions leads to deep formulas and properties. This leads, for example, to a refinement, in this setting, of the usual combinatorial and free Lie algebra analysis of the Baker–Campbell–Hausdorff problem (the computation of the logarithm of a product of exponentials). In practice, it is often the case that algebras of functions provide the simplest way to linearize an object, and this is actually what happens in most correspondences existing between groups and Hopf algebras. For example, consider a finite group G. The vector space k G := Set(G, k) is an algebra for the pointwise product m : kG ⊗ kG → kG m(λ ⊗ κ)(g) = (λ · κ)(g) := λ(g)κ(g) for λ, κ ∈ k G and g ∈ G, and is equipped with a linear map (the coproduct)  : kG → kG ⊗ kG ∼ = k G×G , (λ)(g, h) := λ(gh), so that (λ) =



λ(x y)δx ⊗ δ y ,

(1.1)

x,y∈G

where δx stands for the delta function on G, δx (g) = 1 if g = x and 0 else. Whereas this simple dualization process is quite satisfactory for finite groups, properly defining the linear object dual to more general group structures (infinite groups, continuous groups, algebraic groups, group structures in categories, renormalization groups...) is, in general, less straightforward and requires some care.

1.2 Coalgebras A second key idea is that coalgebras, preferably to algebras, are often the natural framework to encode linearly group-type properties. This is already clear in the example we have just considered: on the algebra of functions k G , the product does not carry any interesting information (any set of functions with values in an algebra is an algebra for the pointwise product). The coproduct instead encodes the product rule on G. There are many technical reasons underlying this broad statement, and some of them, related to duality phenomena, will become clear later in this book.

1.2 Coalgebras

3

A wit, due to Serre, accounts for a related feature of coalgebras: “there is a general principle: every calculation relative to coalgebras is trivial and incomprehensible” (Il y a un principe général : tout calcul relatif aux cogèbres est trivial et incompréhensible).

1.3 Gebras It often occurs that a given Hopf algebra structure goes along with other algebraic structures or, more generally, that a given vector space can be equipped with several interacting algebraic structures. To mention a few, associative, commutative, Lie, pre-Lie, shuffle, quasi-shuffle algebra structures, and the dual coalgebraic notions can coexist on a given space. This phenomenon can be found in many parts of the recent literature on Hopf algebras, some of which are accounted for in this book. The space of tensors T (V ) over a given vector space V will be one of the central objects in this volume and is a good illustration of this phenomenon: it carries several products, coproducts, is the free algebra over V for several algebraic structures, can be viewed as a twisted Hopf algebra, and so on. Similar comments apply to other fundamental objects such as the descent algebra (also known as the algebra of noncommutative symmetric functions), the direct sum of the symmetric group algebras or the vector spaces generated by trees, forests, and other classical families of combinatorial objects. It would be tedious to devise specific names to describe all the possible combinations of structures existing on such objects. On the other hand, referring, for example, to T (V ) as “the tensor algebra” is misleading in that it favors implicitly its free associative algebra structure, not always the most interesting one. It is also actually very convenient to view many vector spaces such as T (V ) as equipped with a family of structures that can be extended progressively. We propose to use the word gebra that was introduced by Serre as unifying the worlds of algebras and coalgebras, in such situations. For example, the tensor gebra (over V ) will refer to T (V ) equipped simultaneously with the various structures alluded to above and other ones that could be defined. This allows, among others, to make unambiguous statements involving several structures. Say, for example, we will see that the tensor gebra can be given a free associative, but also a free commutative and a free commutative shuffle algebra structure.

4

1 Introduction

1.4 Natural Endomorphisms A fourth key idea is, technically, less classical. It will be a central theme in this book. It can be understood as a form of Galois theory: understand the structure of a mathematical object through its automorphisms or endomorphisms. In many situations, groups and gebras arise naturally from endomorphisms of objects, of classes of objects, of endofunctors, or, more generally, of functors from one category to another. The most classical example is provided by linear representations. There is a general philosophy that goes back to the early foundations of group theory, according to which a group is little more than, and essentially given by, the collection of its representations. Tannakian duality is one of its outsprings: it encodes the way in which groups can be constructed from linear categories “behaving as categories of representations,” from the knowledge of the forgetful functor from these categories to vector spaces. There are other, less known, illustrations of this general idea. Some are very interesting for the systematic study of Hopf algebras and generalizations thereof. The fundamental example here, which admits various variants, is the descent gebra of graded connected cocommutative Hopf algebras. Among other properties, it is the subalgebra of the convolution algebra of (natural) linear endomorphisms of this family of Hopf algebras generated by projections on the graded components. It is stable by the composition product of endomorphisms, and a Hopf algebra. Its properties allow to recover Cartier’s structure theorem and many essential results of the theory of free Lie algebras.

1.5 Applications Last but not least, Hopf algebras are useful. They are very present in many application fields, with a strong expansion during the last two decades. Thinking to applications allows to enrich the classical set of tools and results and provides new insights on the theory. Choosing among all application fields those that would be developed here was a difficult task. We considered including chapters on several topics we had independently studied from the Hopf algebraic point of view: algebraic groups over the integers; quantum groups and their applications in Galois theory; symmetric functions and their generalizations; numerical analysis and geometric integration; stochastic integration, rough paths, and regularity structures; and classical and free probabilities. But choices had to be done; the other topics we have chosen to develop here should however give a good idea of the range of possible applications of the theory.

1.5 Applications

5

We should point out that, contrary to the first part of the volume, where the treatment is systematic, the four applications chapters on algebraic groups, algebraic topology, combinatorial Hopf algebras, and renormalization are meant as introductions to the Hopf algebraic point of view, as each topic would deserve a volume on its own.

1.6 Structure of the Book Part I of the book addresses the general theory. Chapter 2 deals with coalgebras, comodules, duality, representative functions, and structural properties of endomorphisms (notions of pseudo- and quasi-coproducts). Chapter 3 introduces fundamental Hopf algebra notions and examples and some direct generalizations thereof, modules and comodules, algebraic and pro-unipotent groups, enveloping algebras, and the Dynkin operator. Chapter 4 is about structure theorems for commutative and cocommutative Hopf algebras, following the approach by properties of their endomorphisms. Chapter 5 studies the properties of endomorphisms in the graded case. It introduces the descent gebra and some of its main applications: Lie idempotents, logarithmic derivatives. Chapter 6 introduces pre-Lie algebras and the equivalent notion of symmetric brace algebras, together with fundamental examples. Special attention is devoted to the two exponential maps on the completion of their enveloping algebras and their relations, together with the associated group-theoretical notions. The chapter also includes a brief introduction to brace algebras, a non-symmetric version of symmetric brace algebras. The notion is illustrated with the example of quasi-shuffle Hopf algebras. Part II is devoted to applications. Chapter 7 illustrates the duality between groups and commutative Hopf algebras with the example of compact Lie groups. The last part explains the mechanisms underlying Tannaka duality—the construction of coalgebras and bialgebras out of suitable linear categories. Chapter 8 treats Hopf algebras in algebraic topology with structure theorems for the homology and cohomology of H -spaces, including the case of nonzero characteristic and the notion of divided powers. The bar construction is introduced and the computation of the homology of Eilenberg–MacLane spaces explained. The chapter ends with the study of the algebra of stable cohomology operations or Steenrod algebra. Chapter 9 considers combinatorial Hopf algebras in the context of species (functors from the category of finite sets and bijections between them). Two equivalent notions of Hopf algebras exist in this context: twisted Hopf algebras and Hopf species. Functors to usual Hopf algebras connect the twisted theory with the classical one. We investigate their combinatorial properties, give examples, and introduce the main constructions.

6

1 Introduction

Chapter 10 concludes the book with the Hopf algebraic approach to renormalization. Various aspects are treated: Wick products, the Hopf algebraic construction of Feynman diagrams, and Hopf algebras of Feynman diagrams. Renormalization proper is presented following the general group-theoretical approach of the exponential method. When amplitudes (concretely, functions on Feynmann diagrams) take values in a Rota–Baxter algebra, the Bogoliubov recursion applies. Its Hopf algebraic Birkhoff–Wiener–Hopf interpretation terminates the chapter. An appendix is dedicated to the language and elementary notions of the theories of categories and operads. Each chapter concludes with separate bibliographical indications. They are aimed at orienting the reader and suggest further or complementary readings on the topics covered in this volume. They do not claim for exhaustivity or completeness, neither contentwise nor historically. We usually point out at some of the works that originated the subject, the references we used, found most useful, and hint at works that complement directly the account given in the chapter.

Part I

General Theory

Chapter 2

Coalgebras, Duality

2.1 Preliminaries on Vector Spaces and Algebras Let V, W be two vector spaces, V ∗ and W ∗ their duals. For v∗ ∈ V ∗ , v ∈ V , we write < v |v∗ >=< v∗ |v > for v∗ (v ) ∈ k. The map from W ⊗ V ∗ to Lin(V, W ), where Lin stands for the category of vector spaces, (w ⊗ v∗ )(v ) := w· < v∗ |v > is an isomorphism when V is finite dimensional. In particular, when V = W , the map V ⊗ V ∗ → End(V ) is an isomorphism when V is finite dimensional, and to the identity map I dV of V corresponds then an element δV ∈ V ⊗ V ∗ . An associative unital algebra over k (or, for short, an algebra) is a triple (A, m, η) where A is a vector space, m : A ⊗ A → A (multiplication) and η : k → A (unit) are linear maps such that the following diagrams, expressing, respectively, the associativity and unit axioms, commute

A⊗ A⊗ A m ⊗ Id ? A⊗ A

I d ⊗m

m

A⊗ A

m? - A,

∼ = k⊗A  η ⊗ Id ? A⊗ A

A

∼ =A⊗k

Id ? Id ⊗ η ? - A  A ⊗ A. m m

One advantage of this diagrammatic definition of algebras is that it generalizes immediately to arbitrary categories equipped with a tensor product (tensor categories, see Appendix A for a definition). Recall also that a Lie algebra is a vector space L equipped with a bilinear product, usually written as a bracket [ , ], such that for any x, y, z ∈ L, • (Antisymmetry) [x, x] = 0 (hence [x, y] = −[y, x]), • (Jacobi identity) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. © Springer Nature Switzerland AG 2021 P. Cartier and F. Patras, Classical Hopf Algebras and Their Applications, Algebra and Applications 29, https://doi.org/10.1007/978-3-030-77845-3_2

9

10

2 Coalgebras, Duality

An associative algebra is always equipped with a canonical Lie bracket defined as the commutator of the product: [x, y] := x y − yx. This construction defines a functor from associative algebras to Lie algebras; we will study later its left adjoint, the enveloping algebra functor, one of the most natural ways to construct Hopf algebras.

2.2 Coalgebras: Definition and First Properties The notion of coalgebra is dual to the one of algebra. Categorically, this means that a coalgebra is simply an algebra in Linop , the opposite category of the category of vector spaces. Concretely, this means that coalgebras are defined diagrammatically by inverting all the arrows in the diagrammatic definition of algebras, see below. To make things precise, the dual notion of a bilinear product on a vector space A, that is, a linear map m : A ⊗ A → A is the one of a coproduct, on a vector space C: a linear map from C to C ⊗ C. It is often written C , or simply  when no confusion can arise  : C → C ⊗ C = C ⊗2 . We will very often use the abbreviated Sweedler notation (c) =: c(1) ⊗ c(2) . One should take care that this notation is really a shortcut for (c) since most often it is not true that an element x of C ⊗ C can be written x1 ⊗ x2 with x1 , x2 ∈ C: in general, it can be written only as a linear combination of such elementary tensor products, as for example (λ) = λ(x y)δx ⊗ δ y in eq. (1.1). It is the reason why, x,y  (1) instead of c(1) ⊗ c(2) , various authors use the original Sweedler notation, cα ⊗ α

cα(2) . However, this last notation is less economic and, as with Einstein’s summation conventions with repeated indices, the use of the abbreviated Sweedler notation does not lead to problems provided one reminds that upper indices (i) refer to the tensor expansion of a coproduct. The coproduct is coassociative if the following identity holds between maps from C to C ⊗3 ( ⊗ I d) ◦  = (I d ⊗ ) ◦ . (2.1) Diagrammatically, the coproduct  is coassociative if the following diagram commutes:

2.2 Coalgebras: Definition and First Properties



C ? C ⊗C

11

- C ⊗C

Id ⊗  ? - C ⊗ C ⊗ C.  ⊗ Id

We denote the map ( ⊗ I d) ◦  by 3 . When the coproduct is coassociative, there is a unique map from C to C ⊗n induced by iterated compositions of coproducts. It is written n and can be defined recursively by 2 := , n := ( ⊗ I d ⊗n−2 ) ◦ n−1 , the coassociativity hypothesis implying that for any i ≤ n − 2, n = (I d ⊗i ⊗  ⊗ I d ⊗n−i−2 ) ◦ n−1 . More generally, one gets by induction n 1 +···+n k = (n 1 ⊗ · · · ⊗ n k ) ◦ k for n 1 , . . . , n k > 0, with the notation 1 := I d. In the Sweedler notation, the identity (2.1) would read, for an arbitrary element c ∈ C, (1) (2) (1) (2) c(1) ⊗ c(1) ⊗ c(2) = c(1) ⊗ c(2) ⊗ c(2) . We will denote this element of C ⊗3 by c(1) ⊗ c(2) ⊗ c(3) and more generally will write n (c) := c(1) ⊗ · · · ⊗ c(n) . Although the notation is slightly ambiguous since c(1) ⊗ c(2) in the last equation could be confused with (c), it does not create ambiguities in practice provided some caution is taken in its use. The coproduct is cocommutative if and only if T ◦  = , where T is the switch map T (a ⊗ b) := b ⊗ a. In the Sweedler notation, c(1) ⊗ c(2) = c(2) ⊗ c(1) and, more generally, it holds that for any permutation σ in Sn , the symmetric group of order n, c(1) ⊗ · · · ⊗ c(n) = c(σ (1)) ⊗ · · · ⊗ c(σ (n)) . In general, T ◦  defines a new coproduct on C, the opposite coproduct.

12

2 Coalgebras, Duality

The dual notion of unit for an algebra is the one of counit: it is a linear map εC (written simply ε when no confusion can arise) from C to k such that (I d ⊗ ε) ◦  = I d = (ε ⊗ I d) ◦ ,

(2.2)

where we use the isomorphisms k ⊗ C ∼ =C ⊗k ∼ = C. Diagrammatically,   C ⊗C  C - C ⊗C ε ⊗ Id ? Id ? Id ⊗ ε ? - C  C ⊗k k⊗C ∼ ∼ = = In the Sweedler notation, c = c(1) ε(c(2) ) = ε(c(1) )c(2) . We set C¯ := ker (ε). Definition 2.2.1 A coalgebra is a vector space C equipped with a coassociative coproduct  and a counit ε. It is cocommutative if  is cocommutative. We will call noncounital coalgebra a k-vector space C equipped with a coassociative coproduct but without a counit. One reason for this convention is that most of the coalgebras we will consider will be equipped with a counit. Example 2.2.1 The ground field k is equipped with the structure of a (cocommutative) coalgebra by the identity map:  = ε = I d : k → k = k ⊗ k. Exercise 2.2.1 (Poset coalgebras) Let (X, ) be a locally finite poset. That is, given x, y ∈ X with x y, there are finitely many z with x z y. Let C be the vector space generated by ordered pairs (x, y) with x ≤ y. Check that (x, y) :=



(x, z) ⊗ (z, y); ε(x, y) = δxy

x≤z≤y

defines a coalgebra structure on C. Given two coalgebras C and D, equipped, respectively, with coproducts C ,  D and counits εC , ε D , a morphism of coalgebras from C to D is a morphism f ∈ Lin(C, D) that commutes to the structure maps (the coproducts and counits). That is, ( f ⊗ f ) ◦ C =  D ◦ f, ε D ◦ f = εC . The category of coalgebras is written Cog (or Cogk when the choice of a ground field has to be emphasized). Later on, we will omit expliciting the definition of morphisms in an algebraic category when it is a straightforward consequence of the very definition of its objects (that is, when morphisms are the structure-preserving maps).

2.2 Coalgebras: Definition and First Properties

13

Remark 2.2.1 Notice that if (C, , ε) is a coalgebra, (C, T ◦ , ε) is also a coalgebra: the opposite coalgebra of C, denoted C op . Algebras and coalgebras are dual objects. The dual of a coalgebra is always an algebra and the dual of a noncounital coalgebra is an algebra without a unit. However in infinite dimension the definition of the dual coalgebra of an algebra requires some care, as we shall see later. Lemma 2.2.1 (Dual of a coalgebra) Let C be a coalgebra, its dual C ∗ is naturally equipped with the structure of an algebra. Proof Indeed, there is a canonical embedding C ∗ ⊗ C ∗ → (C ⊗ C)∗ , defined by < λ ⊗ β|c ⊗ d >=< λ|c >< β|d >, where C ∗ denotes the linear dual of C, λ, β ∈ C ∗ , c, d ∈ C. Composition with the dual map of , from (C ⊗ C)∗ to C ∗ , defines the product on C ∗ . The associativity and unit properties follow by duality, the unit of C ∗ being simply the counit of C viewed as a linear form. The product on C ∗ is written as a convolution product, λ ∗ β := m k ◦ (λ ⊗ β) ◦ C , where m k is multiplication in k. It is an instance of a more general construction defining an associative product on the vector space of linear maps from a coalgebra to an arbitrary algebra, see Sect. 2.12. Remark 2.2.2 When C is finite dimensional, C = (C ∗ )∗ and there is a perfect duality between finite-dimensional algebras and finite-dimensional coalgebras. The categories are anti-equivalent, f : C → D corresponding to f ∗ : D ∗ → C ∗ . The structure maps of the algebras are (as always) dual to the structure maps of the dual coalgebras, and the converse statement also holds. The coproduct C⊗D := (I dC ⊗ TC,D ⊗ I d D ) ◦ (C ⊗  D ), where TC,D (c ⊗ d) := d ⊗ c, and the counit εC⊗D := εC ⊗ ε D equip the tensor product C ⊗ D with a structure of coalgebra. In the Sweedler notation, C⊗D (c ⊗ d) = (c(1) ⊗ d (1) ) ⊗ (c(2) ⊗ d (2) ). Exercise 2.2.2 Show that an algebra A is commutative if and only if the map m : A ⊗ A → A is a morphism of algebras. Dually, show that a coalgebra C is cocommutative if and only if  : C → C ⊗ C is a morphism of coalgebras.

14

2 Coalgebras, Duality

Remark 2.2.3 In the category of cocommutative coalgebras, the tensor product of coalgebras is the categorical product: Cog(C, D) × Cog(C, E) ∼ = Cog(C, D ⊗ E), where the isomorphism is obtained from the composition of maps C

 -

C ⊗C

f ⊗g -

D ⊗ E.

The situation is dual to the case of commutative algebras, where A ⊗ B, the tensor product of A and B equipped with the product law (a ⊗ b) · (a  ⊗ b ) = aa  ⊗ bb , is the coproduct of A and B: Com(A, C) × Com(B, C) ∼ = Com(A ⊗ B, C), where the isomorphism is obtained from the composition of maps A⊗B

f ⊗g -

C ⊗C

mC

C.

Definition 2.2.2 A subcoalgebra of a coalgebra C is a subvector space D of C such that (D) ⊂ D ⊗ D ⊂ C ⊗ C. It is a coalgebra whose counit is the restriction to D of the counit of C, ε D := εC|D . It is called proper in case D = C and D = 0. Recall that an (two-sided) ideal I of an associative algebra A with product m is a subvector space of A such that m(A ⊗ I ) ⊂ I and m(I ⊗ A) ⊂ I . This notion of ideal is dual to the one of subcoalgebra in the following sense. Assume that A = C ∗ and let I ⊂ C ∗ be the annulator of a subcoalgebra D, ∀λ ∈ C ∗ , (λ ∈ I ) ⇐⇒ (∀d ∈ D, λ(d) = 0). Then, for λ ∈ I , β ∈ C ∗ and d ∈ D, λ ∗ β(d) = λ(d (1) )β(d (2) ) = 0 = β(d (1) )λ(d (2) ) = β ∗ λ(d) since (d) = d (1) ⊗ d (2) ∈ D ⊗ D. Hence λ ∗ β and β ∗ λ are in I . Definition 2.2.3 A coideal of C is a subvector space I such that (I ) ⊂ I ⊗ C + C ⊗ I and I ⊂ K er (ε). For such an I , C/I is equipped with a coalgebra structure by the induced maps  : C/I → C/I ⊗ C/I, ε : C/I → k. It is called the quotient coalgebra of C by I .

2.2 Coalgebras: Definition and First Properties

15

As for ideals, our coideals are two-sided. A right (resp., left) coideal is defined by the condition (I ) ⊂ I ⊗ C (resp., (I ) ⊂ C ⊗ I ). A subalgebra of an associative algebra A with product m and unit 1 is a subvector space B of A, not necessarily strict, such that 1 ∈ B and m(B ⊗ B) ⊂ B. The notion of subalgebra is dual to the one of coideal in the following sense. Assume that A = C ∗ and let J ⊂ A be the annulator of a coideal I of C. Then ε ∈ J is the unit of A and for λ, β ∈ J , and d ∈ I , λ ∗ β(d) = λ(d (1) )β(d (2) ) = 0 since (d) = d (1) ⊗ d (2) ∈ I ⊗ C + C ⊗ I . Hence λ ∗ β ∈ J .

2.3 Primitive and Group-Like Elements Definition 2.3.1 (Group coalgebras) The vector space of functions k G on a finite group G is a coalgebra for the coproduct (1.1) and the counit ε(λ) := λ(1), for λ ∈ kG. The dual algebra of k G identifies, through the pairing < λ|g >:= λ(g), with the group algebra kG, the set of linear combinations of elements of G equipped with the bilinear product extending the product of G. The pointwise product of functions in k G dualizes accordingly into the diagonal map on kG, kG (g) := g ⊗ g that, together with the counit εkG (g) := 1, defines on kG the structure of a coalgebra. Summarizing, we have on k G (λ · β)(g) = λ(g)β(g), (δg ) =



δh ⊗ δh  ,

hh  =g

and dually on kG, for g and h in G (g) = g ⊗ g, g · h = gh. The coalgebraic part of this construction holds more generally for all sets: Definition 2.3.2 (Set coalgebras) The maps k S (s) := s ⊗ s, εk S (s) := 1 for s ∈ S equip k S, the linear span of an arbitrary set S, with the structure of a coalgebra. These phenomena suggest the definition:

16

2 Coalgebras, Duality

Definition 2.3.3 (Group-like elements) Let C be a coalgebra. A group-like element of C is an element c such that ε(c) = 1 and (c) = c ⊗ c. The set of group-like elements is written (C). Equivalently, because of counitality, a group-like element is an element of C such that c = 0 and (c) = c ⊗ c. As we shall see later, (C) is actually a group when C is a Hopf algebra. This is the case of kG, for example, with (kG) = G. This last property is a general phenomenon: Exercise 2.3.1 Show that group-like elements of a coalgebra C are linearly independent over the ground field. Deduce that, for an arbitrary set S, (k S) = S. Exercise 2.3.2 Let C be a coalgebra and X, Y be sets. Show that Cog(k X, C) ∼ = Set(X, (C)) and that

Cog(k X, kY ) ∼ = Set(X, Y ).

Lemma 2.3.1 If C is a finite-dimensional coalgebra, there is a natural isomorphism between (C) and characters (algebra morphisms to the ground field) on the dual algebra (C) ∼ = Alg(C ∗ , k). Proof Indeed, in that case Alg(C ∗ , k) ⊂ (C ∗ )∗ ∼ = C and (C ∗ ⊗ C ∗ )∗ ∼ = C ⊗ C. ∗ Therefore, for c ∈ C, c ∈ Alg(C , k) if and only if, for any a, b in C ∗ , < a|c >< b|c >=< ab|c >=< a|c(1) >< b|c(2) >, that is, if and only if (c) = c ⊗ c. Definition 2.3.4 (Coaugmented coalgebra) A coaugmented coalgebra is a coalgebra C equipped with a morphism of coalgebras η : k → C. Equivalently, a coaugmented coalgebra is a coalgebra with a fixed group-like element (η(1)). We will use the notation η(1) = 1 and sometimes also, slightly abusively, η(1) = 1. Definition 2.3.5 (Reduced coproduct) On a coaugmented coalgebra, the reduced ¯ is defined by coproduct  ¯ (x) := (x) − x ⊗ 1 − 1 ⊗ x,

2.3 Primitive and Group-Like Elements

17

¯ : C¯ → C¯ ⊗ C¯ (recall that C¯ = K er ε). One can check so that, as  is counital,  that it makes C¯ a noncounital coalgebra. The iterated reduced coproduct ¯ n : C¯ → C¯ ⊗n  ¯ 2 := , ¯ is defined inductively by  ¯ n+1 = ( ¯ ⊗ I dC⊗n−1 ) ◦  ¯ n.  The notion of coaugmented coalgebra is dual to the one of augmented algebra, that is, an algebra equipped with an algebra map to the ground field. Any finitedimensional algebra gives rise by duality to a finite-dimensional coalgebra, and any finite-dimensional augmented algebra to a finite-dimensional coaugmented coalgebra. Definition 2.3.6 (Primitive elements) Let C be a coaugmented coalgebra. An ele¯ ment c of C¯ is called primitive if and only if (c) = 0 or, equivalently, (c) = c ⊗ 1 + 1 ⊗ c. Notice that the condition (c) = c ⊗ 1 + 1 ⊗ c implies c ∈ C¯ as, from the counit condition, c = ε(c)1 + ε(1)c = c + ε(c)1.

2.4 Tensors Important examples of coalgebras, that will be useful in later chapters of this book, are given by coalgebra structures on tensor spaces. Let V be a vector space, not necessarily finite dimensional. We denote T n (V ) := V ⊗n the tensor product of n copies of V for n ≥ 0, with T 0 (V ) = k. We write T (V ) for the direct sum T (V ) :=



T n (V ),

n≥0

and call T (V ) the tensor gebra over V . We use the word notation and denote v1 . . . vn the tensor product of elements v1 , . . . , vn in V . Elements of T n (V ) are called tensors of length n. Definition 2.4.1 (Deconcatenation coalgebra) The tensor gebra T (V ) equipped with the deconcatenation coproduct d (v1 . . . vn ) :=

n  p=0

v1 . . . v p ⊗ v p+1 . . . vn ,

18

2 Coalgebras, Duality

and the counit ε(1) = 1, ε(v1 . . . vn ) = 0 for n ≥ 1, is a coalgebra called the deconcatenation coalgebra over V . We used the convention v1 . . . v0 := v∅ := 1 =: vn+1 . . . vn . Definition 2.4.2 (Unshuffle tensor coalgebra) The tensor gebra T (V ) equipped with the unshuffle coproduct 

u (v1 . . . vn ) := I



vI ⊗ v J

J =[n]

and the counit ε(1) = 1, ε(v1 . . . vn ) = 0 for n ≥ 1, where the sum runs over partitions of [n] with I = {i 1 , . . . , i p }, i 1 < · · · < i p , J = { j1 , . . . , jn− p }, j1 < · · · < jn− p , and v I := vi1 . . . vi p , is a cocommutative coalgebra called the unshuffle tensor coalgebra over V . The proof that these coproducts d and u indeed define coalgebra structures is left to the reader. Let us assume from now in this section that the ground field k is of characteristic 0. We denote T S n (V ) the vector subspace of T n (V ) of symmetric tensors of length n, that is, tensors in T n (V ) invariant under the natural action of the symmetric group Sn : that is, the action defined by, for σ ∈ Sn , σ (v1 . . . vn ) := vσ −1 (1) . . . vσ −1 (n) . The space T S(V ) :=



T S n (V ) is called the symmetric gebra or gebra of symmetric

n≥0

tensors over V . Symmetric tensors can be studied using the polarization process, according to which T S n (V ) is generated linearly by tensor powers γn (v) := v . . . v = v⊗n . For example, when n = 2, for arbitrary v1 , v2 in V , we have the identity (v1 v2 + v2 v1 ) = γ2 (v1 + v2 ) − γ2 (v1 ) − γ2 (v2 ). The general case follows from the following identity, valid in an arbitrary ring R.1  Let x1 , . . . , xn ∈ R and for H ⊂ [n] set x H := xi . We then have i∈H

(−1)n



xσ (1) . . . xσ (n) =

σ ∈Sn



(−1)Car d(H ) (x H )n .

H ⊂[n]

Lemma 2.4.1 The symmetric gebra T S(V ) is a subcoalgebra of the deconcatenation coalgebra (T (V ), d ) and of the unshuffle tensor coalgebra (T (V ), u ). Proof Since, by polarization, T S(V ) is generated by the tensor powers γn (v), the Lemma follows from the identities: 1

See Bourbaki, Algèbre I, 1970, A I.95, Prop. 2.

2.4 Tensors

19

n    n γ p (v) ⊗ γn− p (v). d (γn (v)) = γ p (v) ⊗ γn− p (v), u (γn (v)) = p p=0 p=0 n 

The space of covariants



T n (V )/Sn identifies with k[V ], the space of poly-

n≥0

nomials over V . Commutative monomials will be also written using the notation v1 . . . vn , this should not create confusion as it will always be clear from the context whether we are referring to commutative monomials or to words (noncommutative monomials). Covariants and invariants are related by the linear isomorphism from T S(V ) to k[V ] induced by the canonical projection from T (V ) to k[V ], and the symmetrization map, from k[V ] to T S(V ): v1 . . . vn −→



vσ (1) . . . vσ (n) .

σ ∈Sn

This map is also an isomorphism of vector spaces (recall we have assumed char (k) = 0). We refer to k[V ] as the polynomial gebra over V . The polynomial gebra is equipped with a coalgebra structure by the polynomial coproduct  p   p (v1 . . . vn ) := vI ⊗ v J I



J =[n]

and the counit ε(1) = 1, ε(v1 . . . vn ) = 0 for n ≥ 1, where notations are as in the definition of the unshuffle coproduct. We call it the polynomial coalgebra or coalgebra of polynomials over V . Notice that the polarization argument applies and we have in k[V ] v1 . . . vn = and

(−1)n  (−1)Car d(H ) (v H )n n! H ⊂[n]

n    n p v ⊗ vn− p .  p (v ) = p p=0 n

Exercise 2.4.1 Use the linear isomorphisms between T S(V ) and k[V ] described above to compare the three coalgebras (T S(V ), u ), (k[V ],  p ), and (T S(V ), d ).

2.5 Endomorphisms For an arbitrary finite-dimensional vector space V , the algebra structure of End(V ) = V ⊗ V ∗ is obtained from the composition

20

2 Coalgebras, Duality

(V ⊗ V ∗ ) ⊗ (V ⊗ V ∗ ) = V ⊗ (V ∗ ⊗ V ) ⊗ V ∗ → V ⊗ k ⊗ V ∗ ∼ = V ⊗ V ∗. Dually, we write End ∨ (V ) := V ∗ ⊗ V the dual coalgebra of the algebra End(V ): End ∨ (V ) → End ∨ (V ) ⊗ End ∨ (V ) v∗ ⊗ w →

n 

v∗ ⊗ ei ⊗ ei∗ ⊗ w,

i=1

where (ei )i≤n stands for an arbitrary basis of V and (ei∗ )i≤n for the dual basis. The counit is defined by ε(v∗ ⊗ w) :=< v∗ |w >. In the basis ci j := ei∗ ⊗ e j , (ci j ) =  cik ⊗ ck j . Notice that End ∨ (V ) = End(V ∗ ). k

Notice also that since δV =

n  i=1

ei ⊗ ei∗ identifies with the identity map of V ,

(v∗ ⊗ w) = v∗ ⊗ δV ⊗ w = v∗ ⊗ I dV ⊗ w. Definition 2.5.1 (Endomorphism coalgebra) We denote Enc(V ) and call endomorphism coalgebra of V the vector space of linear endomorphisms End(V ) = V ⊗ V ∗ viewed as a coalgebra for the coproduct (v ⊗ w∗ ) :=

n 

(ei ⊗ w∗ ) ⊗ (v ⊗ ei∗ ),

i=1

and counit the trace map ε(v ⊗ w∗ ) :=< w∗ |v >. It is the opposite coalgebra to End ∨ (V ∗ )—a choice dictated by the study of right comodules later in this chapter.

2.6 The Structure of Coalgebras A key property of coalgebras is that each element is contained in a finite-dimensional subcoalgebra. This property explains largely why coalgebras are in many situations easier to deal with than algebras, although using coproducts may seem more difficult or less natural than products. This property is usually referred to as the fundamental theorem for coalgebras. Theorem 2.6.1 (Fundamental theorem for coalgebras) Let c ∈ C, where C is a coalgebra. Then c is contained in a finite-dimensional subcoalgebra of C. Proof Indeed, let us expand 3 (c) ∈ C ⊗3 as a finite sum minimal. This implies, in particular, that the tensor

n 

ai i=1 products ai ⊗

⊗ bi ⊗ ci with n bi (resp., ai ⊗ ci ,

2.6 The Structure of Coalgebras

21

bi ⊗ ci ) are linearly independent in C ⊗2 . Else, assuming for example that, up to a reordering of the indices, an ⊗ bn =

n−1 

λi ai ⊗ bi ,

i=1

we would have 3 (c) =

n−1 

ai ⊗ bi ⊗ (ci + λi cn ), a contradiction.

i=1

Let C  denote the linear span of the ai , bi , ci . Then, since the coproduct is coassociative,    (ai ) ⊗ bi ⊗ ci = ai ⊗ (bi ) ⊗ ci = ai ⊗ bi ⊗ (ci ), 4 (c) = i

i

i

from which it follows that 4 (c) ∈ C ⊗ C ⊗ C  ⊗ C  ∩ C  ⊗ C  ⊗ C ⊗ C = C  ⊗ C  ⊗ C  ⊗ C  . The linear independency of the ai ⊗ ci implies then that (bi ) ∈ C  ⊗ C  , and similarly for (ai ) and (ci ). k  Expand now (c) as a finite sum x j ⊗ y j with k minimal. The xi (resp., the j=1

yi ) are then linearly independent and, from 3 (c) =

k 

(x j ) ⊗ y j =

j=1

k 

x j ⊗ (y j ) ∈ C  ⊗ C  ⊗ C 

j=1

we get that the (x j ) and the (y j ) belong to C  ⊗ C  . Finally, the linear span of c, the x j , y j and the ai , bi , ci is a finite-dimensional subcoalgebra of C.

2.7 Representative Functions Let X be a monoid and consider k X := Set(X, k). An element f in k X is called representative if and only if there exist finitely many f i , f i ∈ k X , i ∈ I such that for any x, y ∈ X  f (x y) = f i (x) f i (y). (2.3) i∈I

Linear combinations of representative functions are representative. Let us introduce further notation. For x ∈ X, f ∈ k X , we denote L x f , resp., Rx f the left and right translates of f : L x f (y) := f (x y), R y f (x) := f (x y).

22

2 Coalgebras, Duality

If f is representative, its right and left translates are representative: L x f (yz) = f (x yz) =



L x f i (y) f i (z).

i∈I

Lemma 2.7.1 If f is representative, the space generated by the right translates R y f, y ∈ X is finite dimensional. The same statement holds for left, or left and right translates. Proof The very definition of representative functions implies that R y f can be written as a linear combination of the f i . The lemma follows. A representation π of X is a monoid homomorphism from X to the monoid of linear endomorphisms of a finite–dimensional vector space V , π : X −→ End(V ). Given a basis B = (ei )1≤i≤n of V , we write πi, j (g), 1 ≤ i, j ≤ n for the entries of the matrix of π(g) in B. Since π(x y) = π(x)π(y), πi, j (x y) =



πi,k (x)πk, j (y)

(2.4)

1≤k≤n

and these functions are representative. To π , we associate at last its space of coefficients C(π ), the subvector space of k X spanned linearly by the functions πi, j , 1 ≤ i, j ≤ n. More intrinsically, C(π ) is spanned by the set of coefficient functions cv∗ ,v , cv∗ ,v (g) :=< v∗ |π(g)(v) > for v ∈ V, v∗ ∈ V ∗ . It consists of the set of functions c A,π (g) := T r (A · π(g)) for A ∈ End(V ). The union of the C(π ), where π runs over the linear representations of π is written R(X ). One can deduce that R(X ) is a vector space from the observation that C(π ) + C(π  ) = C(π ⊕ π  ). Equivalently, it is the linear span of the C(π ) in k X . Lemma 2.7.2 If the space of right translates of f is finite dimensional, f ∈ R(X ) and f is representative. The same statement holds for left, or left and right translates. Proof Let V be the space of right translates of f . The map y −→ R y defines a finite-dimensional representation of X on V . Define δ : V → k by δ(g) := g(1). Then, f (y) =< δ, R y ( f ) >: f ∈ R(X ) and f is representative. The following proposition summarizes the previous results. Proposition 2.7.1 For f ∈ k X , the following are equivalent: 1. 2. 3. 4.

The function f is representative. The function f belongs to R(X ). The space generated by the left translates L x f, x ∈ X , is finite dimensional. The space generated by the right translates Rx f, x ∈ X , is finite dimensional.

2.7 Representative Functions

23

Corollary 2.7.1 The functions f i , f i in eq. (2.3) can be chosen to be representative. The coproduct  πi,k ⊗ πk, j (πi, j ) := 1≤k≤n

and the counit (π i, j ) := πi, j (1) are defined on any space of coefficients C(π ), hence on R(X ) = C(π ). They equip R(X ) with the structure of a coassociative coalgebra.

π

In terms of representative functions the coproduct is defined in duality with the product in X ,  ( f )(x ⊗ y) = f (x y), ( f ) = f i ⊗ f i . i∈I

Proof Only the coassociativity assertion requires a proof; it follows from the identity πi, j (x yz) =



πi,k (x)πk,l (y)πl, j (z).

k,l≤n

Example 2.7.1 Matrix elements of representations are clearly, in view of Prop. 2.7.1, canonical examples of representative functions. Another canonical example is provided by finite monoids. In that case, in view of Formula (1.1), that applies to monoids, all functions on X are representative and R(X ) = k X . This may be understood from the representation theoretical point of view by considering the regular representation of X , that is, the action by left translations on k X , the linear span of X . The delta functions δx () generate linearly k X , and δx (yz) =



δa (y)δb (z).

ab=x

2.8 Comodules There is another approach to group representations than representative functions. It relies on duality phenomena and is encapsuled in the notion of comodules, which are the object of the present section. Definition 2.8.1 Let C be a coalgebra. A comodule (or right-comodule) over C is a vector space M equipped with a linear morphism δ : M → M ⊗C such that (I d M ⊗ ) ◦ δ = (δ ⊗ I dC ) ◦ δ

24

2 Coalgebras, Duality

and I d M = (I d M ⊗ ε) ◦ δ. Graphically, the first identity translates into the commutativity of the diagram: δ

M δ? M ⊗C

- M ⊗C

I dM ⊗  ? - M ⊗ C ⊗ C. δ ⊗ I dC

Left comodules are defined similarly, but since we will use only right ones, we call the latter simply comodules. The notion of subcomodule is the canonical one. The category of (resp., finite dimensional) comodules over C is denoted Cmd (resp., Cmd f ) or when the underlying coalgebra has to be explicited CmdC (resp., f CmdC ). Recall from Def. 2.5.1 that for an arbitrary finite-dimensional vector space V , the endomorphism coalgebra Enc(V ) = V ⊗ V ∗ is equipped with the coproduct n  ei ⊗ w∗ ⊗ v ⊗ ei∗ , where (ei )1≤i≤n stands for an arbitrary basis (v ⊗ w∗ ) := i=1

of V and (ei∗ )1≤i≤n for the dual basis. The vector space V is then equipped with a comodule structure over Enc(V ) by V → V ⊗ Enc(V ) v → ∂(v) :=

n 

ei ⊗ (v ⊗ ei∗ ).

i=1

Lemma 2.8.1 The coalgebra Enc(V ) is universal in the sense that it is equivalent to define a C-comodule structure on V or a coalgebra map γC from Enc(V ) to C. Given ∂C : V → V ⊗ C, n  j ei ⊗ ci , ∂C (e j ) = i=1

the image of the basis element e j ⊗ ei∗ of Enc(V ) in C by this coalgebra map is ci . j

Proof The commutativity of the diagram V ∂C ? V ⊗C

∂C

- V ⊗C

I dV ⊗ C ? - V ⊗C ⊗C ∂C ⊗ I dC

translates into the identities (for all 1 ≤ j ≤ n)

2.8 Comodules n 

25

ei ⊗ (cij ) =

i=1

n 

∂C (ek ) ⊗ ckj =

n  n 

k=1

ei ⊗ cki ⊗ ckj .

i=1 k=1

That is, for all 1 ≤ i, j ≤ n: (cij ) =

n 

cki ⊗ ckj .

k=1

The identity (for all 1 ≤ i, j ≤ n) n 

γC ⊗ γC (ek ⊗ ei∗ ⊗ e j ⊗ ek∗ ) =

k=1

n 

cki ⊗ ckj

k=1

translates then into the commutativity of the diagram V ⊗ V∗  Enc(C) ? V ⊗ V∗ ⊗ V ⊗ V∗ We also have

γC

- C

C ? - C ⊗ C. γC ⊗ γC

εC (cij ) = δi = ε Enc(V ) (e j ⊗ ei∗ ). j

The lemma follows. To each comodule M is canonically associated a subcoalgebra C(M) of C. It is the minimal subcoalgebra of C such that δ restricts to a map M → M ⊗ C(M). It can be  defined directly by choosing a basis (m i )i∈I of M. Writing δ(m j ) = i∈I m i ⊗ ci j , C(M) is the vector space spanned by the ci j (we let the reader check it is indeed a coalgebra). Intrinsically, it is the image of the map from Enc(M) = M ⊗ M ∗ to C induced from δ. When M is a subcoalgebra of C, C(M) = M. The proof of the fundamental theorem for coalgebras, Thm 2.6.1, can be mimicked for comodules: Theorem 2.8.1 (Fundamental theorem for comodules) Let C be a coalgebra, M a comodule and m ∈ M. Then, m is contained in a finite-dimensional subcomodule M  of M and there exists a finite-dimensional subcoalgebra C  of C such that δ(m) ∈ M  ⊗ C . It is interesting, and useful for various purposes, to understand how the notion of subcoalgebras is reflected in terms of comodules. If D is a subcoalgebra of C, any f D-comodule is a C-comodule since M ⊗ D → M ⊗ C, and Cmd D is isomorphic f to a subcategory D˜ of CmdC .

26

2 Coalgebras, Duality

Proposition 2.8.1 The map D → D˜ is a bijection between the set of subcoalgebras f of C and the set of subcategories E of the abelian category CmdC such that: f

1. Any object of CmdC which is isomorphic to an object, sub-object or quotient object of an element of E is in E. f 2. The category E is a full subcategory of CmdC (i.e., for A, B in E we have f E(A, B) = CmdC (A, B)). 3. The category E is stable by finite direct sums. Proof For E as in the proposition, let us write C(E) for the coalgebra sum of the ˜ = D for any subcoalgebra D of C: coalgebras C(M) for M ∈ E. We have C( D) ˜ obviously C( D) ⊂ D, and the identity follows by noticing that D is the union of its finite-dimensional subcoalgebras.  = E. Clearly, E ⊂ C(E).  It is therefore enough to show that, Let us show that C(E) given a finite-dimensional C-comodule M in E and F a finite-dimensional C(M)comodule, then there exists a n ≥ 0 such that F is isomorphic to a subcomodule of a quotient of M n . Since the structure map δ : F → F ⊗ C(M) is an injective morphism of comodules and F ⊗ C(M) ∼ = C(M)dim(F) , it is enough to show the property for F = C(M). Let us fix a basis (m i )i∈I of M. We get a sequence of C(M)-comodule maps: M dim(M) ∼ =

 i∈I

M

⊕i∈I -δ



M ⊗ C(M)

⊕i∈I m i∗ ⊗C(M)

- C(M).

i∈I

The total map from M dim(M) ∼ = M ⊗ M ∗ to C(M) is surjective. This follows from the fact that C(M) is the image of the map from M ⊗ M ∗ = Enc(M) to C(M) induced by δ. The proposition follows.

2.9 Representations and Comodules Let now V be a vector space and G a group. We call V -valued representative functions the elements of V G (maps from G to V ) whose left (or right) translates generate a finite-dimensional vector space. The set of V -valued representative functions of G is written R(G, V ) (so that R(G, k) = R(G)). Clearly, R(G, V ) ∼ = V ⊗ R(G). If furthermore V is a locally finite G-module (every v ∈ V is contained in a finitedimensional sub-G-module of V ) with structure map π : G → End(V ), any v ∈ V defines an element written δ(v) of R(G, V ) by δ(v)(g) := π(g)(v). repreWe get a linear map δ from V to V ⊗ R(G). When V is a finite-dimensional  sentation, using the notations of Sect. 2.7, we have δ(ei ) = j≤n e j ⊗ π ji . Since π(gg  ) = π(g)π(g  ) for arbitrary g, g  ∈ G, we have

2.9 Representations and Comodules

27

δ(v)(gg  ) = π(gg  )(v) = π(g)π(g  )(v) = δ(π(g  )(v))(g). In coordinates:

 j,k≤n

ek ⊗ (πk j ⊗ π ji ) =



δ(e j ) ⊗ π ji .

j≤n

Equivalently, (I dV ⊗ ) ◦ δ = (δ ⊗ I d R(G) ) ◦ δ. Corollary 2.9.1 The map δ : V → V ⊗ R(G) just defined equips any locally finite G-module V with the structure of a comodule over R(G). Theorem 2.9.1 The categories of locally finite G-modules and R(G)-comodules are isomorphic. Proof We have seen that any locally finite G-module is naturally equipped with the structure of a comodule over R(G). Conversely, let δ : V → V ⊗ R(G) define a R(G)-comodule structure on V . Then, the map g ∈ G → π(g) ∈ End(V ), π(g)(v) := (I dV ⊗ g) ◦ δ(v), where we view g ∈ G as a linear form on R(G), defines a locally finite G-module structure on V . The local finiteness follows from the fundamental theorem for comodules. We leave the reader check that these constructions are inverse to each other. Although we featured the point of view of groups, the same arguments and results hold more generally for locally finite A-modules V over an associative algebra A, see Sect. 2.12. Moreover, we have the following. Lemma 2.9.1 Under the previous isomorphism the tensor product of two locally finite G-modules, V ⊗ W (with structure map g(v ⊗ w) := g(v) ⊗ g(w)), is reflected in the following comodule structure: V ⊗ W → (V ⊗ R(G)) ⊗ (W ⊗ R(G)) ∼ = (V ⊗ W ) ⊗ (R(G) ⊗ R(G)) → (V ⊗ W ) ⊗ R(G),

where the first map is the tensor product of the comodule structure maps for V and W , the second is induced by the switch map R(G) ⊗ W ∼ = W ⊗ R(G) and the last by the product map on R(G). The tensor product of R(G)-comodules is therefore induced by the product in R(G): this phenomenon will be understood in a broader generality later, using the existence of a Hopf algebra structure on R(G), see in particular Prop. 3.2.2.

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2 Coalgebras, Duality

2.10 Algebra Endomorphisms and Pseudo-coproducts In many situations, it is convenient to extend the notion of representative functions to more general frameworks such as spaces of linear endomorphisms of algebras, coalgebras, and Hopf algebras. This idea will be essential later in this book, especially when studying the fine structure of graded Hopf algebras. Let (A, m, η) be an associative algebra over a field k of characteristic 0, possibly but not necessarily augmented. When this is the case, we denote by ε : A → k the augmentation map and A+ the augmentation ideal of A (the kernel of ε). Recall also that we write then ν := η ◦ ε. Definition 2.10.1 An element f of End(A), the algebra of linear endomorphisms of A is said to be a representative endomorphism of A if there exists a F = f F(1) ⊗ f F(2) ∈ End(A) ⊗ End(A) such that for any a, b ∈ A, f F(1) (a) f F(2) (b) = m ◦ F(a ⊗ b) = f (ab). We then say that F is a pseudo-coproduct for f . We used and will use for pseudo-coproducts the shortcut notation F = f F(1) ⊗ f F(2) . As for the Sweedler notation, this notation does not mean that F can be written as the tensor product of two endomorphisms in End(A) since, in general, F can be written only as a linear combination of such tensor products (see the examples below). Notice that, in general, even when A is a free (associative or commutative) algebra, an element of End(A) may admit several pseudo-coproducts. For example, when A = C[x], equipped with the usual augmentation map (the evaluation at 0 of a polynomial), let f be defined by f (x i ) := x i if i ≥ 2 and f (x) = f (1) = 0. Then, for the maps g, h, k defined by g(x i ) := x i+1 for i = 0, g(1) := 0; h(x i ) := x i−1 for i = 0, h(1) := 0, l(x i ) := x i if i ≥ 1 and l(1) = 0, the two tensor products g ⊗ h + f ⊗ ν + ν ⊗ f and l ⊗ l + f ⊗ ν + ν ⊗ f are pseudo-coproducts for f . Another example is provided by polynomial derivations. For example, x n ∂x admits various coproducts such as x n ∂x ⊗ I d + I d ⊗ x n ∂x , ∂x ⊗ x n I d + x n I d ⊗ ∂x . We will see that the choice of a particular pseudo-coproduct for a map f most often does not matter in practice. In many applications, it will also happen that there is a well-defined natural pseudo-coproduct for f . We will therefore often abbreviate f F(1) ⊗ f F(2) to f (1) ⊗ f (2) excepted when we want to emphasize explicitly the dependency of the pseudo-coproduct of f on a particular choice for F. Example 2.10.1 The map f admits the pseudo-coproduct f ⊗ I d + I d ⊗ f if and only if it is a derivation: f (ab) = f (a)b + a f (b). Example 2.10.2 Let A := C[x1 , ..., xn ]. The linear endomorphisms f of A with pseudo-coproduct f ⊗ I d + I d ⊗ f (the derivations) correspond bijectively to polyn  f i ∂xi with f i ∈ A). nomial vector fields (linear endomorphisms of the form i=1

Definition 2.10.2 If the algebra is augmented and f admits the pseudo-coproduct f ⊗ ν + ν ⊗ f , we say that f is an infinitesimal endomorphism of A.

2.10 Algebra Endomorphisms and Pseudo-coproducts

29

In particular, such a f is the null application on the scalars ( f ◦ ν = 0) since f (1) = f (1)ν(1) + ν(1) f (1) = 2 f (1). A map f is an infinitesimal endomorphism if and only it is null on the scalars and its restriction to (A+ )2 , the square of the augmentation ideal, is the null application (i.e., f is zero on products of elements of A+ ). Definition 2.10.3 If f admits the pseudo-coproduct f ⊗ f and f (1) = 1, we say that f is a group-like element of End(A). An element of End(A) is group-like if and only if it is an algebra endomorphism of A. These notions generalize in a straightforward way to linear maps from A to another algebra B: assume that an algebra map ρ from A to B is fixed, then f ∈ Lin(A, B) is a derivation, resp., group-like, resp., (when A is augmented) infinitesimal if and only if for any a, b ∈ A, f (ab) = f (a)ρ(b) + ρ(a) f (b), resp., f (ab) = f (a) f (b) and f (1) = 1, resp., f (ab) = f (a)η B ◦ ε A (b) + η B ◦ ε A (a) f (b), where η B is the unit map of B, and ε A the augmentation of A. Following standard terminology, when B is commutative, we will also call grouplike morphisms (B-valued) characters and infinitesimal morphisms (B-valued) infinitesimal characters. The concept of pseudo-coproduct is very flexible, as shown by the following result. Theorem 2.10.1 If f, g admit the pseudo-coproducts F, G and α ∈ k, then f + g, α · f, f ◦ g admit, respectively, the pseudo-coproducts F + G, α · F, F ◦ G, where the composition product ◦ is naturally extended from End(A) to End(A) ⊗ End(A). Proof Let F = f (1) ⊗ f (2) and G = g (1) ⊗ g (2) . For x, y ∈ A, we have f ◦ g(x y) = f (g (1) (x)g (2) (y)) = f (1) (g (1) (x)) f (2) (g (2) (y)), so that F ◦ G = f (1) ◦ g (1) ⊗ f (2) ◦ g (2) is a pseudo-coproduct for f ◦ g. The other assertions are straightforward. Corollary 2.10.1 The set of group-like elements in End(A), written EndAlg (A), is a monoid for the composition product. Proof Indeed, the composition of algebra endomorphisms is an algebra endomorphism. Using pseudo-coproducts: let f, g ∈ EndAlg (A), then f ⊗ f and g ⊗ g are pseudo-coproducts for f and g, so that ( f ◦ g) ⊗ ( f ◦ g) is a pseudo-coproduct for f ◦ g. The unit of the monoid is the identity map of A. Take care that in spite of its name, the monoid of group-like elements EndAlg (A) is not a group for the composition product in general: for example, when A is augmented, the map ν belongs to EndAlg (A) but, for any f ∈ EndAlg (A), f ◦ ν = ν. Corollary 2.10.2 The sets of derivations and infinitesimal endomorphisms written, respectively, Der (A) and Endinf (A) are Lie subalgebras of End(A). The corollary is an obvious consequence of the definitions, but we include a proof to illustrate the functioning of pseudo-coproduct calculus.

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2 Coalgebras, Duality

Proof Recall that any associative algebra with product ∗ is equipped naturally with a Lie algebra structure and a Lie bracket usually written [ , ] (or [ , ]∗ , to emphasize what is the underlying associative product, defined by [x, y] := x ∗ y − y ∗ x. The first part of the corollary simply states the property that derivations of an algebra form a Lie algebra. The use of pseudo-coproducts allows to give elementfree proofs of this kind of structural statements. Let f, g be derivations in End(A), a pseudo-coproduct for [ f, g] := f ◦ g − g ◦ f is given by ( f ⊗ I d + I d ⊗ f ) ◦ (g ⊗ I d + I d ⊗ g) − (g ⊗ I d + I d ⊗ g) ◦ ( f ⊗ I d + I d ⊗ f )

= f ◦ g ⊗ Id + Id ⊗ f ◦ g − g ◦ f ⊗ Id − Id ⊗ g ◦ f = [ f, g] ⊗ I d + I d ⊗ [ f, g]. Let f, g be infinitesimal elements in End(A), a pseudo-coproduct for [ f, g] := f ◦ g − g ◦ f is given by (recall that f ◦ ν = 0 and similarly for g): ( f ⊗ ν + ν ⊗ f ) ◦ (g ⊗ ν + ν ⊗ g) − (g ⊗ ν + ν ⊗ g) ◦ ( f ⊗ ν + ν ⊗ f ) = f ◦g⊗ν+ν⊗ f ◦g−g◦ f ⊗ν−ν⊗g◦ f = [ f, g] ⊗ ν + ν ⊗ [ f, g]. Example 2.10.3 For example, when A = C[x1 , ..., xn ], the derivations, which are the polynomial vector fields, form a Lie algebra. The exponential and the logarithm define bijections between a neighborhood of the identity in a Lie group and a neighborhood of 0 in the tangent space to the identity (which is a Lie algebra). The same properties hold true for EndAlg (A) and Der (A); however, some restrictions on A and on the maps under consideration are required to make sure that the formal power series defining the exponential and the logarithm maps are well defined. This kind of ideas will be used later in this book to revisit some parts of the classical theory of free Lie algebras. Let us fix a framework allowing to deal with formal power series. In the remaining  An is a graded connected algebra (the product maps part of this section, A = n≥0  Ai ⊗ A j to Ai+ j and A0 ∼ An ). Notice that we could also = k; we write A+ := n>0

have assumed that A is a complete augmented algebra, a natural framework for the following developments, but for didactical reasons we postpone the formal definition of such an algebra to Section 3.8. An identity-tangent algebra endomorphism of A is a graded algebra endomorphism f such that f (1) = 1 and ( f − I d)(A+ ) ⊂ (A+ )2 . Since ( f − I d)(A+ ) p ⊂ (A+ ) p+1 , ( f − I d)n is the null application on the first n + 1 components, Ai , i = 0, . . . , n, of the graded algebra A and formal power series in ( f − I d) are well defined. Identity-tangent algebra endomorphisms are therefore invertible

2.10 Algebra Endomorphisms and Pseudo-coproducts

f −1 = (I d + ( f − I d))−1 =

31

 (−1)n ( f − I d)n . n≥0

A tangent-to-zero derivation is a graded derivation D such that D(A) ⊂ (A+ )2 . 1 We write EndAlg (A), resp., Der 0 (A) for the group of identity-tangent elements in EndAlg (A), resp., the sub-Lie algebra of tangent-to-zero elements in Der (A). Notice first that, as for Lie groups, it is not true that the exponential and the logarithm maps define a bijection between EndAlg (A) and Der (A): if it was the case, any element of EndAlg (A) would be invertible (since e−x is always the inverse of e x , whenever the corresponding series make sense) but ν is not invertible if A = k. Theorem 2.10.2 The logarithm and the exponential series define inverse bijections 1 between EndAlg (A) and Der 0 (A). 1 (A) to Der 0 (A) Proof Let us show that the logarithm is a map from EndAlg 1 (the reverse property is proved similarly). Any f ∈ EndAlg (A) has the pseudocoproduct f ⊗ f and ( f − I d) has the pseudo-coproduct f ⊗ f − I d ⊗ I d. There (−1)n−1 fore, log( f ) = ( f − I d)n has the pseudo-coproduct: n n≥1

 (−1)n−1 n≥1

n

( f ⊗ f − I d ⊗ I d)n = log( f ⊗ f ) = log(( f ⊗ I d) ◦ (I d ⊗ f ))

which, since f ⊗ I d and I d ⊗ f commute in End(A) ⊗ End(A), equals log( f ⊗ I d) + log(I d ⊗ f ) = log( f ) ⊗ I d + I d ⊗ log( f ), which concludes the proof since log( f )(A) ⊂ (A+ )2 .

2.11 Coalgebra Endomorphisms and Quasi-coproducts Let us show now briefly how the previous constructions dualize. We omit some details since most ideas are similar to the case of algebra endomorphisms. Let (C, , ε) be a coalgebra over a field k of characteristic 0, possibly but not necessarily coaugmented. When this is the case, we denote by η : k → C the coaugmentation map. We recall that C¯ is the kernel of ε and we write 1 for η(1). As usual we set ν = η ◦ ε and we have ε ◦ η = I dk . Definition 2.11.1 An element f of End(C), the algebra of linear endomorphisms of C, is said to be a characteristic endomorphism of C if there exists a F ∈ End(C) ⊗ End(C) such that F ◦  =  ◦ f . We then say that F is a quasi-coproduct for f . Graphically, this amounts to the commutation of the following diagram:

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2 Coalgebras, Duality

 C - C ⊗C F f ? ? C ⊗ C. C  We will use for quasi-coproducts the shortcut notation F = f F(1) ⊗ f F(2) . An element of End(C) may admit several quasi-coproducts. A coderivation of a coalgebra is an endomorphism f such that ( f ⊗ I d + I d ⊗ f ) ◦  =  ◦ f . Therefore, we have Remark 2.11.1 The map f admits the quasi-coproduct f ⊗ I d + I d ⊗ f if and only if it is a coderivation. Definition 2.11.2 If the coalgebra is coaugmented, f (1) = 0, and f admits the quasi-coproduct f ⊗ ν + ν ⊗ f , we say that f is a primitive endomorphism of C. Lemma 2.11.1 The map f is primitive if and only if f (1) = 0 and it maps the coaugmented coalgebra C to Prim(C), the space of primitive elements in C. ¯ any such f , and since  is counital: Proof Indeed, for any c ∈ C, ( f (c)) = ( f ⊗ ν + ν ⊗ f ) ◦ (c) = f (c) ⊗ 1 + 1 ⊗ f (c). Definition 2.11.3 If f admits the quasi-coproduct f ⊗ f and ε ◦ f = ε, we say that f is a group-like element of End(C). An element of End(C) is group-like if and only if it is a coalgebra endomorphism of C. Remark 2.11.2 The set of group-like elements in End(C), written EndCog (C), is a monoid for the composition product: a composite of coalgebra endomorphisms is a coalgebra endomorphism. These notions generalize in a straightforward way to linear maps from C to another coalgebra D. As the concept of pseudo-coproduct, the one of quasi-coproduct is very flexible: Theorem 2.11.1 If f, g admit the quasi-coproducts F, G and α ∈ k, then f + g, α · f, f ◦ g admit, respectively, the quasi-coproducts F + G, α · F, F ◦ G, where the product ◦ is naturally extended to End(C) ⊗ End(C). Proof We have  ◦ ( f ◦ g) = F ◦ (g) = F ◦ G ◦ , so that F ◦ G is a quasicoproduct for f ◦ g. The other assertions are straightforward. Corollary 2.11.1 The sets of coderivations and primitive endomorphisms written, respectively, CoDer (C) and Endprim (C) are Lie subalgebras of End(C).

2.11 Coalgebra Endomorphisms and Quasi-coproducts

33

Proof The first part of the corollary simply states the property that coderivations of a coalgebra form a Lie algebra. Indeed, let f, g be coderivations in End(C), a quasi-coproduct for [ f, g] := f ◦ g − g ◦ f is given by ( f ⊗ I d + I d ⊗ f ) ◦ (g ⊗ I d + I d ⊗ g) − (g ⊗ I d + I d ⊗ g) ◦ ( f ⊗ I d + I d ⊗ f )

= f ◦ g ⊗ Id + Id ⊗ f ◦ g − g ◦ f ⊗ Id − Id ⊗ g ◦ f = [ f, g] ⊗ I d + I d ⊗ [ f, g]. Let f, g be primitive endomorphisms in End(C), a quasi-coproduct for [ f, g] := f ◦ g − g ◦ f is given by (notice that necessarily f ◦ ν = ν ◦ f = 0 and similarly for g): ( f ⊗ ν + ν ⊗ f ) ◦ (g ⊗ ν + ν ⊗ g) − (g ⊗ ν + ν ⊗ g) ◦ ( f ⊗ ν + ν ⊗ f ) = f ◦g⊗ν+ν⊗ f ◦g−g◦ f ⊗ν−ν⊗g◦ f = [ f, g] ⊗ ν + ν ⊗ [ f, g]. When the exponential and logarithmic maps are well defined, the logarithm of a coalgebra endomorphism is a coderivation, and conversely the exponential of a coderivation is a coalgebra endomorphism. The proof is obtained as for the dual case (i.e., for representative endomorphisms of algebras). For example, if f has the quasi-coproduct f ⊗ f , then, provided the logarithmic series is well defined, log( f ) has the quasi-coproduct: log( f ⊗ f ) = log(( f ⊗ I d) ◦ (I d ⊗ f )) and, since f ⊗ I d and I d ⊗ f commute in End(C) ⊗ End(C), log( f ⊗ f ) = log( f ⊗ I d) + log(I d ⊗ f ) = I d ⊗ log( f ) + log( f ) ⊗ I d.

2.12 Duals of Algebras and Convolution We already noticed (Lemma 2.2.1) that the linear dual of a coalgebra is automatically equipped with the structure of an algebra. Another incarnation of the duality between algebras and coalgebras is contained in the notion of convolution product. Proposition 2.12.1 (Convolution) Let (A, m, η) be an algebra and (C, , ε) a coalgebra. The set of linear morphisms Lin(C, A) is equipped with the structure of an algebra (associative, with unit η ◦ ε) by the convolution product: f ∗ g := m ◦ ( f ⊗ g) ◦ .

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2 Coalgebras, Duality

The canonical map A ⊗ C ∗ → Lin(C, A) is a morphism of algebras, and an isomorphism when A or C is finite dimensional. Proof Indeed, for all f, g, h ∈ Lin(C, A), ( f ∗ g) ∗ h = m ◦ (( f ∗ g) ⊗ h) ◦  = m ◦ (m ⊗ I d A ) ◦ ( f ⊗ g ⊗ h) ◦ ( ⊗ I dC ) ◦ 

= m 3 ◦ ( f ⊗ g ⊗ h) ◦ 3 . The same expression is obtained for f ∗ (g ∗ h), and the associativity follows. The fact that η ◦ ε is a unit follows, using that ε is the counit of  and η the unit of m, from the identity f = m ◦ (η ⊗ I d A ) ◦ (I dk ⊗ f ) ◦ (ε ⊗ I dC ) ◦ . For λ ∈ C ∗ , a ∈ A, the map A ⊗ C ∗ → Lin(C, A) which is an isomorphism when A or C is finite dimensional sends a ⊗ λ to the map: c ∈ C −→ aλ(c). The fact that the map is a morphism of algebras follows then from the definition of the product λ ∗ β of two elements in C ∗ (∀c ∈ C, λ ∗ β(c) = λ(c(1) )β(c(2) )). The product of a ⊗ λ with b ⊗ β in A ⊗ C ∗ is mapped to c ∈ C −→ abλ(c(1) )β(c(2) ) that identifies with the convolution product of their images in Lin(C, A). Except in finite dimension, where (A ⊗ A)∗ ∼ = A∗ ⊗ A∗ , the linear dual of an algebra A is not naturally equipped with a coalgebra structure. This is because there is no natural map from (A ⊗ A)∗ to A∗ ⊗ A∗ , so that the dual map of the product from A ⊗ A to A cannot give rise to a map from A∗ to A∗ ⊗ A∗ . The “right” definition of the dual of an algebra is obtained as follows. We write A× for A viewed as a monoid for its product m. Definition 2.12.1 Let A be an algebra. The restricted dual of A, written A◦ , is the set of elements in A∗ that are representative functions on A× : A◦ := A∗ ∩ R(A× ). Lemma 2.12.1 The restricted dual A◦ is the subspace of elements in A∗ whose kernel contains a two-sided ideal I such that A/I is finite dimensional. In particular, when A is finite dimensional, A◦ = A∗ . Proof Indeed, let f ∈ A◦ : since the left and right translates of f generate a finitedimensional vector space, the intersection I of their kernels for x running over A is

2.12 Duals of Algebras and Convolution

35

of finite codimension. It is a two-sided ideal (∀a, x, x  ∈ A, ∀y ∈ I, L x Rx  f (ay) = L xa Rx  f (y) = 0 = L x Rax  f (y) = L x Rx  f (ya)). Conversely, if I is a two-sided ideal contained in K er ( f ), f ∈ A∗ , and of finite codimension in A, I is also contained in the kernels of the left and right translates of f . Their intersection has finite codimension (it is bounded by dim(A/I )). It follows that the left and right translates generate a finite-dimensional vector space. Theorem 2.12.1 (Restricted dual coalgebra) Let A be an algebra. Its restricted dual A◦ is naturally equipped with the structure of coalgebra obtained by restricting to A◦ the map from A∗ to (A ⊗ A)∗ dual to the product. Proof Let f ∈ A◦ ⊂ A∗ , the map f : A → k factorizes then through A/I , where I is a two-sided ideal and A/I is finite dimensional. In particular, f belongs to the dual coalgebra of the finite-dimensional algebra A/I . Its coproduct is the restriction to a map from (A/I )∗ to (A/I ⊗ A/I )∗ ∼ = (A/I )∗ ⊗ (A/I )∗ of the map from A∗ to ∗ (A ⊗ A) dual to the product. Exercise 2.12.1 The notion of restricted dual allows to generalize various results of Sect. 2.8 for G-modules and R(G)-comodules to arbitrary associative algebras. Consider, for example, a locally finite A-module V over an associative algebra A (a module such that any v ∈ V is contained in a finite-dimensional submodule). Show that the A-module structure π : A → End(V ) gives rise to a linear map δ : V → V ⊗ A◦ , where A◦ stands for the restricted dual of A. Show the following proposition generalizing Prop. 2.9.1: Proposition 2.12.2 The map δ : V → V ⊗ A◦ just defined equips any locally finite A-module V with the structure of a comodule over A◦ . Furthermore, under this correspondence, the categories of locally finite A-modules and A◦ -comodules are isomorphic. Exercise 2.12.2 Show that the map A −→ A◦ is a contravariant functor from the category of algebras to the category of coalgebras. Exercise 2.12.3 (Adjunction for restricted duals) Since A◦ is a coalgebra, its linear dual (A◦ )∗ is an algebra and the canonical embedding A ⊂ (A∗ )∗ induces an algebra map from A to (A◦ )∗ . Conversely, show that the embedding of a coalgebra C into its bidual (C ∗ )∗ restricts to a coalgebra map from C to (C ∗ )0 . Deduce from these maps the existence of an adjunction (natural isomorphism) Cog(C, A◦ ) ∼ = Alg(A, C ∗ ). An application of restricted duals is the construction of the cofree coalgebra over a finite-dimensional vector space V (so that V = (V ∗ )∗ ). Recall that the free algebra functor is the left adjoint to the forgetful functor from algebras to vector spaces. In particular, the free algebra over a vector space V is the (unique up to isomorphism) algebra F equipped with a linear morphism i : V → F such that for any algebra A and any linear map f : V → A, there exists

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2 Coalgebras, Duality

a unique extension of f to an algebra map f  from F to A ( f  ◦ i = f ). The most classical realization  ⊗n of F is the tensor algebra over V , which is the tensor gebra V equipped with the concatenation product T (V ) := n∈N

v1 . . . vn · w1 . . . wm := v1 . . . vn w1 . . . wm . The unit map of the tensor algebra is the inclusion k = V ⊗0 ⊂ T (V ). Setting f  (v1 . . . vn ) : = f (v1 ) . . . f (vn ), one gets the adjunction Lin(V, A) ∼ = Alg (T (V ), A). Dually to the algebra case, the cofree coalgebra functor is right adjoint to the forgetful functor from coalgebras to vector spaces. That is, the cofree coalgebra over a vector space V is the (unique up to isomorphism) coalgebra C equipped with a linear morphism j : C → V such that for any coalgebra D and any linear map g : D → V , there exists a unique lift of g to a coalgebra map g  from D to C ( j ◦ g  = g):

- g ∃!

g D - V 6 j C. When V is finite dimensional, a simple realization of C results from the construction of the free algebra. Indeed, from the following sequence of isomorphisms (the first uses V = (V ∗ )∗ ): Lin(D, V ) ∼ = Alg(T (V ∗ ), D ∗ ) ∼ = Cog(D, T (V ∗ )0 ), = Lin(V ∗ , D ∗ ) ∼ we get that T (V ∗ )0 , the coalgebra of representative linear forms on T (V ∗ ), is a cofree coalgebra over V . For a given morphism g : D → V , the corresponding morphism g  from D to T (V ∗ )0 is given by g  (d) := (ε(d), g(d), g ⊗2 ◦ (d), . . . , g ⊗n ◦ n (d), . . . ), where we identify V ⊗n with ((V ∗ )⊗n )∗ and T (V ∗ )∗ with



Tn (V ).

n≥0

We will see later that the construction of cofree coalgebras simplifies on various subcategories of Cog (e.g., when considering connected graded coalgebras).

2.13 Graded and Conilpotent Coalgebras Very often coalgebras arising in combinatorics, representation theory, Lie theory and most of the time those appearing in application domains such as physics, dynamical systems, or control come equipped with extra structures making their study much easier. The existence of a grading is the simplest and the most frequent. The conilpo-

2.13 Graded and Conilpotent Coalgebras

37

tency property is more general and often carried, in applications, by coalgebras that are not graded. Recall  that a graded vector space is a vector space decomposing as a directgsum Vn . It is said to be reduced if V0 = 0. The corresponding category Lin has V = n∈N  fn , for morphisms linear morphisms f : V → W that respect the grading ( f = n∈N

f n : Vn → Wn ). Thecategory Ling is a symmetric tensor  category for the tensor (V ⊗ W )n , where (V ⊗ W )n := V p ⊗ Wq . The ground product: V ⊗ W = p+q=n

n∈N

field k identifies with the graded vector space, still written k, with only one nonzero component, k0 = k. It is the unit of the tensor product (V ⊗ k = V = V ⊗ k). Definition 2.13.1 A graded algebra (resp., graded coalgebra) is an algebra (resp., coalgebra) in the tensor category Ling . That is, in other terms, graded coalgebras and other structures are defined as usual except for the fact that the structure morphism, for example, the coproduct  of a graded coalgebra C, has to be a morphism of graded vector spaces, so that  : Cn →



C p ⊗ Cq .

p+q=n

The only point that deserves to be noticed here is that unit and counit maps have to be null maps except in degree 0, due to the representation of the ground field as a graded vector space with all nonzero degree components equal to the null vector space. Definition 2.13.2 A graded coalgebra C is connected if C0 = k. It is automatically coaugmented by the inclusion k = C0 → C. Similarly a graded algebra A is connected if A0 = k and it is augmented by the canonical projection on A0 . The graded coalgebra C (resp., the graded algebra A) is said to be locally finite dimensional if all the Cn (resp., An ) are finite-dimensional vector spaces. Many notions of the theory of coalgebras simplify in the connected and/or locally finite-dimensional graded case, which is frequently encountered. For example, the graded dual, still written V ∗ , of a graded vector space V (the dual of V in the category  ∗ Vn Ling ) is simply the direct sum of the duals of the graded components: V ∗ = n∈N

and is locally finite dimensional when V is. Exercise 2.13.1 Show that duality defines an anti-equivalence of categories between the category of locally finite-dimensional graded algebras and the category of locally finite-dimensional graded coalgebras. Exercise 2.13.2 Show that, if C is a graded connected coalgebra, (C) = {1}. Exercise 2.13.3 (Cofree graded coalgebras) Let V be a reduced graded vector space. A cofree graded coalgebra over V is a connected graded coalgebra C

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equipped with a morphism π : C → V in Ling such that, for each connected graded coalgebra D, any morphism φ from D to V in Ling lifts uniquely to a morphism  of graded coalgebras from D to C. • Show that (as usual for free and cofree objects) any two cofree graded coalgebras over V are isomorphic (by a unique isomorphism). This justifies to call (slightly abusively) any cofree graded coalgebra over V , “the” cofree graded coalgebra over V .  ⊗n V , the tensor gebra over V , equipped with • Show that Coalg(V ) := k ⊕ n∈N∗

the deconcatenation coproduct (v1 . . . vn ) :=

n 

v1 . . . vi ⊗ vi+1 . . . vn , together

i=0

with the projection to V induced by the direct sum decomposition is a cofree graded coalgebra over V . Show also that, with the previous notation, the morphism  ¯ n , where  ¯ n stands for the iterated reduced coproduct from D to V ⊗n is φ ⊗n ◦  (Def. 2.3.5). Let now (C, , ε) be a coaugmented coalgebra with coaugmentation η : k → C. ¯ where n+1 stands for the iterated reduced coproduct Let Fn := K er n+1 ⊂ C, ¯ n+2 = ( ¯ ⊗ I d ⊗n ¯ n+1 , Fn ⊂ Fn+1 . )◦ acting on C. Notice that since  C¯ Definition 2.13.3 The coproduct  and the coaugmented coalgebra C are called conilpotent if C¯ = Fn , that is, if for every c ∈ C¯ there exists an integer n such n

that n (c) = 0. When the coalgebra C is graded and connected, the coproduct is automatically conilpotent. Indeed, for degree reasons, n , n ≥ 2, vanishes on Ck for 0 ≤ k ≤ n − 1. Exercise 2.13.4 (Cofree cocommutative graded coalgebras) Show that, in the category of connected graded cocommutative coalgebras over a field of characteristic 0, the coalgebra (T S(V ), d ) is a cofree cocommutative coalgebra over the reduced graded vector space V . Remark 2.13.1 The results in Exercises 2.13.3 and 2.13.4 also hold in the categories of conilpotent and conilpotent cocommutative coalgebras. In that setting, V is a vector space and one considers the extension of morphisms φ from D to V to morphisms of conilpotent coalgebras (resp., conilpotent cocommutative coalgebras) to C. These properties can be understood from the dual point of view. We consider, for example, the cocommutative case. Polynomial algebras k[V ] are free augmented commutative algebras. That is, for an arbitrary augmented commutative algebra A, Lin(V, A+ ) ∼ = Alca (k[V ], A). Here, + is used to denote the augmentation ideal and Alca the category of augmented commutative algebras. By duality, for an arbitrary conilpotent cocommutative coalgebra C = k ⊕ C¯ there is a natural bijection (an adjunction) ¯ V) ∼ Lin(C, = Cocc (C, T S(V )),

2.13 Graded and Conilpotent Coalgebras

39

where Cocc denotes the category of conilpotent cocommutative coalgebras. In particular, a conilpotent cocommutative coalgebra morphism to T S(V ) is entirely characterized by its corestriction to V . Concretely, the inverse bijection is obtained by dualizing the isomorphism ¯ V ), the corresponding element in Lin(V, A+ ) ∼ = Alca (k[V ], A): for f ∈ Lin(C, Cocc (C, T S(V )) is given on k ⊕ C by I dk ⊕



f ⊗n ◦ n ,

(2.5)

n∈N∗

¯ ⊗n ) Sn ⊂ (C) ¯ ⊗n . The where n is the iterated reduced coproduct from C¯ to ((C) conilpotency hypothesis insures that this map is well defined. Exercise 2.13.5 With the same notation as above, use Exercise 2.4.1 to show that (k[V ],  p ) is a cofree coalgebra in the categories of connected graded cocommutative and of conilpotent cocommutative coalgebras over a field of characteristic 0. ¯ V ). Compute the element in Cocc (C, k[V ]) associated to f ∈ Lin(C,

2.14 Bibliographical Indications The theory of coalgebras originates with the study of duality in algebraic topology and of algebraic groups and is closely connected to the theory of Hopf algebras; about the latter, we refer to the bibliographical indications in the next chapter. Indeed, the idea underlying coalgebras probably was born with Hopf’s fundamental insight that one should use duality to investigate the algebra structures on the homology and cohomology of topological spaces equipped with a product [Hop]. In his work, this product was not necessarily associative, and the fact that the algebras arising in homology and cohomology are graded simplified the investigation of duality properties. The theory developed with the study of algebraic groups and associated Lie algebra structures. Working with ungraded coalgebras and/or over a field of finite characteristic made the theory more complex. Coalgebras and comodules were introduced by Cartier in this context [Car]. Still in the 50s, the links between group representations and representative functions were investigated systematically by Hochschild and Mostow in a series of articles starting with [HM], see also [Hoc]. Applications developed steadily and were extended progressively to a wide variety of fields: combinatorics, the theory of fields... more recently quantum groups and quantum field theory. The theory of coalgebras contains, of course, many developments that go beyond the scope of this book and have therefore not been addressed in this chapter. A detailed exposition of the history and theory of coalgebras is contained in [Mic], to which we refer for details and further insights. The extension of the mechanism of representative functions to algebra, coalgebra, and Hopf algebra endomorphisms together with the definition of pseudo-coproducts

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that dualizes to the one of quasi-coproducts is less classical than the other topics in this chapter. It was introduced in the joint works of Patras and Reutenauer on Lie idempotents viewed as linear endomorphisms of Hopf algebras [PR].

References [Car] Cartier, P.: Hyperalgèbres et groupes de Lie formels, Séminaire “Sophus Lie" 2e année: 1955/56 (1957) [Hoc] Hochschild, G.: The Structure of Lie Groups. Holden-Day Inc., San Francisco (1965) [HM] Hochschild, G., Mostow, G.D.: Representations and representative functions of Lie groups. Ann. Math. 495–542 (1957) [Hop] Hopf, H.: Über die Topologie der Gruppen-Mannifaltigkeiten und ihrer Verallgemeinerungen. Ann. Math. 42, 22–52 (1941) [Mic] Michaelis, W.: Coassociative coalgebras. In: Hazewinkel, M. (ed.) Handbook of Algebra vol. 3, pp. 587–788 (2003) [PR] Patras, F., Reutenauer, Ch.: On Dynkin and Klyachko idempotents in graded bialgebras. Adv. Appl. Math. 28(3), 560–579 (2002)

Chapter 3

Hopf Algebras and Groups

In this chapter, we introduce bialgebras and Hopf algebras, as well as various fundamental related notions and constructions. We discuss briefly their relations to algebraic groups and enveloping algebras that will be detailed later in this book. Many examples of Hopf algebras and of their application domains will be introduced later. For the time being, building on the previous chapter, we can mention as a first motivation for their introduction the phenomena already discussed, whose understanding will be deepened through the introduction of Hopf algebra structures: group theory, representative functions, and characteristic and representative endomorphims. Concretely, the notions of bialgebras and Hopf algebras result from the interplay of algebra and coalgebra structures. Finite groups provide a simple but fundamental example of these phenomena. Recall that for a finite group G, kG and k G are simultaneously algebras and coalgebras and are, as such, in duality. Since k G has finite dimension, its elements are representative functions and the coalgebra k G identifies with the coalgebra generated by coefficients of finite-dimensional representations of G. The group G can be recovered from the identifications (Exercise 2.3.1 and Lemma 2.3.1) G = (kG) ∼ = Alg(k G , k). When G is infinite, and when, together with its representations, it carries extra structures (continuity or compactness properties, for example), this simple correspondence between groups and representative functions breaks down. Understanding why and how such a correspondence can be restored is one of the historical reasons, together with algebraic topology, for the introduction of Hopf algebras and bialgebras. In the continuous case, new phenomena show up, such as the classical relations between Lie groups and Lie algebras that are obtained through the logarithm and exponential maps. Less familiar maps also appear in relation to Lie idempotents

© Springer Nature Switzerland AG 2021 P. Cartier and F. Patras, Classical Hopf Algebras and Their Applications, Algebra and Applications 29, https://doi.org/10.1007/978-3-030-77845-3_3

41

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such as Dynkin’s. These phenomena also are often best understood through the use of Hopf algebras. Let us mention at last that there are various ways to generalize naturally the notion of Hopf algebras; simple and, however, important examples will be given from Sect. 3.8 onward.

3.1 Bialgebras, Hopf Algebras Definition 3.1.1 A bialgebra is a 5-tuple (B, m, η, , ε) such that: • (B, m, η) is an algebra (associative, with unit η : k → B); • (B, , ε) is a coalgebra (coassociative, with counit ε : B → k); • The coproduct  and the counit ε are morphisms of algebras, or, equivalently, the product m and the unit η are morphisms of coalgebras. The equivalence in the last item of the definition is best expressed diagrammatically: the two equivalent conditions translate into the commutativity of the diagrams B⊗B ⊗ ? B ⊗4 m B⊗B - B

m

- B Id ⊗ T ⊗ Id k

 - B⊗B m⊗m 6

η B

- B ⊗4 , k

ηB

ε⊗ε ? ? =? ε? ε? ∼ =? η ⊗ η = - B⊗B k⊗k ∼ k⊗k k - k, = k where for a, b ∈ B, T (a ⊗ b) := b ⊗ a. The first and second (resp., third and fourth) express that m (resp., η) is a morphism of coalgebras, whereas the first and third (resp., second and fourth) express that  (resp., ε) is a morphism of algebras. Notice that these diagrams are self-dual: they are left invariant when inverting the sense of arrows and exchanging m and , resp., ε and η. We set ν := η ◦ ε η ε (ν : B - k - B). The unit is a coaugmentation for (B, , ε) and the counit an augmentation for (B, m, η). A morphism of bialgebras is a linear map that commutes with the structure maps: that is, it is an algebra morphism and a coalgebra morphism. Remark 3.1.1 (Graded bialgebra) A bialgebra B is graded if it is graded as an algebra and as a coalgebra. The unit η maps k to B0 , the degree 0 component of B, and ε is the null map on the Bi , i > 0. It is connected if B0 ∼ = k. In a graded connected bialgebra B, the unit and the counit are the inclusion k = B0 → B and the canonical projection from B to B0 .

3.1 Bialgebras, Hopf Algebras

43

A fundamental feature of the theory of bialgebras and Hopf algebras is its close connection to the theory of groups, but also to the one of Lie algebras. The latter is grounded in the following lemma. Lemma 3.1.1 The vector space of primitive elements of a bialgebra B, written Prim(B), is a Lie subalgebra of B for the Lie bracket [x, y] = x y − yx. Proof Indeed, since  is an algebra map, we have for x, y ∈ Prim(B), ([x, y]) = (x)(y) − (y)(x) = (x ⊗ 1 + 1 ⊗ x)(y ⊗ 1 + 1 ⊗ y) − (y ⊗ 1 + 1 ⊗ y)(x ⊗ 1 + 1 ⊗ x) = (x y − yx) ⊗ 1 + 1 ⊗ (x y − yx) = [x, y] ⊗ 1 + 1 ⊗ [x, y]. Recall from Section 2.12 that, if C is a coalgebra and A an algebra, Lin(C, A) is an algebra for the convolution product ∗. In particular, the vector space of linear endomorphisms EndLin (B) of a bialgebra B is an algebra for the convolution product, with unit ν = η ◦ ε. Definition 3.1.2 A Hopf algebra is a bialgebra H such that I d has a left and right inverse S, called the antipode, for the convolution product: for all h ∈ H , S(h (1) )h (2) = ν(h) = h (1) S(h (2) ). Notice that if I d has a left inverse S  and a right inverse S  , then the two agree (and define the antipode) since S  = S  ∗ ν = S  ∗ I d ∗ S  = ν ∗ S  = S  . The same argument implies the uniqueness of the antipode, when it exists. Exercise 3.1.1 (Naturality of the antipode) Show that if H and H  are two Hopf algebras with antipodes S and S  , and φ : H → H  is a morphism of bialgebras, then it is a Hopf algebra map, that is, it also holds that S  ◦ φ = φ ◦ S. Proposition 3.1.1 We have, in a Hopf algebra H ,  ◦ S = (S ⊗ S) ◦ T ◦ , ε ◦ S = ε and dually S ◦ m = m ◦ T ◦ (S ⊗ S), S ◦ η = η. Equivalently, on elements h, l of H : (S(h)) = S(h (2) ) ⊗ S(h (1) ), ε(S(h)) = ε(h), S(h · l) = S(l) · S(h), S(1) = 1.

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In other terms, S maps H to the opposite algebra and the opposite coalgebra. Proof Let us prove, for example, the identity S(h · l) = S(l) · S(h), the others are left as exercises. For l, h ∈ H , using the properties of convolution and iterated coproducts and the fact that ν(h) = h (1) S(h (2) ), h = h (1) ν(h (2) ), S(h) = S(h (1) )ν(h (2) ), and ν(hl) = ν(h)ν(l) (recall ν = η ◦ ε, so that the image of ν is central in H ), we have S(hl) = S((hl)(1) )ν(h (2) )ν(l (2) ) = S(h (1)l (1) )h (2) ν(l (2) )S(h (3) ) = S(h (1) l (1) )h (2)l (2) S(l (3) )S(h (3) ) = ν((hl)(1) )S(l (2) )S(h (2) ) = ν(l (1) )S(l (2) )ν(h (1) )S(h (2) ) = S(l)S(h). Lemma 3.1.2 If H is commutative or cocommutative, S 2 = I d. Proof Assume that H is commutative, we have, using the previous lemma, for h ∈ H (S 2 ∗ S)(h) = S 2 (h (1) )S(h (2) ) = S(h (2) S(h (1) )) = S(S(h (1) )h (2) ) = S ◦ ν(h) = ν(h),

so that S 2 is a convolution inverse of S and so S 2 = I d. When H is cocommutative, use instead that S(h (2) S(h (1) )) = S(h (1) S(h (2) )). The argument (and the property) breaks down if H is neither commutative nor cocommutative. The following theorem is the analog for bialgebras and Hopf algebras of the fundamental theorem for coalgebras (Thm. 2.6.1). Theorem 3.1.1 Let H be a bialgebra (resp., Hopf algebra). It is equal to the union of its finitely generated subbialgebras (resp., sub-Hopf algebras). Proof Here, finitely generated means finitely generated as an algebra. Indeed, by the fundamental theorem for coalgebras, any h ∈ H is contained in a finite-dimensional subcoalgebra of H . The subalgebra generated by this subcoalgebra is a subbialgebra since  is an algebra homomorphism. The theorem follows. In the Hopf algebra case, one has to take the action of the antipode into account; the same argument holds mutatis mutandis using that it is linear and an antihomomorphism of algebras and coalgebras (Lemma 3.1.1). Example 3.1.1 (Polynomial Hopf algebras) Let V be a vector space. Recall that we write k[V ] for the algebra of polynomials over V . By universal properties of polynomials, a linear map f : V → A to an arbitrary commutative algebra A extends uniquely to an algebra map from k[V ] to A. In particular, the map V → k[V ] ⊗ k[V ] v −→ v ⊗ 1 + 1 ⊗ v

3.1 Bialgebras, Hopf Algebras

45

extends uniquely to an algebra map  k[V ] - k[V ] ⊗ k[V ].

This coproduct together with the product of polynomials equips the polynomial gebra k[V ] with a graded connected commutative and cocommutative Hopf algebra structure. The graduation is the usual one for polynomials. As such k[V ] will be called the polynomial Hopf algebra over V . Since for v ∈ V (v) = v ⊗ 1 + 1 ⊗ v, the action of the antipode is given on V by S(v) = −v and, in general, by S(v1 . . . vn ) = (−1)n v1 . . . vn . We let the reader check that  =  p , the polynomial coproduct: 

(v1 . . . vn ) = I



vI ⊗ v J ,

J =[n]

where for S = {s1 , . . . , sk } ⊂ N and s1 < · · · < sk , v S := vs1 . . . vsk . Let us conclude this section with another elementary but fundamental example: Hopf algebras of tensors.  n Example 3.1.2 (Tensor Hopf algebra) The tensor gebra T (V ) = T (V ) = n∈N  ⊗n V over a vector space V , equipped with the concatenation product n∈N

v1 . . . vn · vn+1 . . . vn+m := v1 . . . vn+m and the unshuffle coproduct u (Def. 2.4.2), is a graded connected Hopf algebra. Since for v ∈ V (v) = v ⊗ 1 + 1 ⊗ v, the antipode, being an antihomomorphism of algebras, is given on V by S(v) = −v and in general by S(v1 . . . vn ) := (−1)n vn . . . v1 . That u , the unshuffle coproduct, is an algebra morphism, follows from the following construction. Let us consider the map δu from V to T (V ) ⊗ T (V ) defined by δu (v) := v ⊗ 1 + 1 ⊗ v. Since T (V ) is the free associative algebra over V , δu extends uniquely to an algebra map  from T (V ) to T (V ) ⊗ T (V ) ((v1 . . . vn ) := (v1 ) . . . (vn )), where T (V ) ⊗ T (V ) is equipped with the product (a ⊗ b) · (c ⊗ d) = ac ⊗ bd. The reader will check that this construction provides an alternative definition of the unshuffle coproduct:  = u .

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Example 3.1.3 (Shuffle tensor Hopf algebra) Let us define the shuffle product on T (V ) recursively by 1 1 := 1, 1 v = v 1 := v and v1 . . . vn

w1 . . . wm := v1 (v2 . . . vn

For example, v1 v1 v2

w1 . . . wm ) + w1 (v1 . . . vn

w2 . . . wm ).

w1 = v1 w1 + w1 v1 , w1 = v1 (v2

w1 ) + w1 v1 v2 = v1 v2 w1 + v1 w1 v2 + w1 v1 v2 .

follows from the closed formula for

The commutativity of

v1 . . . vn

vn+1 . . . vn+m :=



:

vT ,

(3.1)

T ∈T

where T stands for the set of all total orders on [n + m] that extend the order1 1 ≤ · · · ≤ n; n + 1 ≤ · · · ≤ n + m, and where, given an ordered sequence I of integers i 1 · · · i k (not necessarily in the natural ordering), we write v I for vi1 . . . vik (for example, if I is the ordered sequence 2 1 3, v I = v2 v1 v3 ). Equivalently, v1 . . . vn

vn+1 . . . vn+m =



vσ (1) . . . vσ (n+m) ,

σ ∈Sh n,m

where the sum runs over the set Sh n,m of (n, m)-shuffles.2 The associativity of follows by observing that (v1 . . . vn

vn+1 . . . vn+m )

vn+m+1 . . . vn+m+k =



vT ,

T ∈T 

and that the same identity holds for v1 . . . vn (vn+1 . . . vn+m vn+m+1 . . . vn+m+k ), where T  stands for the set of total orders on [n + m + k] that extend the order 1 ≤ · · · ≤ n; n + 1 ≤ · · · ≤ n + m; n + m + 1 ≤ · · · ≤ n + m + k. The triple (T (V ), , d ), where d stands for the deconcatenation coproduct d (v1 . . . vn ) =

n 

v1 . . . vi ⊗ vi+1 . . . vn

i=0

is a connected graded commutative Hopf algebra called the shuffle tensor Hopf algebra. The antipode is given by S(v1 . . . vn ) = (−1)n vn . . . v1 . 1

Order means partial order; a set equipped with an order is a poset. A permutation σ in the symmetric group Sn+m is called a (n, m)-shuffle if and only if σ −1 (1) < · · · < σ −1 (n), σ −1 (n + 1) < · · · < σ −1 (n + m).

2

3.1 Bialgebras, Hopf Algebras

47

It is an interesting exercise to deduce this formula from an explicit computation using the recursive definition of the shuffle product in the formula S ∗ I d(v1 . . . vn ) = 0 for n ≥ 1. Remark 3.1.2 When V is finite dimensional, identifying T n (V ∗ ) with (T n (V ))∗ , one shows easily that the shuffle product on T (V ) and the unshuffle coproduct, resp., the concatenation product and deconcatenation coproduct are in duality in the category of locally finite-dimensional graded vector spaces. The tensor gebra T (V ), equipped with the shuffle product and the deconcatenation coproduct, is therefore a Hopf algebra, dual (in the graded sense) to the tensor Hopf algebra T (V ∗ ). This duality property can be used to deduce abstractly that the tensor and shuffle tensor Hopf algebras have the same antipode. Notice that even when V is not finite dimensional, the shuffle tensor Hopf algebra T (V ) is a Hopf algebra, the union of the T (W ) where W runs over the finitedimensional subspaces of V . Example 3.1.4 (Restricted dual Hopf algebra) If B is a bialgebra (resp., H is a Hopf algebra), the restricted dual B ◦ (resp., H ◦ ) of the algebra B (resp., H ) is naturally a bialgebra (resp., a Hopf algebra). The product is inherited from the canonical product on the dual B ∗ of the coalgebra, the antipode on H ◦ from the antipode on H by duality.

3.2 Modules and Comodules An important property of bialgebras and Hopf algebras, especially in view of their applications to group theory, is the behavior of their categories of modules and comodules. In the rest of this section, let again B be a bialgebra. A module (i.e., with our conventions, a left module) over a bialgebra B or B-module is simply a module over B viewed as an algebra. Similarly, a (right) comodule over B or B-comodule is simply a comodule over B viewed as a coalgebra. From the fact that  is a morphism of algebras ((ab) = (a)(b)), we get the proposition: Proposition 3.2.1 Let M, N two B-modules with structure maps m M , m N from B ⊗ M and B ⊗ N to M and N . Their tensor product M ⊗ N is equipped with a B-module structure by the map B⊗M⊗N

⊗I d M ⊗IdN

B⊗B⊗M⊗N

I d B ⊗T ⊗IdN

B⊗M⊗B⊗N

m M ⊗m -N

M ⊗ N.

In terms of elements, b · (m ⊗ n) = b(1) · m ⊗ b(2) · n. Equivalently, the tensor product of B-modules can be defined by observing that M ⊗ N is a B ⊗ B-module and that the coproduct  : B → B ⊗ B pulls back the category of B ⊗ B-modules to the category of B-modules.

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Remark 3.2.1 When B is cocommutative, M ⊗ N ∼ = N ⊗ M as B-modules since b · (n ⊗ m) = b(1) · n ⊗ b(2) · m = b(2) · n ⊗ b(1) · m. When H is a Hopf algebra and M a H -module, the latter carries a right module structure defined by m · h := S(h) · m or, in terms of arrows M⊗H

T -

H⊗M

S⊗I dM

H⊗M

mM

M.

Indeed, (m · h) · h  = S(h  ) · (S(h) · m) = S(hh  ) · m = m · hh  . Conversely a right module carries a left module structure defined by h · m := m.S(h). When S 2 = I d, which is the case for commutative or commutative Hopf algebras, the two constructions are inverse to each other. Due to the self-dual properties of the definition of bialgebras and Hopf algebras these properties dualize. The following construction has already been used to define the tensor product of two comodules over coalgebras of representative functions. Proposition 3.2.2 Let M, N two B-comodules with structure maps ∂ M , ∂ N from M and N to M ⊗ B and N ⊗ B. Their tensor product M ⊗ N is equipped with a B-comodule structure by the composition: M⊗N

∂ M ⊗∂ -N

M⊗B⊗N⊗B

I d M ⊗T ⊗IdB

M⊗N⊗B⊗B

I d M ⊗I d N ⊗m -B

M ⊗ N ⊗ B.

Remark 3.2.2 When B is commutative, the switch M ⊗ N ∼ = N ⊗ M is an isomorphism of B-comodules. When H a Hopf algebra, the antipode S allows to equip left-comodules with a right-comodule structure and conversely by the same process as for modules.

3.3 Characteristic Endomorphisms and the Dynkin Operator Let us consider now more in detail the convolution algebra End(B) of linear endomorphisms of a commutative or cocommutative bialgebra (B, m, η, , ε). We develop the example of cocommutative bialgebras, dual results would hold in the commutative case. Recall from Section 2.11 that f ∈ End(B) is a characteristic endomorphism of B if and only if there exists F = f (1) ⊗ f (2) ∈ End(B) ⊗ End(B) such that F ◦ = ◦ f; the map F is called a quasi-coproduct for f .

3.3 Characteristic Endomorphisms and the Dynkin Operator

49

Proposition 3.3.1 Let B be a cocommutative bialgebra and let us write Char (B) the subspace of characteristic endomorphisms in End(B). Then, Char (B) is a subalgebra of (End(B), ◦) and of (End(B), ∗). Moreover, if F and G are quasi-coproducts for f and g in End(B), then F ◦ G and F ∗ G are, respectively, quasi-coproducts for f ◦ g and f ∗ g, where ◦ and ∗ are extended to End(B) ⊗ End(B). Proof We have already proven the properties relative to the composition product; they depend only on B being a coalgebra. Let F = f (1) ⊗ f (2) and G = g (1) ⊗ g (2) be quasi-coproducts for f and g, then we have F ∗ G = ( f (1) ⊗ f (2) ) ∗ (g (1) ⊗ g (2) ) = ( f (1) ∗ g (1) ) ⊗ ( f (2) ∗ g (2) ) = (m ◦ ( f (1) ⊗ g (1) ) ◦ ) ⊗ (m ◦ ( f (2) ⊗ g (2) ) ◦ ) = (m ⊗ m) ◦ ( f (1) ⊗ g (1) ⊗ f (2) ⊗ g (2) ) ◦ ( ⊗ ) = (m ⊗ m) ◦ (I d ⊗ T ⊗ I d) ◦ ( f (1) ⊗ f (2) ⊗ g (1) ⊗ g (2) ) ◦ (I d ⊗ T ⊗ I d) ◦ ( ⊗ ),

where T (a ⊗ b) = (b ⊗ a). Thus F ∗ G = (m ⊗ m) ◦ (I d ⊗ T ⊗ I d) ◦ (F ⊗ G) ◦ (I d ⊗ T ⊗ I d) ◦ ( ⊗ ). Since B is cocommutative, we have (I d ⊗ T ⊗ I d) ◦ ( ⊗ ) ◦  = ( ⊗ ) ◦ . Thus (F ∗ G) ◦  = (m ⊗ m) ◦ (I d ⊗ T ⊗ I d) ◦ (F ⊗ G) ◦ ( ⊗ ) ◦  = (m ⊗ m) ◦ (I d ⊗ T ⊗ I d) ◦ ((F ◦ ) ⊗ (G ◦ )) ◦  = (m ⊗ m) ◦ (I d ⊗ T ⊗ I d) ◦ (( ◦ f ) ⊗ ( ◦ g)) ◦ , since f, g have quasi-coproducts F, G. Thus (F ∗ G) ◦  = (m ⊗ m) ◦ (I d ⊗ T ⊗ I d) ◦ ( ⊗ ) ◦ ( f ⊗ g) ◦  =  ◦ m ◦ ( f ⊗ g) ◦ ,

since  is an algebra morphism. Finally (F ∗ G) ◦  =  ◦ ( f ∗ g). Remark 3.3.1 The proposition, fundamental for the study of cocommutative bialgebras, dualizes as follows. Let B be a commutative bialgebra and let us write Rep(B) for the subalgebra of representative endomorphisms of (End(B), ◦)—recall that f is representative if there exists a F = f (1) ⊗ f (2) ∈ End(A) ⊗ End(A) (in the Sweedler notation) such that ∀a, b ∈ B, f (ab) = f (1) (a) f (2) (b).

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The element F is called a pseudo-coproduct for f . Proposition 3.3.2 The space of representative endomorphisms Rep(B) is also a subalgebra of (End(B), ∗). Moreover, if F is a pseudo-coproduct for f and G a pseudo-coproduct for g, F ∗ G is a pseudo-coproduct for f ∗ g. Proof Indeed, given a, b ∈ B and f, g ∈ Rep(B) with pseudo-coproducts F = f (1) ⊗ f (2) , G = g (1) ⊗ g (2) : f ∗ g(ab) = f (a (1) b(1) )g(a (2) b(2) ) = f (1) (a (1) )g (1) (a (2) ) f (2) (b(1) )g (2) (b(2) ) = f (1) ∗ g (1) (a) f (2) ∗ g (2) (b). Let us give a first application of the theory of characteristic endomorphisms: the study of the Dynkin operator. Lemma 3.3.1 Let H be a cocommutative Hopf algebra and D be a coderivation:  ◦ D = (D ⊗ I d + I d ⊗ D) ◦ . Then, S ∗ D is a map from H to Prim(H ). Proof By Lemma 3.1.1, S is group-like:  ◦ S = (S ⊗ S) ◦ . Therefore,  ◦ (S ∗ D) = ((S ⊗ S) ∗ (D ⊗ I d + I d ⊗ D)) ◦  = (S ∗ D ⊗ S ∗ I d + S ∗ I d ⊗ S ∗ D) ◦  = (S ∗ D ⊗ ν + ν ⊗ S ∗ D) ◦ . Thus, S ∗ D and maps H to Prim(H ) since  is counital. Definition 3.3.1 (Generalized Dynkin operator) Let H be a graded connected cocommutative Hopf algebra. Let D be the coderivation D(h) := n · h for h ∈ Hn . Then, Dyn := S ∗ D is called the Dynkin operator. Lemma 3.3.2 For x ∈ Prim(H ) and h ∈ H , Dyn(hx) = [Dyn(h), x] + ν(h) · D(x). Proof Indeed, since S(x) = −x,

3.3 Characteristic Endomorphisms and the Dynkin Operator

51

Dyn(hx) = S ∗ D(hx) = S(h (1) x)D(h (2) ) + S(h (1) )D(h (2) x) = −x S(h (1) )D(h (2) ) + S(h (1) )D(h (2) )x + S(h (1) )h (2) D(x) = [Dyn(h), x] + ν(h) · D(x). Proposition 3.3.3 The Dynkin operator maps H to Prim(H ) and satisfies Dyn ◦ Dyn = D ◦ Dyn. It is surjective when the ground field if of characteristic 0. Furthermore, for primitive elements h 1 , . . . , h n , Dyn(h 1 . . . h n ) = [. . . [[h 1 , h 2 ], h 3 ], . . . , h n ]. In particular, when H = T (V ), the tensor Hopf algebra, for v1 , . . . , vn ∈ V , Dyn(v1 . . . vn ) = [. . . [[v1 , v2 ], v3 ], . . . , vn ]. Proof By Lemma 3.3.1, the Dynkin operator maps H to Prim(H ). This map is surjective when the positive integers are invertible since, for h ∈ Prim(H ) ∩ Hn : S ∗ D(h) = S(h)D(1) + S(1)D(h) = D(h) = n · h. Since the image of Dyn is in Prim(H ), the same calculation shows that Dyn ◦ Dyn = D ◦ Dyn. The last statements follow from Lemma 3.3.2.

3.4 Hopf Algebras and Groups Before entering further the subject of relations between Hopf algebras and groups in Chapter 7, let us mention the simplest ones. Recall that products of representative functions are representative. Example 3.4.1 (Monoid bialgebra) Let M be a monoid. The monoid algebra k M equipped with the coproduct (g) := g ⊗ g for g in M is a bialgebra. Example 3.4.2 (Group Hopf algebra) Let G be a group. The group algebra kG equipped with the coproduct (g) := g ⊗ g for g in G is a Hopf algebra. The action of the antipode S on elements of G is given by S(g) := g −1 . Indeed, setting S(g) := g −1 , we get S ∗ I d(g) = g −1 g = 1 = ν(g). Example 3.4.3 (Hopf algebra of representative functions) The algebra R(M) (resp., R(G)) of representative functions on a monoid M (resp., a group G) is a bialgebra (resp., Hopf algebra). It identifies with the restricted dual of the monoid bialgebra

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3 Hopf Algebras and Groups

(resp., group Hopf algebra). The action of the antipode on f ∈ R(G) is given by f i (x) f i (y), for g ∈ G, S( f )(g) = f (g −1 ). Indeed, if f (x y) = i∈I



f i (g −1 ) f i (g) = f (1) = ν( f )(g).

i∈I

Lemma 3.4.1 Let H be a Hopf algebra (resp., bialgebra), then (H ), the set of group-like elements in H , is a group (resp., monoid). Indeed, for h, l ∈ (H ), we get first (hl) = (h)(l) = (h ⊗ h)(l ⊗ l) = hl ⊗ hl, so that (H ) is a monoid (with neutral element 1) whenever H is a bialgebra. When H has an antipode, we get S(h)h = (S ∗ I d)(h) = ν(h) = 1, and (S(h)) = (S ⊗ S) ◦ + ◦ (h) = (S ⊗ S)(h ⊗ h) = S(h) ⊗ S(h), so that S(h) ∈ (H ). Exercise 3.4.1 Show that the functor H −→ (H ) from Hopf algebras to groups is right adjoint to the group algebra functor G −→ kG. Recall now that, since a Hopf algebra H is, in particular, a coalgebra, H ∗ = Lin(H, k) is an algebra for the convolution product ∗ of linear forms, with unit the counit ε of H . A character of a Hopf algebra H is a character of H as an algebra: an algebra map to the ground field. Lemma 3.4.2 (Group of characters) The subset G(H ) := Alg(H, k) of H ∗ of characters of a Hopf algebra (resp., bialgebra) H is a group (resp., monoid) under the convolution product ∗. More generally, for A a commutative algebra over k, let us call A-valued characters of H the elements of Alg(H, A) =: G(H, A). Then, G(H, ) is a functor from Com to Grp (resp., to Mon)—from commutative algebras to groups (resp., monoids). The convolution inverse of an element φ is obtained by right composition with the antipode: φ −1 = φ ◦ S. Proof Indeed, first, ε ∈ G(H ). The set G(H ) is stable under ∗ since the coproduct of H is a map of algebras from H to H ⊗ H . Similarly, G(H, A) is stable under ∗ since the product m A : A ⊗ A → A is also a morphism of algebras since A is commutative. When H is a Hopf algebra, we have furthermore, for φ ∈ G(H ), that φ ◦ S ∈ G(H ) since S is an algebra antihomomorphism of H . Moreover, since φ is an algebra morphism, (φ ◦ S) ∗ φ = φ ◦ (S ∗ I d) = φ ◦ ν = . The lemma follows. Recall that f ∈ H ∗ (resp., f ∈ Lin(H, A), where A is a commutative algebra over the same ground field) is an infinitesimal character if and only if f is null on

3.4 Hopf Algebras and Groups

53

the scalars and on the square of the augmentation ideal of H , that is, f (H ) = 0 and for h, h  ∈ H , f (hh  ) = f (h)ε(h  ) + ε(h) f (h  ). Lemma 3.4.3 The subset I(H ) of H ∗ (resp., I(H, A) of Lin(H, A)) of infinitesimal characters of a bialgebra is a Lie algebra for the bracket [ f, g]∗ := f ∗ g − g ∗ f. Proof We have to show that [ f, g]∗ ∈ I(H ). Indeed, [ f, g]∗ (H ) = 0 and f ∗ g(hh  ) = f (h (1) h (1) )g(h (2) h (2) ) = ( f (h )ε(h ) + ε(h (1) ) f (h (1) ))(g(h (2) )ε(h (2) ) + g(h (2) )ε(h (2) )) = f (h (1) )g(h (2) )ε(h (1) )ε(h (2) ) + ε(h (1) )g(h (2) ) f (h (1) )ε(h (2) ) + f (h (1) )ε(h (2) )ε(h (1) )g(h (2) ) + ε(h (1) )ε(h (2) ) f (h (1) )g(h (2) ) = f ∗ g(h)ε(h  ) + ε(h) f ∗ g(h  ) + f (h)g(h  ) + g(h) f (h  ), (1)

so that

(1)

f ∗ g − g ∗ f (hh  ) = [ f, g]∗ (h)ε(h  ) + ε(h)[ f, g]∗ (h  ).

Notice that the same calculation could be performed using the formalism of pseudo-coproducts. Indeed, assume that for f, g ∈ Lin(H, A) there exist F = f (1) ⊗ f (2) and G = g (1) ⊗ g (2) in Lin(H, A) ⊗ Lin(H, A) (in the Sweedler notation) such that F(h ⊗ h  ) = f (hh  ), G(h ⊗ h  ) = g(hh  ), then f ∗ g(hh  ) = f (h (1) h (1) )g(h (2) h (2) ) = f (1) (h (1) )g (1) (h (2) ) f (2) (h (1) )g (2) (h (2) ) = f (1) ∗ g (1) (h) f (2) ∗ g (2) (h  ) or

f ∗ g(hh  ) = F ∗ G(h ⊗ h  ).

When f and g are infinitesimal, we have ( f ⊗ ε + ε ⊗ f ) ∗ (g ⊗ ε + ε ⊗ g) = f ∗ g ⊗ ε + f ⊗ g + g ⊗ f + ε ⊗ f ∗ g and recover f ∗ g(hh  ) = f ∗ g(h)ε(h  ) + ε(h) f ∗ g(h  ) + f (h)g(h  ) + g(h) f (h  ). Assume now that the ground field is of characteristic 0 and that H is conilpotent, that is, for any x in H + the augmentation ideal of H , there exists n ∈ N∗ such that m  (x) = 0 for m ≥ n. Then, given f ∈ Lin(H, A) such that f ◦ ν = 0 and x ∈ H + ,

54

3 Hopf Algebras and Groups ⊗m ⊗m f ∗m (x) = m [m] ◦ [m] (x) = m [m] ◦ A ◦ f A ⊗ f

[m]

(x) = 0

for m large enough. In particular, exp∗ f is well defined and, conversely if f = ε + g with g ◦ ν = 0, log∗ ( f ) is well defined. Corollary 3.4.1 If H is a conilpotent bialgebra over a field of characteristic 0 (e.g., if H is graded and connected), the group of characters and the Lie algebra of infinitesimal characters are in bijection: log∗ : G(H, A)  I(H, A) : exp∗ . Proof Indeed, let f ∈ I(H, A), then exp∗ ( f )(hh  ) = exp∗ ( f ⊗ ε + ε ⊗ f )(h ⊗ h  ) = (exp ( f ) ⊗ ε) ∗ (ε ⊗ exp∗ ( f ))(h ⊗ h  ) = (exp∗ ( f ) ⊗ exp∗ ( f ))(h ⊗ h  ), ∗

and similarly for the inverse bijection: given g ∈ G(H, A), log∗ (g)(hh  ) = log∗ (g ⊗ g)(h ⊗ h  ) = log∗ ((g ⊗ ε) ∗ (ε ⊗ g))(h ⊗ h  ) = (log∗ (g) ⊗ ε + ε ⊗ log∗ (g))(h ⊗ h  ).

3.5 Algebraic Groups A Hopf algebra is mapped to k G(H ) by h −→ ( f h : φ −→ φ(h)) in such a way that f hh  = f h · f h  . Moreover, let φ, ψ ∈ G(H ) and h ∈ H : f h (φ ∗ ψ) = φ ∗ ψ(h) = φ(h (1) )ψ(h (2) ), and f h is a representative function on G(H ). There is thus a natural Hopf algebra map from H to the Hopf algebra of representative functions R(G(H )) on G(H ). Definition 3.5.1 (Affine algebraic group) A pro-affine algebraic group is the group G(H ) of characters of a commutative Hopf algebra H . The group is called affine if H is finitely generated. Notice that the Hopf algebra H is part of the data defining a pro-affine algebraic group. Since a Hopf algebra is the union of its finitely generated Hopf subalgebras, a pro-affine algebraic group is always an inverse (projective) limit of affine algebraic group.

3.5 Algebraic Groups

55

Example 3.5.1 The additive group Ga (k) (of the affine line k) is the group of characters of the Hopf algebra k[X ], (X ) := X ⊗ 1 + 1 ⊗ X, S(X ) = −X, ε(X ) = 0. Example 3.5.2 The multiplicative group Gm (k) is the group of characters of the Hopf algebra k[X, X −1 ], (X ) := X ⊗ X, S(X ) = X −1 , ε(X ) = 1. Example 3.5.3 The general linear group G L n (k) is the group of characters of the Hopf algebra H = k[Ti j , det (T )−1 ]1≤i, j≤n , where T stands for the matrix (Ti j )1≤i, j,≤n and (Ti j ) :=

n 

Tik ⊗ Tk j , S(Ti j ) = det (T )−1 A ji , ε(Ti j ) = δi, j ,

k=1

where A ji is the cofactor of T ji in the matrix T . Example 3.5.4 (Linear algebraic groups) Let G be a subgroup of the general linear group G L(d, k). We say that G is a linear algebraic group if there exists a family (Pα ) of polynomials in d 2 variables xi j with coefficients in k such that a matrix g = (gi j )1≤i, j≤d belongs to G iff the equations Pα (. . . gi j . . . ) = 0 hold. The coordinate ring O(G) of G consists of rational functions on G regular at every point of G, namely, the functions of the form u(g) = Q(. . . gi j . . . )/(det g) N , where Q is a polynomial and N ≥ 0 an integer. The multiplication rule det (gg  ) = det (g)det (g  ) implies that such a function u is in the algebra of representative functions R(G) and Cramer’s rule for the inversion of matrices implies that S(u) is in O(G) for any u in O(G). Therefore, • The algebra of regular functions O(G) is a sub-Hopf algebra of R(G), • The group G is the spectrum (the set of characters) of O(G), G = G(O(G)), • The algebra O(G) is finitely generated (by the coordinate functions gi j and the inverse of the determinant), and G is an affine algebraic group. Remark 3.5.1 There is another point of view on algebraic groups, the one of group schemes: a commutative Hopf algebra H and the representable functor G(H, ) : Com → Grp. The functor G(H, ) is called an affine group scheme when H is finitely generated.

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3 Hopf Algebras and Groups

3.6 Unipotent and Pro-unipotent Groups In this section, let us assume that the ground field k is of characteristic 0 in order to use freely logarithms and exponentials. Among the various families of algebraic groups, one is particularly important in applications of the theory of Hopf algebras: the one of unipotent groups. Pro-unipotent groups (inverse limits of unipotent groups) appear naturally when one considers graded Hopf algebras for example. Let us consider first the group Tn (k) of strict upper triangular matrices g = (gi j )1≤i, j≤n with entries in k, where gii = 1, gi j = 0 for i > j. The corresponding Lie algebra tn (k) consists of matrices x = (xi j )1≤i, j≤n with xi j = 0 for i ≥ j. The product of n matrices in tn (k) is always 0, and Tn (k) is the set of matrices In + x, with In the identity matrix and x ∈ tn (k). We get inverse maps log : Tn (k)  tn (k) : exp, where log and exp stand for the logarithm and exponential series truncated at order n, and are therefore polynomial maps. This isomorphism generalizes as follows. A finite-dimensional Lie algebra g is nilpotent if the adjoint map adx : y → [x, y] is nilpotent for any x ∈ g. According to theorems of Ado and Engel, g is always isomorphic to a Lie subalgebra of a tn (k), for some n ≥ 1. A linear algebraic group G is unipotent if all its elements are unipotent. By Kolchin’s theorem, in some basis all elements of G are strictly upper triangular.3 From the properties of triangular matrix groups and Lie algebras, one gets Proposition 3.6.1 For any unipotent linear algebraic group G with nilpotent Lie algebra g, the exponential map is an isomorphism of g with G as algebraic varieties. Representative functions in O(G) correspond to the polynomial functions on g, and O(G) is a polynomial algebra and a conilpotent coalgebra. Proof The conilpotency property follows from the unipotency of G: for f ∈ O(G) and g1 , g2 ∈ G, ( f )(g1 , g2 ) = f (g1 g2 ) and, when f (1) = 0, ¯ f )(g1 , g2 ) = f (g1 g2 ) − f (g1 ) − f (g2 ) =< f, (g1 − 1)(g2 − 1) > . ( In general, when f (1) = 0, one gets ¯ n ( f )(g1 , . . . , gn ) =< f, 

n 

(gi − 1) > .

i=1

3

The hypothesis that G is a matrix group is not restrictive in practice, see, e.g., Chap. 8, Sections 8.2 and 8.3 in W. Waterhouse, Introduction to Affine Group Schemes, Springer 1969.

3.6 Unipotent and Pro-unipotent Groups

57

From the unipotency hypothesis, we can always assume that f is in the algebra spanned by the coefficients of a linear representation G → T p (k), from which it follows that n ( f ) = 0 for n large enough. These results can be alternatively understood as follows. The Baker–Campbell– Hausdorff formula computes the product of two exponentials in an associative algebra as the exponential of a sum of commutators: e x e y = e BC H (x,y) . When the Lie bracket is nilpotent of order n (i.e., n-fold iterated brackets vanish), BC H (x, y) = n−1  Hi (x, y), where Hi (x, y) is made of iterated Lie brackets of order i. For instance i=1

H1 (x, y) = x + y, H2 (x, y) =

1 1 [x, y], H3 (x, y) = ([x, [x, y]] + [y, [y, x]]). 2 12

In other words, a unipotent linear algebraic group G with nilpotent Lie algebra g is isomorphic to the exponential group (g, · BC H ) with unit 0, inverse x −→ −x and product law induced by the BCH formula, x · BC H y := BC H (x, y). The same results hold essentially for pro-unipotent groups Gˆ and pro-nilpotent Lie algebras gˆ . A Lie algebra gˆ is pro-nilpotent if it is isomorphic to an inverse limit of finite-dimensional nilpotent Lie algebras: for a tower of Lie algebras, g1  g2 · · ·  gn  . . . , gˆ = lim gn . ←

A pro-unipotent algebraic group Gˆ is an inverse limit of unipotent algebraic groups, Gˆ = lim G n . ←

The exponential and logarithm maps define (set-theoretical) isomorphisms of ˆ of representative functions on Gˆ is the towers. By duality, the Hopf algebra O(G) direct limit of the algebras of polynomial functions O(G n ) and a pro-unipotent group Gˆ is, up to isomorphism, an inverse limit of exponential groups.

3.7 Enveloping Algebras, Groups, Tangent Spaces Another important example of Hopf algebra is provided by the enveloping algebra U (g) of a Lie algebra. This is an associative algebra (recall associative means with unit excepted otherwise stated), equipped with a map ι : g → U (g) (we shall prove later that it is an embedding) and characterized, up to isomorphism, by the following properties: 1. As an algebra, U (g) is generated by the image of g by ι. 2. For a, b ∈ g, ι([a, b]) = ι(a)ι(b) − ι(b)ι(a).

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3 Hopf Algebras and Groups

3. (Universal property) If A is any associative algebra and ρ : g → A is any linear map such that ρ([a, b]) = ρ(a)ρ(b) − ρ(b)ρ(a), then ρ extends uniquely to a morphism of unital algebras ρ¯ : U (g) → A (ρ = ρ¯ ◦ ι). In categorical terms, U is a functor, the enveloping algebra functor, from Lie algebras to unital associative algebras. It is the left adjoint of the forgetful functor from unital associative algebras to Lie algebras (recall that an arbitrary associative algebra is equipped with a Lie algebra structure by the bracket [x, y] := x y − yx). The map ι is the unit of the adjunction. Enveloping algebras are defined only up to isomorphism, however one can construct explicitly a representative of the isomorphism class as follows. Let g be a Lie algebra and consider a map ρ : g → A as above. Then, since T (g) equipped with the concatenation product is the free associative algebra over g, ρ extends uniquely to an algebra map ρˆ from T (g) to A. The condition ρ([a, b]) = ρ(a)ρ(b) − ρ(b)ρ(a) implies that ρˆ vanishes on the ideal I of T (g) generated by [g, g  ] − gg  + g  g, g, g  ∈ g. The associative algebra T (g)/I equipped with the map from g to T (g)/I is an enveloping algebra for g. Slightly abusively, we will call it “the enveloping algebra” of g and write U (g) = T (g)/I . We will call any other representative E in the isomorphism class of T (g)/I “an enveloping algebra” of g, and will write E ∼ = U (g). One defines then a linear map δ : g → U (g) ⊗ U (g) by δ(x) := x ⊗ 1 + 1 ⊗ x. It is easily checked that it maps [x, y] to δ(x)δ(y) − δ(y)δ(x) = [δ(x), δ(y)], hence δ extends uniquely to an algebra morphism  from U (g) to U (g) ⊗ U (g). There exists also a morphism S from U (g) to the opposite algebra U (g)op mapping x to −x for every x in g, and a morphism ε : U (g) → k vanishing identically on g. These properties follow from the universal property of the enveloping algebra. With all its structure, U (g) is a Hopf algebra. We will come back in the next chapter on the fine structure of enveloping algebras. The present section aims at explaining their relations to the “infinitesimal structure” of Lie groups.4 We only state, therefore, in this section an algebraic recognition principle for enveloping algebras; its proof will be given in Chapter 4 (Thm. 4.3.1): if H is a conilpotent cocommutative Hopf algebra over a field of characteristic 0, it is isomorphic to the enveloping algebra of its Lie algebra of primitive elements, Prim(H ). Remark 3.7.1 Recall that a Lie algebra representation is a map ρ from g to End(V ), where V is a vector space such that, for a, b ∈ g, ρ([a, b]) = ρ(a)ρ(b) − ρ(b)ρ(a) = [ρ(a), ρ(b)]. Taking A = End(V ) in the definition of the enveloping algebra by universal properties, we see that representations of the Lie algebra g coincide with linear representations of the associative algebra U (g). 4

Recall Lie algebras were initially called infinitesimal groups because they describe, among others, the structure of tangent spaces to Lie groups.

3.7 Enveloping Algebras, Groups, Tangent Spaces

59

Let H be a Hopf algebra with coproduct . If πi is a linear representation of the algebra H in a space Vi (for i = 1, 2), then we can define a representation π1 ⊗ π2 of H in V1 ⊗ V2 by H

 -

or, on elements:

H⊗H

π1 ⊗π -2

End(V1 ) ⊗ End(V2 ) → End(V1 ⊗ V2 )

(π1 ⊗ π2 )(h) := π1 (h (1) ) ⊗ π2 (h (2) ).

(3.2)

If H is of the form kG for a group G, or U (g) for a Lie algebra g, we get the construction of the tensor product of two representations of a group or a Lie algebra. Similarly, the antipode S gives a definition of the contragradient representation, and the counit ε that of the unit representation (in both cases, for a group G or a Lie algebra g). Let now G be a Lie group. We denote by C ∞ (G) the algebra of real-valued smooth functions on G, with pointwise multiplication. If we endow it with the topology of uniform convergence of all derivatives on all compact subsets of G, the dual space is the space Cc−∞ (G) of distributions on G with compact support. Let T1 and T2 be two such distributions and u ∈ C ∞ (G). For a given element g2 of G, the right translate Rg2 u : g1 → u(g1 g2 ) is in C ∞ (G). It can therefore be coupled to T1 , giving rise to a smooth function v : g2 →< T1 , Rg2 u >. We can then couple T2 to v and define the distribution T1 ∗ T2 by < T1 ∗ T2 , u >:=< T2 , v > .   In integral notation, G T2 (g2 )dg2 G T1 (g1 )u(g1 g2 )dg1 . With this definition of the convolution product, Cc−∞ (G) is an algebra. Distributions supported by the unit 1 of G form a subalgebra C1−∞ (G). It is a folklore theorem in mathematical physics, proved by Schwartz in his 1950/51 treatise5 , that any distribution which vanishes outside a point is a sum of higher order derivatives of a Dirac δ-function. More precisely, choose a coordinate system (u 1 , . . . , u N ) on G centered at the unit 1. Use the standard notation (where α = (α1 , . . . , α N ) belongs to (N) N ): ∂ j = ∂/∂u j , ∂ α =

N  j=1

(∂ j )α j , α! =

N 

α j !.

j=1

If we set < Z α , f >:= (∂ α f )(1)/α!, the distributions Z α form a basis of the vector space C1−∞ (G) of distributions supported by 1. Among these, the distributions (∂ j f )(1) also form an algebraic basis of the Lie algebra g of G. By duality, C1−∞ (G) inherits from C ∞ (G) a coproduct . The Leibniz rule gives

5

L. Schwartz, Théorie des distributions, Hermann, 2 vol., 1950/51.

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3 Hopf Algebras and Groups

(Z α ) =



Zβ ⊗ Zγ .

(3.3)

β+γ =α

This coproduct is graded (setting C1−∞ (G)n :=

 |α|=n

R · Z α , |α| := α1 + · · · + α N ),

therefore conilpotent, cocommutative, and Prim(C1−∞ (G)) = g, from which follows a theorem of Schwartz, elaborating on old results of Poincaré: Theorem 3.7.1 Let G be a Lie group. The distributions in C1−∞ (G), supported by the unit 1 of G, form a Hopf algebra. It is an enveloping algebra (C1−∞ (G) ∼ = U (g)) of the Lie algebra of G. Remark 3.7.2 (Formal groups) The algebra structure coefficients for C1−∞ (G) (i.e., closed formulas for the products Z α ∗ Z β ) can be computed as follows. Let us express analytically the multiplication in G at the neighborhood of 1 by power series φ j (x, y) = φ j (x 1 , . . . , x N ; y1 , . . . , y N ) (for 1 ≤ j ≤ N ) giving the coordinates of N  the point z = x · y. If we develop φ γ = φ j (x, y)γ j in a Taylor series j=1

φ γ (x, y) =



γ

cα,β x α y β ,

α,β

we get by duality Zα ∗ Zβ =

 γ

γ

cα,β Z γ .

(3.4)

This calculation rests on the examination of the power series φ = (φ j ) j≤N representing the product in the group. They satisfy the identities φ(φ(x, y), z) = φ(x, φ(y, z)), (associativity) φ(x, 0) = φ(0, x) = x. (unit) A formal group over a field k is more generally a collection of formal power series satisfying these identities. Let O be the ring of formal power series k[[x 1 , . . . , x N ]], and let Z α be the linear form on O associating to a series f the coefficient of the monomial x α in f . The Z α form a basis for an algebra C, where the multiplication is defined again by Formula (3.4). The coproduct of Formula 3.3 equips C with the structure of a Hopf algebra. When the field is of characteristic zero we get that C is the enveloping algebra of its Lie algebra of primitive elements. It encodes the formal group. Remark 3.7.3 The restricted dual coalgebra of the algebra C ∞ (G) is the space H (G) = C −∞ f inite (G) of distributions with a finite support in G. It is immediate that H (G) is stable under the convolution product of distributions, hence is a Hopf algebra. According to the previous theorem, U (g) is a sub-Hopf algebra of H (G). For

3.7 Enveloping Algebras, Groups, Tangent Spaces

61

every element g of G, the distribution δg is defined by < δg , f >= f (g) for any function f in C ∞ (G). It satisfies the convolution equation δg ∗ δg = δgg and the coproduct rule (δg ) = δg ⊗ δg . Hence the group algebra RG associated to G considered as a discrete group is a sub-Hopf algebra of H (G). As an algebra, H (G) is the twisted tensor product G  U (g) where G acts on g by the adjoint representation: G  U (g) := U (g) ⊗ kG with the product (x ⊗ g) · (y ⊗ g  ) := x Ad(g)(y) ⊗ gg  (see also the Cartier–Gabriel theorem 4.5.1).

3.8 Filtered and Complete Hopf Algebras Definition 3.8.1 (Filtered algebra) A filtered algebra A is an algebra equipped with a decreasing filtration by subvector spaces A = A(0) ⊃ A(1) ⊃ · · · ⊃ A(n) . . . such that A(n) · A(m) ⊂ A(n + m). In particular, each A(i) is an ideal of A and the associated graded vector space gr A =



grn A :=

n∈N



A(n)/A(n + 1)

n∈N

is naturally equipped by the induced product with the structure of a graded algebra. Definition 3.8.2 (Locally finite filtered algebra) A filtered algebra A is locally finite if the graded vector space gr A is locally finite, that is, if the grn A are finitedimensional vector spaces. Definition 3.8.3 (Complete algebras) A complete algebra is a filtered algebra such that A = lim A/A(n). ←

The complete algebra A is called augmented if furthermore it is augmented and A(1) = A¯ := K er (ε), where ε : A → k stands for the augmentation map. The category of complete augmented algebras is written Algc . A complete augmented algebra A is standard if furthermore gr A is generated as an algebra by gr1 A. Exercise 3.8.1 Show that A ∈ Algc is standard if and only if one of the following equivalent conditions hold:

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• A¯ n + A(r ) = A(n) for any r ≥ n, • A(n) is the closure of A¯ n for the topology defined by the filtration. Example 3.8.1 (Grading and completion) The successive quotients of the filtration in a complete algebra give rise to a graded algebra.  Conversely, a graded algebra  An gives rise to a filtered algebra A(n) := Am . The completion of A, A= m≥n

n∈N

written Aˆ is, by definition,

Aˆ := lim A/A(n), ←

ˆ with the filtration A(k) := lim A(k)/A(n), and identifies with ←



An .

n∈N

The completions of the free graded commutative (resp., noncommutative) polynomial algebras k[X 1 , . . . , X n ] (resp., k < X 1 , . . . , X n >) equipped with the grading by the degree of monomials are the formal power series algebras k[[X 1 , . . . , X n ]] (resp., k  X 1 , . . . , X n ). Definition 3.8.4 (Complete tensor products) Let V and W be two  filtered vector spaces. The tensor product V ⊗ W is filtered by V ⊗ W (n) := V (i) ⊗ W ( j). i+ j=n

ˆ , is the completion The complete tensor product of V and W , denoted by V ⊗W of the filtered vector space V ⊗ W : ˆ := lim(V ⊗ W )/(V ⊗ W )(n). V ⊗W ←

These constructions are functorial: filtered and complete filtered vector spaces are symmetric tensor categories, the unit of the tensor product is the ground field equipped with the trivial filtration and the complete tensor product defines a symmetric tensor category structure on Algc . Notice the canonical isomorphism of graded vector spaces ˆ ). gr (V ) ⊗ gr (W ) ∼ = gr (V ⊗ W ) ∼ = gr (V ⊗W Complete tensor products of complete augmented (resp., standard) algebras are complete augmented (resp., standard) algebras. Definition 3.8.5 A complete bialgebra is a complete augmented algebra A equipped with a coassociative and counital coproduct in Algc : ˆ  : A → A⊗A, which is a map of complete augmented algebras. Any complete bialgebra has an antipode, and the two notions of complete bialgebra and complete Hopf algebra (i.e., complete bialgebra equipped with an antipode) identify. Equivalently, complete bialgebras may be defined diagrammatically as usual bialgebras by replacing the category of vector space by the category of complete filtered

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Filtered and Complete Hopf Algebras

63

vector spaces, the usual tensor product by the complete tensor product, and requiring the augmentation conditions. The existence of an antipode is guaranteed by the fact that the identity map of a complete bialgebra can be inverted in the convolution algebra of complete filtered vector space endomorphisms of A: S=

 ν (−1)k (I d − ν)∗k =ν+ ν + (I d − ν) k≥1

that restricts to a finite sum on A/A(n) since A is a complete augmented algebra. It insures the equivalence of the two notions of complete bialgebra and complete Hopf algebra. Complete Hopf algebras form a category written Hopc .

3.9 Signed Hopf Algebras Algebras, coalgebras, bialgebras, and Hopf algebras can be defined in symmetric tensor categories by mimicking the usual diagrammatic definitions. For example, the diagrammatic definition of an associative algebra in Section 2.1 extends to any tensor category. Complete Hopf algebras provide a first example: in their definition (Def. 3.8.5), the usual tensor product of vector spaces is replaced by the complete tensor product of complete filtered vector spaces. Other interesting examples of categories include signed graded vector spaces where the switch map contains a sign taking into account the grading, certain categories of comodules, and vector species (functors from finite sets and bijections to vector spaces). There are many other examples (based, for example, on chain and cochain complexes, various notions of topological vector spaces, certain categories of functors...), but these four are the most meaningful for our purposes and illustrate three different kinds of examples, respectively, in topology, algebra, and combinatorics. We survey briefly the corresponding theory of generalized Hopf algebras through some of these examples in this section and the following one. The example of vector species will be addressed separately in Chapter 9 dealing with applications of Hopf algebras in combinatorics. Signed Hopf algebras are one of the simplest generalizations of the notion of Hopf algebra that one can figure out. This is actually the kind of Hopf algebras that first appeared historically, in the works of Borel, Hopf, and others. Consider the category Ling of graded vector spaces V = (Vn )n∈N , equipped with the usual graded tensor product (V ⊗ W )n := (

p≤n

V p ⊗ Vn− p )n ,

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but with the symmetric monoidal structure induced by the signed switch map T s defined, for v ∈ V p , w ∈ Wn− p , by T s (v ⊗ w) := (−1) p(n− p) w ⊗ v. To notationnally distinguish between the tensor product of graded vector space that does not involve signs in the switch map and this new one, the signed tensor product, we write the latter ⊗s . Let us illustrate on two examples how the new tensor structure on Ling impacts the definition of algebraic structures. We call “signed” these structures. Definitions that do not involve the switch map remain unchanged, for example, since the associativity relation does not involve it, a signed associative algebra is simply a graded associative algebra, that is, an associative algebra in Ling in the usual sense. A signed commutative algebra instead is a graded vector space V equipped with an associative product map m : V ⊗s V → V, such that moreover for all v ∈ V p , w ∈ Vq , we have m(v ⊗s w) = (−1) pq m(w ⊗s v). Similarly, a signed Lie algebra is a graded vector space V equipped with a product map [ , ] : V ⊗s V → V, such that moreover for all x ∈ V p , y ∈ Vq , z ∈ Vr , we have [x, y] = −(−1) pq [y, x]; (−1) pr [x, [y, z]] + (−1) pq [y, [z, x]] + (−1)qr [z, [x, y]] = 0.

Signed bialgebras and Hopf algebras with possibly commutativity or cocommutativity properties are defined similarly. The structure results that hold for usual bialgebras and Hopf algebras, as explained in the next chapter, hold for their signed counterparts. For example, the homology and cohomology of compact connected Lie groups over a field are signed Hopf algebras: the homology is a signed connected cocommutative Hopf algebra, the cohomology is a signed connected commutative Hopf algebra, and the two signed Hopf algebras are in duality. Another important example is given by the homology and cohomology of loop spaces over simply connected manifolds or topogical spaces: their signed Hopf algebra structure is one essential ingredient of modern homotopy theory, and in particular of rational homotopy theory.

3.10

Module Algebras and Coalgebras

65

3.10 Module Algebras and Coalgebras Recall that the categories of (left) modules and (right) comodules over a bialgebra have a monoidal structure: the tensor product of two modules or comodules is a module or a comodule over the bialgebra. Notions of algebras, coalgebras, and bialgebras can be defined in these categories. It happens that various bialgebras have such a structure: they are actually also modules or comodules over another bialgebra (or Hopf algebra) with compatibility relations between the various products and coproducts. In this section, B is a bialgebra and H a Hopf algebra; the structure maps are written with an index to avoid ambiguities, for example, the product map of an algebra A is written m A and its unit map η A . Definition 3.10.1 A B-module algebra is an algebra A which is a left B-module such that m A and η A are B-module maps, that is, b · (ac) = (b(1) · a)(b(2) · c), b · 1 A = ε B (b)1 A , where b ∈ B, a, c ∈ A. This definition can be understood in the light of the notion of representative endomorphisms and pseudo-coproducts of Section 2.10: an element b of B gives rise to a linear endomorphism of A by left multiplication and b(1) ⊗ b(2) is then a pseudo-coproduct for b. Example 3.10.1 Let H = kG and A be an H -module algebra. From (g) = g ⊗ g for g ∈ G, we get g(ab) = (ga) · (gb) for all a, b ∈ A, so that g acts as an algebra automorphism of A. We get a morphism of groups G → Aut (A). Conversely, any such map turns A into a kG-module algebra. Exercise 3.10.1 Define a B-module structure on B ◦ (resp., B ∗ ) by b · h(b ) := h(b b) for b, b ∈ B, h ∈ B ◦ . Show that B ◦ (resp., B ∗ ) is a B-module algebra. Exercise 3.10.2 Let g be a Lie algebra. Recall that M is a representation of g if and only if it is a U (g)-module. Show that if M is furthermore an algebra, it is a U (g)-module algebra if and only if the map g → End(M) factors through Der (M). Definition 3.10.2 A B-module coalgebra is a coalgebra C which is a left B-module and such that C and εC are B-module maps, that is, (b · c)(1) ⊗ (b · c)(2) = b(1) · c(1) ⊗ b(2) · c(2) , εC (b · c) = ε B (b)εC (c), where b ∈ B, c ∈ C. Equivalently, C is a B-module coalgebra if the map B ⊗ C → C, b ⊗ c −→ b · c is a map of coalgebras. Exercise 3.10.3 Let f : B → B  be a map of bialgebras, show that it makes B  a B-module and a B-module coalgebra. In particular, a bialgebra B is canonically a B-module coalgebra. Show that f is a map of B-module coalgebras.

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Definition 3.10.3 If K is a bialgebra (resp., Hopf algebra) which is a B-module and its bialgebra (resp., Hopf algebra) structure maps (product, coproduct, unit, counit, resp., and antipode) are maps of B-modules, K is said to be a B-module bialgebra (resp., Hopf algebra). Exercise 3.10.4 (Adjoint action) Let H be a Hopf algebra, A an algebra, and f ∈ Alg(H, A). Show that the f -adjoint action of H on A defined by ad f (b)(a) := f (b(1) )a f (S(b(2) )), a ∈ A, b ∈ H makes A a H -module and a H -module algebra. When A = H and f = I d H , the action is called simply the adjoint action. Show that if A is a bialgebra, f a bialgebra map, and H is cocommutative, then A is an H -module bialgebra. Example 3.10.2 The terminology “adjoint action” is justified by the following example. Consider a group G and the Hopf algebra kG. The action of G on itself by conjugacy, g(g  ) := gg  g −1 , induces an action of kG on itself that identifies with the adjoint action in the bialgebraic sense when f = I d since S(g) = g −1 . Similarly, if L is a Lie algebra, the adjoint action of L on itself (ad(l)(l  ) := [l, l  ]) extends to the adjoint action of the enveloping algebra on itself (as a bialgebra). Indeed ad I d (l)(l  ) = ll  − l l for l ∈ L , l  ∈ U (L) since l ∈ Prim(U (L)). These various notions dualize, we limit the account of comodule structures to giving the basic definitions. Definition 3.10.4 A B-comodule algebra is an algebra A which is a B-comodule such that m A and η A are B-comodule maps. Equivalently, A is a B-comodule algebra if the comodule structure map ∂ A : A → A ⊗ B is a map of algebras. Definition 3.10.5 A B-comodule coalgebra is a coalgebra C which is a B-comodule such that C and εC are B-comodule maps. Definition 3.10.6 If K is a bialgebra (resp., Hopf algebra) which is a B-comodule and its bialgebra (resp., Hopf algebra) structure maps (product, coproduct, unit, counit, resp., and antipode) are maps of B-comodules, K is said to be a B-comodule bialgebra (resp., Hopf algebra).

3.11 Bibliographical Indications General theory Let us start with some general historical indications and references on the beginnings of the theory of Hopf algebras. We postpone to the corresponding chapter details on

3.11

Bibliographical Indications

67

specific topics (algebraic topology, algebraic groups...) and merely sketch the general picture. Further details can be found in [AF] that studies carefully the period from the 40s to the 60s. The history of Hopf algebras originates in two different fields: algebraic topology and algebraic groups. In algebraic topology, from the mid-1930s, various results on the structure and cohomology of compact Lie groups and their homogeneous spaces had been obtained by E. Cartan, Ch. Ehresmann, L. S. Pontrjagin, A. Borel, and others. The key idea which gave birth to the theory of Hopf algebras, due to H. Hopf in 1941 [Hop], was to enrich the study of duality between homology and cohomology over a field k using algebra structures. Considering a topological space M equipped with a product map m : M × M → M (not necessarily associative or commutative, even up to homotopy), he noticed that the induced algebra map m ∗ : H∗ (M) ⊗ H∗ (M) → H∗ (M) had compatibility properties with the algebra structure on the cohomology of M (induced by the diagonal map from M to M × M) that enforce various properties of H ∗ (M). This implies, for example, that the rational cohomology of a compact Lie group is an exterior algebra over odd generators: the group has therefore the same cohomology as a product of spheres of odd dimensions, a property that would before have been obtained following a case-by-case approach and using the classification of compact Lie groups. The name “Hopf algebra” itself was coined by A. Borel in 1953 [Bor], as a tribute to the seminal idea of Hopf. Besides the fact that in topology structure maps are “signed” (commutation rules between two odd cohomology classes involve a minus sign), a Hopf algebra in Borel’s sense is not the modern one: the coproduct is not required to be coassociative nor to have an antipode. Another line of development of these ideas originates in the work of J. A. Dieudonné on formal Lie groups over a field of characteristic p > 0 [Die]. In finite characteristic, the classical dictionary between groups and Lie algebras does not hold, and he defined a notion of “hyperalgebra” that identifies with an enveloping algebra over a field of characteristic 0 but has a much more complex structure in general. The formal definition of a hyperalgebra was given by P. Cartier in 1956 [Car] (the existence of the antipode is not assumed explicitly but follows from a conilpotency assumption). This was, in practice, essentially the first formal definition of a Hopf algebra in the modern sense, see [AF] for a detailed analysis. The theory of Hopf algebras in relation to algebraic groups developed then, especially in the works of G. Hochschild and G. D. Mostow, culminating in Hochschild’s 1965 book “The structure of Lie Groups” [Hoc] and then, later, in the theory of affine group schemes [DG,GD]. The formalism and terminology stabilized in this period of the mid-1960s. A key role was then played by Sweedler’s 1969 monograph [Swe] that remained for long “the” reference on the subject of Hopf algebras and contributed largely to the development of the subject of Hopf algebras in algebra (as opposed here to algebraic topology and algebraic geometry). In algebraic topology, a similar role was played by the Milnor–Moore article [MM] (published in 1965, but

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that circulated since 1959) that accounted for the various structure theorems for Hopf algebras in topology and developed their very important application to loop spaces (a key to modern homotopy theory, and in particular to rational homotopy theory). Characteristic endomorphisms The definition of characteristic endomorphisms and their application to the study of the Dynkin operator goes back to Patras–Reutenauer [PR]. These ideas were developed further in [EGP] and [MP]. The properties of characteristic endomorphisms underly the study of graded Hopf algebras in the two forthcoming chapters. Algebraic groups, enveloping algebras We already mentioned the seminal contribution of Hochschild and Mostow to the Hopf algebraic approach to the theory of algebraic groups. Besides [DG,GD], other historical references on the subject are, on the closely related subject of formal groups [Haz], on enveloping algebras [Dix]. These topics will be addressed again later in this book, further details and references will be given in the corresponding chapters. Signed Hopf algebras The first Hopf algebras that have been considered historically came from algebraic topology and were signed (with a minus sign involved in the commutation rules between odd degree elements). We will come back later, in Chapter 8, to applications of Hopf algebras in topology. Let us mention here only, besides the references already given, the book of Kane [Kan], dedicated to the homology of Hopf spaces. In the characteristic 0 case on which the present book focuses, Hopf algebras play a key role in relation to loop spaces and iterated loop spaces, particularly in rational homotopy theory. See, for instance, [Qui], [FHT]. Module and comodule Hopf algebras Algebra, coalgebra, and bialgebra structures on modules or comodules over a bialgebra have been studied in detail by Molnar [Mol], from which the corresponding section of this chapter and many examples are taken. He studied the lift to Hopf algebras of the notion of semi-direct products of groups, algebraic groups, and their Lie algebra analogs. We refer to his article for more details. We won’t treat further the subject in this book but mention it and gave the basic definitions as these structures have been used recently in combinatorics, numerical analysis, and the theory of regularity structures for stochastic partial differential equations. We refer to Manchon’s survey article [Man] for details. General references Various books have been dedicated to Hopf algebras after Sweedler’s and the Milnor– Moore article. They often focus on a given application domain, and develop the theory accordingly, including in these developments topics that we have decided not to cover in the present book such as finite-dimensional Hopf algebras, applications in number theory or quantum groups (Hopf algebras that are not commutative nor cocommutative and often deformations of Hopf algebras of functions on finite-dimensional continuous groups). Abe’s monograph [Abe] develops the theory of integrals, irreducibility, algebraic groups, and applications to the theory of fields. Algebraic number the-

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69

ory together with Galois module theory is addressed in [Und]. Quantum groups, finite-dimensional Hopf algebras, and quasi-triangular Hopf algebras are studied in [DNR], [Kas], [Maj], [Mon], [Rad]. Hopf algebras, with an emphasis on the finitedimensional case, are studied from the point of view of tensor categories in [Eal]. The monograph [HGK] starts with a survey on the theory of Lie algebras but the main developments concern algebraic combinatorics and more precisely symmetric functions, representations of symmetric groups, related structures, and various noncommutative or noncocommutative Hopf algebraic generalizations. A detailed and more general study of combinatorial Hopf algebras from the point of view of Hopf algebras in the category of vector species (see Chapter 9 of this book) is contained in Aguiar-Mahajan’s [AM]. The same authors investigated more recently connections of several combinatorial aspects of the theory of Hopf algebras with the theory of real hyperplane arrangements [AM2].

References [Abe] Abe, E.: Hopf algebras, Cambridge Tracts in Mathematics, vol. 74. Cambridge University Press (1980) [AF] Andruskiewitsch, N., Ferrer Santos, W.: The beginnings of the theory of Hopf algebras. Acta applicandae mathematicae 108(1), 3–17 (2009) [AM] Aguiar, M., Mahajan, S.A.: Monoidal functors, species and Hopf algebras. American Mathematical Society, Providence, RI (2010) [AM2] Aguiar, M., Mahajan, S.A.: Topics in hyperplane arrangements. Mathematical Surveys and Monographs of the AMS 226,(2017) [Bor] Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts. Ann. Math. 87, 115–207 (1953) [Car] Cartier, P.: Hyperalgèbres et groupes de Lie formels, Séminaire “Sophus Lie” 2e année: 1955/56, (1957) [DG] Demazure, M., Gabriel, P.: Introduction to algebraic geometry and algebraic groups (Vol. 39), Elsevier, (1980) [DNR] Dascalescu, S., Nastasescu, C., Raianu, S.: Hopf algebra: An introduction. CRC Press (2000) [Die] Dieudonné, J.A.: Groupes de Lie et hyperalgèbres de Lie sur un corps de caractéristique p>0. Comm. Math. Helv. 28, 87–117 (1954) [Dix] Dixmier, J.: Enveloping algebras (Vol. 14). Newnes, 1977 [EGP] Ebrahimi-Fard, K., Gracia-Bondia, J., Patras, F.: A Lie theoretic approach to renormalization. Comm. Math. Phys. 276, 519–549 (2007) [Eal] Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. Vol. 205. American Mathematical Soc., 2016 [FHT] Félix, Y., Halperin, S., Thomas, J.–C.: Rational homotopy theory (Vol. 205). Springer (2012) [GD] Grothendieck, A., Demazure, M.: Schémas en groupes. Lecture Notes in Math. 151,(1962) [Haz] Hazewinkel, M.: Formal groups and applications (Vol. 197). New York: Academic press, (1978) [HGK] Hazewinkel, M., Gubareni, N. M., Kirichenko, V. V.: Algebras, rings, and modules: Lie algebras and Hopf algebras. Mathematical Surveys and Monographs No. 168. American Mathematical Soc. (2010) [Hoc] Hochschild, G.: The structure of Lie groups. Holden-Day Inc., San Francisco (1965) [Hop] Hopf, H.: Über die Topologie der Gruppen-Mannifaltigkeiten und ihrer Verallgemeinerungen. Ann. Math. 42, 22–52 (1941)

70 [Kan] [Kas] [Maj] [Man]

[MP] [MM] [Mol] [Mon] [PR] [Qui] [Rad] [Swe] [Und]

3 Hopf Algebras and Groups Kane, R.M.: The homology of Hopf spaces. North-Holland, Amsterdam (1988) Kassel, Ch.: Quantum groups, Graduate Texts in Mathematics, vol. 155. Springer (1995) Majid, S.: Foundations of quantum group theory. Cambridge University Press (2000) Manchon, D.: A Review on Comodule-Bialgebras. In: Celledoni et al. (eds), Computation and Combinatorics in Dynamics, Stochastics and Control. Abel Symposium 2016. Abel Symposia, vol 13. Springer, Cham (2018) Menous, F., Patras, F.: Logarithmic Derivatives and Generalized Dynkin Operators. Journal of Algebraic Combinatorics: Volume 38, Issue 4, 901–913 (2013) Milnor, J. W., Moore, J. C.: On the structure of Hopf algebras, Ann. of Math. (2), 81, 211–264 (1965) Molnar, R.K.: Semi-direct products of Hopf algebras. Journal of Algebra 47(1), 29–51 (1977) Montgomery, S.: Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. 82. American Mathematical Society, (1993) Patras, F., Reutenauer, Ch.: On Dynkin and Klyachko idempotents in graded bialgebras. Advances in Applied Mathematics 28(3–4), 560–579 (2002) Quillen, D.: Rational homotopy theory. Annals of Mathematics , 205–295 (1969) Radford, D.E.: Hopf algebras, Series on knots and everything 49. World Scientific (2012) Sweedler, M.: Hopf algebras. Benjamin, New York (1969) Underwood, R.G.: An introduction to Hopf algebras. Springer (2011)

Chapter 4

Structure Theorems

This chapter is dedicated to the structure theorems for Hopf algebras, from Poincaré– Birkhoff–Witt to the theorems of Samelson–Leray, Hopf–Leray, Cartier, Cartier– Gabriel, and Fresse. The ground field (written K in this chapter) is of characteristic 0 excepted otherwise stated. The chapter states and proves the structure theorems under weak hypotheses (such as unipotency). These results are obtained by using the approach of natural operations; in other terms, the structure of Hopf algebras is investigated using the action of natural transformations of the forgetful functors from the various categories of Hopf algebras under consideration to vector spaces. Together with the use of the techniques related to characteristic endomorphisms, this approach simplifies and, in our opinion, enlightens the structure theory of Hopf algebras. It also provides various tools that can be used in their application fields. The structure of graded Hopf algebras, closely related to the combinatorics of free Lie algebras, will be investigated separately in the following chapter. Although we state the theorems in the context of classical Hopf algebras, the arguments based on the use of natural operations such as dilations, as well as their applications to structure theorems for Hopf algebras, depend only on the fact that one considers a unipotent Hopf algebra in a symmetric tensor category (with, in particular, a linear structure such that the structure maps defining algebra and coalgebra structures are multilinear with respect to the ground field). They hold, therefore, in this more general setting. Applications of this remark include signed Hopf algebras, twisted Hopf algebras (also known as Hopf species), and cogroups in algebras over operads.

© Springer Nature Switzerland AG 2021 P. Cartier and F. Patras, Classical Hopf Algebras and Their Applications, Algebra and Applications 29, https://doi.org/10.1007/978-3-030-77845-3_4

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4.1 Dilations, Unipotent Bialgebras, and Weight Decompositions We first introduce various operations and constructions that will be instrumental in our presentation of the proofs of the classical structure theorems for Hopf algebras. Definition 4.1.1 Let (B, m, , ε, η) be a bialgebra. The k-th dilation (or k-th Adams operation) is the linear endomorphism of B defined as the k-th convolution power of the identity  k := I d ∗ · · · ∗ I d = I d ∗k = m k ◦ k (resp.,  0 := η ◦ ε). The terminology is inspired by topology: when acting on the cohomology of a topological group, the  k are induced by the power maps x −→ x k generalizing the action of dilations (x −→ k · x) on Euclidean spaces. The combinatorics of the  k is actually closely related to the action of dilations on convex Euclidean polytopes, see also Remark 4.1.1. We refer to the bibliographical indications at the end of the chapter for details. Example 4.1.1 On a group-like element x (for example, on the elements of a group G in the group algebra K [G] viewed as a Hopf algebra)  k (x) = m k ◦ k (x) = m k (x ⊗ · · · ⊗ x) = x k . On a primitive element y of a bialgebra,  k (y) = m k (y ⊗ 1 · · · ⊗ 1 + · · · + 1 ⊗ · · · ⊗ 1 ⊗ y) = k · y. Lemma 4.1.1 The dilations satisfy  k ∗  l =  k+l .

(4.1)

Furthermore, if B is commutative (resp., cocommutative) they are algebra (resp., coalgebra) endomorphisms of B and satisfy  k ◦  l =  kl .

(4.2)

Distributivity relations follow in the commutative (resp., cocommutative) case, such as  k ◦ ( p ∗  q ) = ( k ◦  p ) ∗ ( k ◦  q ) = ( p ∗  q ) ◦  k . Only the second assertion requires a proof. It follows from the general statement:

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Lemma 4.1.2 Let f, g be two algebra endomorphisms of a commutative bialgebra B (resp., coalgebra endomorphisms of a cocommutative bialgebra). Then, f ∗ g is an algebra endomorphism (resp., a coalgebra endomorphism). Proof. Indeed, assume, for example, that B is commutative. Then the product map m : B ⊗ B → B is an algebra morphism. The convolution product f ∗ g = m ◦ ( f ⊗ g) ◦  is therefore a composition of algebra morphims. The proof dualizes in the cocommutative case. The lemma also implies the identity (4.2). Let us assume, for example, that B is commutative (the dual proof holds in the cocommutative case). Let us write m l(k) for the iterated l fold product on the algebra B ⊗k (the product map from (B ⊗k )⊗l to B ⊗k ). We get, since m and its iterates m l are coalgebra morphisms  k ◦  l = m k ◦ k ◦ m l ◦ l = m k ◦ m l(k) ◦ (k )⊗l ◦ l = m kl ◦ kl , where the second equality expresses the commutativity of the diagram B ⊗l

ml-

⊗l k

B

k

? ? ⊗k B ⊗kl B , (k) ml

whereas the last equality follows from the commutativity of the product (m k ◦ m l(k) = m kl ) and the coassociativity of the coproduct ((k )⊗l ◦ l = kl ). Example 4.1.2 If B = K [V ], the polynomial Hopf algebra over V ,  k acts as k n on K [V ]n : indeed, on elements v of V (which are primitive)  k (v) = k · v and, since K [V ] is commutative, on a product v1 . . . vn of elements of V we get, since  k is multiplicative  k (v1 . . . vn ) = k n v1 . . . vn . Remark 4.1.1 When B is commutative, the  k are algebra endomorphisms satisfying  k ◦  l =  kl : they define a -ring and, since structures are defined over a field of characteristic 0, a λ-ring structure on B. This is one of the reasons why the  k can be called alternatively Adams operations1 . This terminology is not justified On λ-rings, see F. Patras, Lambda-rings. Handbook of Algebra, Vol. 3. Ed. M. Hazewinkel. Amsterdam: North-Holland. 2003, 961–986 or M. Hazewinkel, N. M. Gubareni, V. V. Kirichenko, Algebras, rings, and modules: Lie algebras and Hopf algebras. Mathematical Surveys and Monographs No. 168. American Mathematical Soc. (2010).

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in the noncommutative case, this is the reason why we prefer to use here the one of dilations that also carries a geometric and group-theoretical semantics. Recall that ν := η ◦ ε is the unit of the convolution product in End(B). Definition 4.1.2 (Unipotent bialgebras) Let J := I d − ν ∈ End(B) and let us define the operations J k by J 0 := I d, J 1 := J and, in general: J k := J ∗k = (I d − ν)∗k = m k ◦ k . The bialgebra B is called unipotent if and only if the identity map is locally unipotent for the convolution product. That is, if ∀b ∈ B, ∃N / ∀k ≥ N , J k (b) = 0. Remark 4.1.2 Due to the local unipotency of I d, unipotent bialgebras B are always Hopf algebras: the antipode is given by S = I d ∗−1 =



 ν = (−1)i J i . ν+J i=0

Here, computations are performed in the endomorphism algebra End(B) equipped with the convolution product. The unipotency assumption insures that the series restricts to a finite sum when acting on an arbitrary element of the bialgebra.  Example 4.1.3 • Conilpotent bialgebras (B=K ⊕ Fn , where Fn : =K er n+1 ) n

are unipotent bialgebras. • The bialgebras of regular functions on unipotent groups are unipotent (they are always conilpotent). • Graded connected bialgebras are unipotent (they are conilpotent). • Enveloping algebras of Lie algebras are unipotent (they are conilpotent). We will assume from now on in this section that the bialgebra B under consideration is unipotent (equivalently that it is a unipotent Hopf algebra). We get  = (J + ν) k

1

∗k

=

k    k j=0

j

J = j

∞    k j=0

j

J j.

Using the Stirling numbers of the first kind s( j, i) defined implicitly by x(x − 1) . . . (x − j + 1) =

j  i=1

s( j, i)x i ,

4.1 Dilations, Unipotent Bialgebras, and Weight Decompositions

75

we obtain finally the polynomial or weight decomposition j ∞  ∞   Jj  =ν+ [s( j, i) ]k i =: ei k i , j! j=1 i=1 i=0 k

where e0 = ν. In particular, since s( j, 1) = (−1) j−1 ( j − 1)!, ∞  (I d − ν)∗ j e = (−1) j−1 = log∗ (I d), j j=1 1

and the weight decomposition rewrites  k = exp∗ ◦ log∗ (I d ∗k ) = exp∗ (k · e1 ) =

∞  k i (e1 )∗i i=0

(4.3)

i!

Because of the unipotency assumption, all finite sums under consideration have only finitely many non-vanishing terms. Theorem 4.1.1 We have: I d =

∞ 

∞ 

ei and  k =

i=0

k i ei , where e1 = log∗ (I d) and

i=0 ∞

ei =

On



 (e1 )∗i (I d − ν)∗ j = . s( j, i) i! j! j=i

K er J m , the ei , i > n vanish and the formula reduces to  k =

m≥n+1

n 

k i ei .

i=0

Theorem 4.1.2 When furthermore we have  k ◦  l =  kl , an hypothesis that holds k simultaneously when the bialgebra is commutative or cocommutative, the  are i K er J m , k i , i ≤ n). The diagonalizable with eigenvalues k , i ∈ N (resp., on m≥n+1

ei are the spectral projections. The spectral decomposition of B is called the weight decomposition. Proof. Indeed,  k ◦  k =  kl implies (

∞  i=0

k i ei ) ◦ (

∞ 

l jej) =

∞  (kl)i ei ,

i=0

i=0

and therefore, by identification of the coefficients, ei ◦ e j = δ ij ei .

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4 Structure Theorems

Definition 4.1.3 When B is commutative or cocommutative, we write B(n) for the eigenspace associated to the eigenvalue k n of  k . The spectral projection e1 from B to B(1) (resp., ei from B to B(i)) is called the canonical projection (resp., the i-th canonical projection).

4.2 Enveloping Algebras By Ado’s theorem, every finite-dimensional Lie algebra g over K has a faithful finitedimensional representation. That is, there exists an embedding ρ : g → Mn (K ) into a finite-dimensional Lie algebra of matrices. This embedding factorizes through a morphism of algebras ρˆ : U (g) → Mn (K ): ρ = ρˆ ◦ ι, where ι is the canonical map from g to its enveloping algebra. The injectivity of ρ implies the one of ι. We admit these results that pertain to the general theory of Lie algebras 2 and get the lemma (that holds also in the infinite-dimensional case3 ). Lemma 4.2.1 The canonical map ι : g → U (g) is an injection. We will therefore identify from now on x ∈ g with its image in U (g). For a subspace V of T (g), let us write t ≡ t  [V ] when t − t  ∈ V . Let us set n  T k (g). Recall that we write I for the ideal of T (g) generated by also T ≤n (g) := k=0

∼ yx [g ⊕ I ] and since transpositions the [g, g  ] − gg  + g  g, g, g  ∈ g. From x y = generate the symmetric group Sn , we get for all x1 , . . . , xn ∈ g ∀σ ∈ Sn , x1 . . . xn ∼ = xσ (1) . . . xσ (n) [T ≤n−1 (g) + I ], so that

1  xσ (1) . . . xσ (n) [T ≤n−1 (g) + I ]. x1 . . . xn ∼ = n! σ ∈S n

By induction on the length of tensors, any element in U (g) can be written as the class modulo I of a symmetric tensor (an element of T S(g)). Therefore Lemma 4.2.2 The canonical map g : T S(g) → U (g) = T (g)/I induced by the embedding T S(g) → T (g) is surjective. Lemma 4.2.3 When T S(g) is equipped with the unshuffle coproduct u , the map g is a coalgebra map.

2

See, for example, N. Bourbaki, Groupes et algèbres de Lie: chapitres 1-8. Hermann, (1975) and J. Dixmier, Enveloping algebras (Vol. 14) Newnes (1977). 3 This follows, for example, from arguments given in the proof of the Poincaré–Birkhoff–Witt theorem in Bourbaki or Dixmier, op. cit.

4.2 Enveloping Algebras

77

Proof. Recall that T S n (g) is generated by the γn (v) = v ⊗n , and that, since v is primitive, n    n γi (v) ⊗ γn−i (v). u (γn (v)) = i i=0 Since the product in U (g) is inherited from the concatenation in T (g), g(γn (v)) = v n , where v n is the n-fold product of copies of v in U (g). Since v is primitive in U (g), we have finally, in U (g) (v n ) = (v ⊗ 1 + 1 ⊗ v)n =

n    n i=0

i

v i ⊗ v n−i .

The lemma follows. Since U (g) is a conilpotent cocommutative Hopf algebra, the results of the previous section apply. In particular, it decomposes as U (g) =



U (g)(i),

i=0

where U (g)(0) = K . From (v n ) =

n n   n i n n n−i 2 n v v = ⊗ v , we get  (v ) = i i

i=0

i=0

2n v n . It follows that g maps T S n (g) to U (g)(n). Finally, since g injects in U (g) and g is surjective: Corollary 4.2.1 The map g induces an isomorphism of vector spaces g:g∼ = U (g)(1). Theorem 4.2.1 (Poincaré–Birkhoff–Witt (PBW)) The canonical map g : T S(g) → U (g) is a coalgebra isomorphism compatible with the weight decomposition of the enveloping algebra: g : T S n (g) ∼ = U (g)(n). Proof. To prove the theorem, it is enough to find a left inverse to g. However, by Lemma 4.2.2, the v n , v ∈ g, generate linearly U (g) and a direct computation ¯ n (v n ) = n!v ⊗n ∈ T S n (g) ⊂ g⊗n ⊂ U (g)⊗n . In other using that v is primitive yields  terms, the iterated reduced coproduct maps U (g)(n) to T S n (g) and is a left inverse to g up to a scalar factor n!. Remark 4.2.1 The coalgebra isomorphism part of our statement of the PBW theorem holds more generally over a field of arbitrary characteristic4 .  n Remark 4.2.2 The coalgebra T S(g) = T S (g) is a graded cocommutative coaln

gebra. It is the cofree graded cocommutative coalgebra over g (viewed as a 4

See, e.g., Dixmier, op. cit.

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4 Structure Theorems

graded vector space concentrated in degree 1). From this point of view, the map 1 ⊗n ¯ n )n∈N is the canonical lift to a map from (U (g), ) to (T S(g), u ) of the ( (e n!) ◦  canonical projection e1 : U (g)  g.

4.3 Cocommutative Unipotent Hopf Algebras Let H = (H, m, , ε, η) be a cocommutative unipotent Hopf algebra over a field of characteristic 0. Recall that H decomposes then into eigenspaces under the action ∞  of dilations: H = H (n). We already know that g := Prim(H ) is a Lie algebra. n=0

Lemma 4.3.1 The Lie algebra of primitive elements identifies with the first weight space: g = H (1). Proof. The inclusion Prim(H ) ⊂ H (1) is obvious since ∀h ∈ H, (h) = h ⊗ 1 + 1 ⊗ h =⇒  2 (h) = m 2 ◦ (h) = 2h. Conversely, since the dilations  k are coalgebra endomorphisms of H , the coproduct restricts to maps between eigenspaces (for  k , resp.,  k ⊗  k ):  : H (n) →



H ( p) ⊗ H (q).

p+q=n

In particular, (H (1)) ⊂ H (1) ⊗ H (0) ⊕ H (0) ⊗ H (1) = H (1) ⊗ K ⊕ K ⊗ H (1). For any x ∈ H (1) (recall that the coproduct is cocommutative), there exists therefore y ∈ H (1) s.t. (x) = y ⊗ 1 + 1 ⊗ y. Since  2 (x) = 2x, we get 2x = 2y and x = y, that is, x ∈ Prim(H ). Theorem 4.3.1 (Cartier) Let H be a cocommutative unipotent Hopf algebra over a field of characteristic 0. Then, g = Prim(H ) is a Lie algebra and the inclusion of g into H extends to an isomorphism of Hopf algebras  : U (g) → H. For notational clarity, given x ∈ g, we will write xUn (resp., x Hn ) for the n-th power of x in U (g) (resp., H ). Let us consider now the diagram h T S(g) - H g ?  U (g) ,

4.3 Cocommutative Unipotent Hopf Algebras

79

where 1. the map  is the algebra map induced by the Lie algebra embedding of g into H . It is a Hopf algebra map since U (g) is generated linearly by the xUn , x ∈ g and ( ⊗ ) ◦

(xUn )

= ( ⊗ )(

n    n p=0

=

n    n p=0

p

p

n− p

xH ⊗ xH

p

p

n− p

xU ⊗ xU )

= (x Hn ) =  ◦ (xUn ).

2. the map g is the coalgebra isomorphism introduced in the previous section (g(γn (v)) := vUn ). 3. the map h is defined by h(γn (v)) := v nH . Equivalently, it is the restriction to T S(g) ⊂ T (g) of the algebra map from T (g), the free associative algebra over g, to H induced by the embedding of g into H . The commutativity of the diagram, and the fact that h is a coalgebra map, follows from the identity  ◦ g(γn (x)) = (xUn ) = x Hn = h(γn (x)). Since  is a Hopf algebra map and g a coalgebra isomorphism, the theorem will follow if we prove that h is an isomorphism of vector spaces. Notice first that h maps T S n (g) to H (n) since ∀x ∈ g,  (h(γn (x)) =  2

2

(x Hn )

=m◦

(x Hn )

n    n )x Hn = 2n x Hn . =( k k=0

Recall also that the spectral projection en from H to H (n) satisfies en =

(e1 )∗n (e1 )⊗n (e1 )⊗n ¯ n, = mn ◦ ◦ n = m n ◦ ◦ n! n! n!

the last identity since e1 is null on H (0) = K . The restriction to T S n (g) ⊂ g⊗n ⊂ H ⊗n of the iterated product map m n identifies 1 ⊗n ¯ n maps H to with h. On the other hand, since e1 maps H to H (1) = g, (e n!) ◦  1 ⊗n ¯n T S n (g) (recall that the coproduct is cocommutative). We get finally that (e n!) ◦  n is a right inverse to the restriction of h to a map from T S (g) to H (n). On the other hand, since n (x Hn ) = n!x ⊗n , (e1 )⊗n (e1 )⊗n ◦ n (x Hn ) = ◦ n (x Hn ) = x ⊗n ∈ T S n (g) ⊂ H ⊗n , n! n!

80

4 Structure Theorems (e1 )⊗n n!

◦ n is also a left inverse to the restriction of h to T S n (g). Globally, since ∞  (e1 )⊗n ¯ n is a left and right inverse to h. The ◦ ek is null on the H (n), n = k, n!

and

n=0

theorem follows. Example 4.3.1 (Free Lie algebras) Consider T (X ), the tensor gebra over a set X equipped with the concatenation product and graded by the length of words over X . It is an exercise in universal algebra that T (X ) is the enveloping algebra of the free Lie algebra Lie(X ) over X . This can be deduced for example from the fact that the enveloping algebra functor is left adjoint to the forgetful functor from associative to Lie algebras. We refer to Reutenauer’s monograph on free Lie algebras for details5 . From the Hopf algebraic point of view, Lie(X ) identifies then with the Lie algebra of primitive elements in T (X ) equipped with the unshuffle coproduct. The Theorem 4.3.1 dualizes as follows. Theorem 4.3.2 Let H ∗ be the dual algebra to a cocommutative unipotent Hopf algebra H over a field K of characteristic zero. Let us assume for simplicity that (g1 , . . . , gn ) is a finite-dimensional basis of Prim(H ). Then, the mapping associating to f ∈ H ∗ the power series F(x1 , . . . , xn ) :=< f,

n

exp (xi gi ) >

i=1

is an algebra isomorphism from H ∗ to K [[x1 , . . . , xn ]]. n

Proof. Recall that exp (xi gi ) := exp (x1 g1 ) . . . exp (xn gn ). In characteristic 0, the i=1

theorem of Poincaré–Birkhoff–Witt as we stated it also implies, by an easy induction argument, that the products g1k1 . . . gnkn form a basis of U (Prim(H )), and therefore of H . That the mapping is a linear bijection between H ∗ and K [[x1 , . . . , xn ]] follows. The algebra isomorphism statement follows then from the observation that the exp(xi gi ) are group-like since the gi are primitive. When H is also commutative, Prim(H ) is a commutative Lie algebra: the bracket is the null map. Theorem 4.3.1 specializes to Theorem 4.3.3 (Samelson–Leray) Let H be a bicommutative (i.e., commutative and cocommutative) unipotent Hopf algebra over a field of characteristic 0. Then, V = Prim(H ) is a commutative Lie algebra and the inclusion of V into H extends to an isomorphism of Hopf algebras  : K [V ] → H, where the gebra of polynomials K [V ] is equipped with the product and coproduct of polynomials: the one for which the elements of V are primitive, 5

Ch. Reutenauer, Free Lie Algebras, Oxford, 1993.

4.3 Cocommutative Unipotent Hopf Algebras

81



(v1 . . . vn ) = I



vI ⊗ v J ,

J =[n]

where for I = {i 1 , . . . , i k }, v I := vi1 . . . vik .

4.4 Commutative Unipotent Hopf Algebras Let (H, m, , ε, η) be a commutative unipotent Hopf algebra over a field of characteristic 0. Recall that H = K ⊕ H + decomposes into eigenspaces under the action of ∞ ∞   dilations: H = H (n) with H (0) = K , H + = H (n). Since the  k are algen=0

n=1

bra endomorphisms, this weight decomposition is multiplicative: the product map maps H (n) ⊗ H (m) to H (n + m). Lemma 4.4.1 Let Q(H ) be the vector space of indecomposable elements in H + : Q(H ) := H + /(H + )2 ; then the canonical map from H (1) to Q(H ) is an isomorphism.  Indeed, from the multiplicativity of the weight decomposition, (H + )2 ⊂ H (n) n>1

and Q(H ) decomposes into a direct sum of eigenspaces under the action of the ∞   k : Q(H ) = Q(H )(n). In particular, the canonical map q : H (1) → Q(H ) is n=1

injective. Conversely, for x ∈ H + , write x¯ for its class in Q(H ). We have ¯ =  2 (x) = m(x ⊗ 1 + 1 ⊗ x + r ),  2 (x) ¯ = 2 x. ¯ Therefore where the remainder term r belongs to H + ⊗ H + . Finally,  2 (x) Q(H )(n) = 0 for n > 1 and q is surjective. Theorem 4.4.1 Let H be a commutative unipotent Hopf algebra over a field of characteristic 0. The injection of H (1) into H extends to an isomorphism of algebras  : K [H (1)] → H. Proof. Since H is commutative and since K [H (1)] =

∞  n=0

K [H (1)]n is the free com-

mutative algebra generated by H (1), the embedding of H (1) into H extends uniquely to a map  from K [H (1)] to H . From Q(H ) ∼ = H (1) and the multiplicativity of the weight decomposition, we get that H is generated by H (1) and that K [H (1)]n maps surjectively to H (n). 1 ∗n On the other hand, the n-th canonical projection en = (en!) acts as the identity on H (n). Using the identification of polynomials with symmetric tensors using the symmetrization map,

82

4 Structure Theorems

K [H (1)]n ∼ = (H (1)⊗n ) Sn , the commutativity of m and the identity ⎛ ⎞ 1 ⊗n 1 ⊗n  1 ) ) (e (e ⎠ ◦ n , ◦ n = m n ◦ ⎝ en = m n ◦ σ◦ n! n! σ ∈S n! n

 where σ acts by permutations on tensor products of length n and n!1 σ is the σ ∈Sn  1 ⊗n ¯ n is a section of the projection σ ◦ (e n!) ◦  symmetrization map, it results that n!1 σ ∈Sn

¯ from K [H (1)]n to H (n). For v ∈ H (1), (v) ∈ H + ⊗ H + and we get by induction n ⊗n ¯ n (v ) = n!v + r , where the remainder term r is in the kernel of (e1 )⊗n . that  Using that K [H (1)]n is spanned linearly by the v n , the injectivity of  and the theorem follow. Theorem 4.4.1 is classically stated in a less intrinsic form that does not take advantage of the weight decomposition. In its most classical form, Theorem 4.4.2 (Hopf–Leray) Let H be a commutative graded connected Hopf algebra over a field of characteristic 0. Let φ : Q(H ) → H + be a section of the canonical projection from H + to Q(H ) in the category of graded vector spaces. The induced map K [Q(H )] → H is then an isomorphism of commutative graded algebras. In particular, H is a free commutative algebra. Remark 4.4.1 The theory of commutative unipotent bialgebras is dual to the one of cocommutative unipotent bialgebras. One can show that H (1) is a Lie coalgebra (the structure dual to the one of a Lie algebra). Lie coalgebras have coenveloping (coassociative) coalgebras. However, the structure of coenveloping coalgebras is more involved than the one of enveloping algebras. This reflects the fact that the combinatorial structure of cofree coassociative coalgebras is, in general, more involved than the one of free associative algebras6 . In practice, the duality is usually stated in the locally finite graded and connected case: all ingredients of the proofs of the Cartier theorem dualize then immediately and H is then, up to isomorphism and without any restriction, the coenveloping coalgebra of H (1) in the category of graded connected coalgebras.

4.5 Cocommutative Hopf Algebras Let H be a cocommutative Hopf algebra over an algebraically closed field of characteristic 0. Consider again the Lie algebra g = Prim(H ) and its enveloping algebra U (g), viewed as a Hopf algebra. Let also  be the group of group-like elements with 6

See W. Michaelis, Lie coalgebras, Adv. in Math. 38, 1–54 (1980).

4.5 Cocommutative Hopf Algebras

83

group algebra K , also viewed as a Hopf algebra. For x ∈ g and g ∈ , x g := gxg −1 is also primitive and belongs to g; this action by conjugacy by g is actually a Lie algebra automorphism of g since, for every x, y ∈ g, [x, y]g = [x g , y g ]. Hence, the group  acts on the Lie algebra g and its enveloping algebra U (g). We define the twisted tensor product Hopf algebra   U (g) as the tensor product U (g) ⊗ K  with coproduct the one induced by the tensor product of coalgebras and multiplication given by (x ⊗ g) · (y ⊗ g  ) = x y g ⊗ gg  . Theorem 4.5.1 (Cartier–Gabriel) Assume that H is a cocommutative Hopf algebra over an algebraically closed field of characteristic 0. Let g be the Lie algebra of primitive elements and  the group of group-like elements. Then, there is a Hopf algebra isomorphism from   U (g) to H inducing the identity on  and g. ¯ its iterates  ¯ n , and the increasing filtration Proof. Consider the reduced coproduct ,  ¯ n . Set H¯ 1 := Fn and H1 = H¯ 1 ⊕ K · 1. Then, H1 of their kernels, Fn := K er  n≥2

is, by construction, a cocommutative conilpotent sub-Hopf algebra of H , isomorphic to U (g). Let now g ∈ . Since (g) = g ⊗ g and ε(g) = 1, H = H¯ ⊕ K · g, where H¯ := ¯ g in H¯ by K er ε. Define a new reduced coproduct  ¯ g (x) := (x) − x ⊗ g − g ⊗ x;  ¯ g in a sequence of maps  ¯ gn : H¯ → H¯ ⊗n . From the it maps H¯ into H¯ ⊗2 . Iterate  relation ¯ gn (xg) =  ¯ n (x) · (g ⊗ · · · ⊗ g),  ¯ gn . A product xg yg  , it follows that H¯ 1 · g is the union of the kernels of the maps   g  where x, y ∈ H1 , g, g ∈ , rewrites x y gg and belongs to H1 · gg  . Recall then that, as a coalgebra, H is the union of its finite-dimensional subcoalgebras C. For any such coalgebra C, consider its dual algebra C ∗ . It is a commutative finite-dimensional algebra over an algebraically closed field, and therefore a direct product C ∗ = E 1 × · · · × Er , where E i possesses a unique maximal ideal m i s.t. E i /m i is isomorphic to k and m i is nilpotent: m iN = 0 for some N . Algebra homomorphisms from C ∗ to k correspond to the group-like elements in C. By duality, this decomposition of C ∗ corresponds to a direct sum decomposition C = C1 ⊕ · · · ⊕ Cr , where each Ci contains a unique element gi in . Furthermore, gi from the nilpotency of m i , it follows that Ci ∩ H¯ is annihilated by  N for a large r  enough N , hence Ci ⊂ Hgi := H1 · gi and C = (C ∩ Hgi ). Since H is the union i=1  Hg , hence the theorem. of such coalgebras, the previous relation entails H = g∈

84

4 Structure Theorems

Corollary 4.5.1 Assume that K is algebraically closed of characteristic 0. Then, every finite-dimensional cocommutative Hopf algebra over K is a group algebra K G.

4.6 Complete Cocommutative Hopf Algebras As a motivation to the study of complete cocommutative Hopf algebras, let us start by recalling the construction of the Malcev completion. Example 4.6.1 (Malcev completion) Let G be a group, K a field of characteristic 0, K G its group Hopf algebra with augmentation ideal K G + . Powers of K G + induce a filtered algebra structure on K G: K G =: K G(0) ⊃ K G + =: K G(1) ⊃ · · · ⊃ (K G + )n =: K G(n) ⊃ . . . The corresponding standard complete augmented algebra (see Definition 3.8.3) denoted by  K G inherits from the Hopf algebra structure of K G the structure of a complete Hopf algebra. Indeed, since the coproduct  is a morphism of augmented algebras from K G to K G ⊗ K G, it stabilizes the augmentation ideal and its higher powers: (K G(n)) ⊂ (K G ⊗ K G)(n). Definition 4.6.1 The group ( K G) is called the Malcev completion of G. There are various reasons for the introduction of the Malcev completion. For example, in the study of fundamental groups, one can define a complete Lie algebra associated to such a group. The exponential map is then a bijection from this Lie algebra to the Malcev completion of the group. Two results are particularly important to understand why the classical theory as exposed earlier in this chapter extends almost automatically to the complete setting. First, a map f : A → B between complete filtered vector spaces is an isomorphism if the corresponding map f gr between the associated graded vector spaces is an isoˆ Bˆ from a tensor product morphism. Second, the canonical morphism A ⊗ B → Aˆ ⊗ of filtered vector spaces to the complete tensor product of their completions induces an isomorphism of complete filtered vector spaces ˆ  ˆ B. A ⊗B∼ = Aˆ ⊗ Hereafter we use implicitly the fact that various arguments we previously used (definition of convolution products, of convolution logarithms and exponentials, definition

4.6 Complete Cocommutative Hopf Algebras

85

and properties of characteristic endomorphisms...) extend mutatis mutandis when ˆ 7. tensor products ⊗ are replaced by complete tensor products ⊗ Let H be a cocommutative complete Hopf algebra, H = lim H/H (n). Recall ← from Definition 3.8.5 that this means, inparticular, that H (1) = K er (ε). The associated graded Hopf algebra H gr := H (n)/H (n + 1) is connected since n

H (0)/H (1) = K and satisfies therefore the structure Theorem 4.3.1. We denote ˆ (H ) := {x ∈ 1 + H (1)|(x) = x ⊗x} the set of group-like elements. There is furthermore a decreasing filtration of Lie algebras Prim(H ) ⊃ Prim(H (1)) ⊃ · · · ⊃ Prim(H (n)) ⊃ . . . , ˆ + 1⊗h ˆ ∈ H ⊗H ˆ } and Prim(H (n)) where Prim(H ) := {h ∈ H (1)|(h) = h ⊗1 := Prim(H ) ∩ H (n). The Lie algebra of primitive elements in H is complete: Prim(H ) = lim Prim(H )/Prim(H (n)). ←

Because of the completeness assumption the map e1 := log∗ (I d) is well defined on H and restricts to an endomorphism of H (n) for each n ≥ 0. It also induces on H gr the projection map, still written e1 , onto Prim(H gr ). Lemma 4.6.1 The map e1 is a projection from H onto Prim(H ). Proof. That log∗ (I d) acts as the identity on Prim(H ) follows from the definition of the convolution product. Let us apply the technique of characteristic endomorphisms. We have ˆ d) ◦ .  ◦ I d = (I d ⊗I Besides, provided the action of the logarithmic map is well defined, when a and b commute in an algebra, it holds that log(ab) = log(a) + log(b). Therefore

ˆ d) ◦   ◦ (log∗ (I d)) = log∗ (I d ⊗I

ˆ ∗ (ν ⊗I ˆ d)) ◦  = (log∗ (I d)⊗ν ˆ + ν⊗ ˆ log∗ (I d)) ◦ , = log∗ ((I d ⊗ν) from which it follows, since  is counital, that e1 maps to Prim(H ). 7

Hereafter we sketch the arguments and refer to B. Fresse, Homotopy of Operads and GrothendieckTeichmüller Groups: Parts 1 and 2, Mathematical Surveys and Monographs 217, AMS, 2017, for the results on filtered vector spaces and completions that we use and, more generally, for a detailed study of the topics covered in this section.

86

4 Structure Theorems

Lemma 4.6.2 The canonical map Prim(H )gr → Prim(H gr ) is an isomorphism. Proof. The map is easily seen to be an injection since the filtration of Prim(H ) is gr induced by the one of H . Conversely, given x ∈ H (n), write x for its class in Hn ˆ + 1⊗x ˆ modulo H ⊗H ˆ (n + 1). By definition of e1 , and assume that (x) = x ⊗1 ˆ + 1⊗x) ˆ ∈ H ⊗H ˆ (n + 1), x = e1 (x) modulo H (n + 1), so that since (x) − (x ⊗1 we can assume that x ∈ Prim(H ). The map is therefore surjective and the lemma follows. Lemma 4.6.3 The exponential and logarithmic maps are inverse isomorphisms exp : Prim(H )  (H ) : log . Proof. Indeed, if x ∈ Prim(H ), ˆ + 1⊗x ˆ (x) = x ⊗1 and ˆ + 1⊗x) ˆ = exp(x ⊗1) ˆ · exp(1⊗x) ˆ (exp(x)) = exp((x)) = exp(x ⊗1 ˆ · (1⊗ ˆ exp(x)) = exp(x)⊗ ˆ exp(x), = (exp(x)⊗1) thus exp(x) ∈ (H ), and conversely. The filtration on tensor products of filtered vector spaces and the filtration on Prim(H ) induce a filtration on T S(Prim(H )). Let us write TS(Prim(H )) for the coalgebra in the category of complete vector spaces defined as the completion of the filtered coalgebra (T S(Prim(H )), u ). Theorem 4.6.1 (PBW theorem, complete case) Let H be a cocommutative complete Hopf algebra; the canonical map TS(Prim(H )) → H induced by the map from T S(Prim(H )) to H 

xσ (1) ⊗ · · · ⊗ xσ (n) −→

σ ∈Sn

is an isomorphism of coalgebras.

 σ ∈Sn

xσ (1) . . . xσ (n)

4.6 Complete Cocommutative Hopf Algebras

87

Proof. The map is a morphism of filtered vector spaces and of coalgebras since the elements of Prim(H ) are primitive in H . Passing to the associated graded structures, we get the map: TS(Prim(H ))gr ∼ = T S(Prim(H gr )) → H gr , = T S(Prim(H )gr ) ∼ which is an isomorphism of graded coalgebras by the classical Poincaré–Birkhoff– Witt and Cartier theorems. The theorem follows. Consider now the enveloping algebra U (Prim(H )) as a quotient of the filtered vector space T (Prim(H )), with the filtration induced by the one of Prim(H ). Denote Uˆ (Prim(H )) its completion. The isomorphism of filtered coalgebras T S(Prim(H )) ∼ = U (Prim(H )) extends to an isomorphism of complete coalgebras TˆS(Prim(H )) ∼ = Uˆ (Prim(H )). We get, using the previous PBW theorem or applying Theorem 4.3.1 to the associated graded Hopf algebras: Theorem 4.6.2 (Structure of cocommutative complete Hopf algebras) The canonical map Uˆ (Prim(H )) → H is an isomorphism of cocommutative complete Hopf algebras. Example 4.6.2 (Completion of cocommutative graded connectedHopf algeHn filtered bras) Consider a graded connected cocommutative Hopf algebra H = n 

by H (m) = Hn and its completion Hˆ = Hn . Then, n≥m

n

Hˆ ∼ = Uˆ (Prim( Hˆ )), Prim( Hˆ ) =



(Prim(H ) ∩ Hn )

n

and ( Hˆ ) is the group of characters of the graded dual H ∗ =

 n

we have the isomorphism exp : Prim( Hˆ )  ( Hˆ ) : log .

Hn∗ of H . Finally,

88

4 Structure Theorems

4.7 Remarks and Complements Cogroups in Categories of Algebras By the Hopf–Leray theorem, a commutative connected graded Hopf algebra is a free commutative algebra. This result generalizes as follows to arbitrary categories of algebras over operads. Let P be a unital operad, that is, an operad such that P(1) = k. Recall that we always assume that P(0) = 0, and from now on in this section, P-algebras are algebras without a unit, even for the commutative and associative operads, see Section B.4. Categories of algebras over operads have coproducts, see in Appendix Proposition B.4.1 and Examples B.4.1 and B.4.2. We refer also to the appendix for the general definitions and constructions used hereafter. Coproducts are denoted , excepted for associative algebras for which it is denoted ∗, as the coproduct of associative algebras is the free product. Let us detail this particular example and explain its connections with tensor gebras. We write Algn the category of nonunital associative algebras and Alggn the category of positively graded nonunital associative algebras. The coproduct or free product, ∗, in the category Algn is obtained as follows: let H1 , H2 be two associative algebras, then (H1 ∗ H2 )(n) := [(1, H ⊗n ) ⊕ (2, H ⊗n )], H1 ∗ H2 := n∈N∗

n∈N∗

where (1, H ⊗n ) (resp., (2, H ⊗n )) denotes alternating tensor products of H1 and H2 of length n starting with H1 (resp., H2 ). For example, (2, H ⊗4 ) = H2 ⊗ H1 ⊗ H2 ⊗ H1 . The product of two tensors h 1 ⊗ ... ⊗ h n and h 1 ⊗ ... ⊗ h m in H1 ∗ H2 is defined as the concatenation product h 1 ⊗ ... ⊗ h n ⊗ h 1 ⊗ ... ⊗ h m when h n and h 1 belong, respectively, to H1 and H2 (or to H2 and H1 ), and else as h 1 ⊗ ... ⊗ (h n · h 1 ) ⊗ ... ⊗ h m . Definition 4.7.1 A comonoid in the category of P-algebras is a P-algebra C equipped with a map, called the coproduct, :C →C C which is coassoaciative and counital: (  I dC ) ◦  = (I dC  ) ◦ , (I dC  0) ◦  = (0  I dC ) ◦  = I dC , where 0 stands for the null map. Notice the form of the counit assumption in the absence of units in algebras. Consider again the example of associative algebras. When H1 = T + (V1 ) and H2 = T + (V2 ), the nonunital tensor algebras, one gets H1 ∗ H2 = T + (V1 ⊕ V2 ).

4.7 Remarks and Complements

89

Moreover, by universal properties of free algebras, the linear map ι from V to T + (V ) ∗ T + (V ) defined by ι(v) := (1, v) + (2, v)

(4.4)

induces an algebra map  from T + (V ) to T + (V ) ∗ T + (V ) which is coassociative,  (H1 ∗ cocommutative, and counital (that is, (x) = (1, x) + (2, x) + z with z ∈ n≥2

H2 )(n) ). Notice that the elements of V are the primitive elements, that is, in this context, they are the sole elements such that (x) = (1, x) + (2, x).

(4.5)

Equivalently, T + (V ) is a cocommutative comonoid in Algn and, when V is a positively graded vector space, in Alggn . Setting S(v) := −v on V and extending S to an algebra endomorphism of T + (V ) makes T + (V ) a cocommutative cogroup in Algn and Alggn in the sense of the following definition. Definition 4.7.2 A cogroup in the category of P-algebras is a comonoid H equipped with a map S : H → H , called the antipode, such that furthermore m H ◦ (I d H  S) ◦  = m H ◦ (S  I d H ) ◦  = 0, where we write m H the canonical map from H  H to H . A comonoid (resp., cogroup) is cocommutative if and only if the coproduct map is cocommutative: T ◦  = , where T stand for the switch map from H1  H2 to H2  H1 , where we distinguish notationally between the two copies of H in H  H . Cogroups are less exotic objects as they may seem at first sight. For example, in the category of pointed topological spaces, suspensions are cogroups and the fact that their fundamental groups are free in intimately related to their cogroup structure. A key result due to Berstein8 is a structure theorem for cocommutative cogroups in Alggn . Namely, he proved that a cocommutative cogroup C in Alggn is always isomorphic as a cocommutative comonoid to T + (Prim(C)), where Prim(C) is defined by Eq. (4.5). The following exercise9 allows to better grasp the functioning of comonoids and cogroups as a natural generalization of commutative bialgebras and Hopf algebras.

8

I. Berstein, On co-groups in the category of graded algebras. Transactions of the American Mathematical Society, 115, (1965) 257–269. 9 Taken from M. Livernet, A rigidity theorem for pre-Lie algebras. Journal of Pure and Applied Algebra, 207(1), (2006) 1–18, to which we refer for further insights.

90

4 Structure Theorems

Exercise 4.7.1 Let H be a cocommutative comonoid H in Algn with coproduct . Let δ be the composition of the canonical projection from H ∗ H to (1, H ⊗2 ) with . • Show that δ is coassociative coproduct (it is noncounital). • Show that, with the notation δ(x) = x (1) ⊗ x (2) , the coproduct δ is infinitesimal, that is, (4.6) δ(x · y) = x ⊗ y + x · y (1) ⊗ y (2) + x (1) ⊗ x (2) · y. • Write δ (k) for the iteration of δ to a map from H to H ⊗k and use the Sweedler-type convention δ (n) (x) =: x(1) ⊗ . . . ⊗ x(n). Deduce from Eq. (4.6) and the coassociativity of δ that δ (k) (x · y) =

k−1 

x(1) ⊗ ... ⊗ x(i) ⊗ y(1) ⊗ ... ⊗ y(k − i)

i=1

+

k 

x(1) ⊗ ... ⊗ x(i) · y(1) ⊗ ... ⊗ y(k + 1 − i).

i=1

• Show that  can be recovered from δ by the formula (h) =

 (1, δ (n) (h)) + (2, δ (n) (h)),

(4.7)

n≥1

where δ (1) (h) := h. • Show that, conversely, if H is an infinitesimal bialgebra, that is, an associative nonunital algebra equipped with a noncounital coproduct δ satisfying (4.6), then , as defined by Eq. (4.7), equips H with the structure of a cocommutative comonoid in Algn . The theorem of Hopf–Leray and the results of Berstein on cogroups in the category of associative algebras generalize, in fact, to all categories of algebras over operads: Theorem 4.7.1 (Theorem of Fresse) Let H be a cogroup in the category of complete P-algebras. Then H is the completion of a free P-algebra. Similarly, if H is a positively graded P-algebra equipped with a cogroup structure, then H is a free graded P-algebra. Fresse’s proof of the theorem relies, among others, on the observation that the combinatorial proof of Cartier’s structure theorem used in this monograph holds for Hopf algebras in an arbitrary symmetric tensor category10 . Fresse’s structure theorem

10

B. Fresse, Cogroups in algebras over an operad are free algebras, Commentarii Mathematici Helvetici 73.4 (1998), 637–676. See also B. Fresse, Algèbre des descentes et cogroupes dans les algèbres sur une opérade. Bulletin de la Société Mathématique de France 126.3 (1998), 107–134.

4.7 Remarks and Complements

91

was extended by Oudom to connected counital comagmas in the category of Palgebras11 . Oudom’s proof is direct and uses filtration and graduation arguments. In another direction, over a field of finite characteristic, the theorem has been extended by Patras to cogroups in categories of algebras with divided powers over an operad P12 .

Structure Theorems for Generalized Hopf Algebras A category of algebras can be defined abstractly by means of the associated (monadic) free algebra functor. Algebras over operads enter, for example, in this framework. Dually, coalgebras can be defined by means of a (comonadic) cofree coalgebra functor. Monadicity and comonadicity express categorically the associativity and coassociativity of the composition of operations that define an algebra and a coalgebra. The notion of categories of bialgebras and Hopf algebras has a broad categorical generalization in terms of monads, comonads, and compatibility relations between them13 . This approach allows to extend structure theorems for Hopf algebras in a categorical setting. We refer to the works of Livernet et al. for precise definitions, statements, details, and further references14 .

4.8 Bibliographical Indications The technique of dilations and weight decompositions, as well and their applications to the proof of structure theorems for Hopf algebras, were developed by the second author in his doctoral thesis [Pat1, Pat2, Pat3], see also [Car2]. The Poincaré–Birkhoff–Witt (PBW) theorem goes back to Poincaré’s Sur les groupes continus15 [Poi].

11

J.-M. Oudom, Théorème de Leray dans la catégorie des algèbres sur une opérade. Comptes Rendus de l’Académie des Sciences, Série I, 29.2 (1999), 101-106. 12 F. Patras, A Leray theorem for the generalization to operads of Hopf algebras with divided powers, Journal of Algebra 218.2 (1999), 528–542. 13 I. Moerdijk, Monads on tensor categories, Category theory 1999 (Coimbra), J. Pure Appl. Algebra, 168, (2002), 2-3, 189–208, B. Mesablishvili and R. Wisbauer, Bimonads and Hopf monads on categories, Journal of K-Theory 7.2 (2011), 349–388. 14 M. Livernet, B. Mesablishvili, and R. Wisbauer, Generalised bialgebras and entwined monads and comonads, Journal of Pure and Applied Algebra, 219.8 (2015), 3263–3278. 15 Poincaré’s contribution has been reevaluated recently, and historical studies have confirmed the primacy of his discovery of universal enveloping algebras and of the PBW theorem. See TonThat, Tuong, and Thai-Duong Tran. Poincaré’s proof of the so-called Birkhoff–Witt theorem [La démonstration de Poincaré du théorème dit de Birkhoff-Witt]. Revue d’histoire des mathématiques 5.2 (1999): 249-284.

92

4 Structure Theorems

The structure Theorem 4.3.1 for cocommutative Hopf algebras was obtained by Cartier in [Car1] under assumptions essentially equivalent to the ones we used in this chapter. The theorem was popularized for signed Hopf algebras by Milnor and Moore [MM]. Milnor and Moore also considered the dual statement in the context of commutative graded connected Hopf algebras over a field of characteristic 0. We followed their attribution of Theorem 4.4.2 to Hopf and Leray and of Theorem 4.3.3 jointly to Samelson and Leray [Sam, Ler]. The decomposition of Theorem 4.5.1 was obtained by Cartier and Gabriel during the autumn 1962, in the context of Grothendieck-Demazure’s Séminaire de Géométrie Algébrique du Bois Marie (SGA3, Schémas en groupes), but not included in the published version of the seminar. The proof appeared only later in [Die]; we followed [Car2]. The theorem was obtained independently by Kostant (unpublished). The structure of complete Hopf algebras was investigated by Quillen [Qui] in relation to Malcev groups, Malcev Lie algebras, and groups defined by the Baker– Campbell–Hausdorff formula [Laz]. A detailed and self-contained study of complete Hopf algebras and Malcev groups can be found in Fresse’s monograph [Fre].

References [Car1] Cartier, P.: Hyperalgèbres et groupes de Lie formels. Séminaire Sophus Lie. 2.1955/56 (1957) [Car2] Cartier, P. : A primer of Hopf algebras. In Frontiers in number theory, physics, and geometry II. Springer, Berlin, Heidelberg, 537–615 (2007) [Die] Dieudonné, J.-A.: Introduction to the theory of formal groups, vol. 20. CRC Press (1973) [Fre] Fresse, B.: Homotopy of Operads and Grothendieck-Teichmüller Groups: Parts 1 and 2, Mathematical Surveys and Monographs 217, AMS, 2017 [Laz] Lazard, M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Ec. Norm. Sup. 71, 101–190 (1954) [Ler] Leray, J.: Sur la forme des espaces topologiques et sur les points fixes des représentations. J. Math. Pures Appl 24(9), 95–167 (1945) [MM] Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Annals of Mathematics 211– 264 (1965) [Pat1] Patras, F.: Homothéties simpliciales. January 1992. PhD Thesis. Université Paris 7 [Pat2] Patras, F.: La décomposition en poids des algèbres de Hopf. Ann. Inst. Fourier. 43(4), 1067– 1087 (1993) [Pat3] Patras, F.: L’algèbre des descentes d’une bigèbre graduée. J. Algebra 170(2), 547–566 (1994) [Poi] Poincaré, H.: Sur les groupes continus. Cambridge Philosophical Transaction vol 18, 220–255 (1899). (Oeuvres complètes t. III, Gauthier-Villars 1965, 173-212) [Qui] Quillen, D.: Rational homotopy theory. Annals of Mathematics 205–295 (1969) [Sam] Samelson, H.: Beiträge zur Topologie der Gruppen-Mannigfaltigkeiten. Annals of Mathematics 1091–1137 (1941)

Chapter 5

Graded Hopf Algebras and the Descent Gebra

This chapter is dedicated to the structure of graded connected commutative or cocommutative Hopf algebras. Our approach is based on the properties of their descent gebras, from which many results pertaining to the classical theory of free Lie algebras can be recovered and extended to statements on the structure of graded Hopf algebras. Various statements follow almost immediately from the ideas developed in the previous chapters or from easy computations and are left as exercises. Let us introduce some notation. A composition of n, μ = (μ1 , . . . , μk ) is a sequence of positive integers summing to n; we write μ |= n. Compositions are ordered by refinement: the order > is defined as the transitive closure of the relation >> defined by (μ1 , . . . , μk ) >> (μ1 , . . . , μi + μi+1 , . . . , μn ), 1 ≤ i ≤ k1 ; where (μ1 , . . . , μk ) is called a refinement of (μ1 , . . . , μi + μi+1 , . . . , μn ). In other terms, given two compositions β and μ, β < μ if and only if β can be obtained from μ by summing consecutive terms. For example, (μ1 + μ2 + μ3 , μ4 , μ5 + μ6 ) < (μ1 , μ2 , μ3 , μ4 , μ5 , μ6 ); the minimal composition is the trivial one, (n), and the maximal one is (1, . . . , 1), which is written 1n . A generalized composition of n is a sequence ν = (ν1 , . . . , νk ) of nonnegative integers (that is, possibly equal to 0) and summing to n. We write ν |=0 n. The length k of a sequence μ = (μ1 , . . . , μk ) is written l(μ), the sum of its terms s(μ) := μ1 + · · · + μk . The sum of two sequences α, β is the sequence obtained by concatenating them: α + β := (α1 , . . . , αl(α) , β1 , . . . , βl(β) ).

© Springer Nature Switzerland AG 2021 P. Cartier and F. Patras, Classical Hopf Algebras and Their Applications, Algebra and Applications 29, https://doi.org/10.1007/978-3-030-77845-3_5

93

94

5 Graded Hopf Algebras and the Descent Gebra

In this chapter, the ground field k is of characteristic 0. Various results in the chapter do not require this hypothesis, however most of the interesting ones are obtained making it. From the structure theorems of the previous chapter, we recall that if H is a graded connected cocommutative Hopf algebra, it is the enveloping algebra of Prim(H ), the reduced (Prim(H )0 = 0) graded Lie algebra of its primitive elements. If H is instead a graded connected commutative Hopf algebra, it is a polynomial algebra over Q(H ), the vector space of indecomposables (that we can identify with the eigenspace H (1)).

5.1 Descent Gebras of Graded Bialgebras Let us consider first a graded connected Hopf algebra (A, m, , η, ε). We write In for the canonical graded projection from A to An (so that I0 = η ◦ ε = ν). Notice that since   (I p ⊗ Iq ) ◦ , m ◦ (I p ⊗ Iq ) = In ◦ m,  ◦ In = p+q=n



p+q=n

(I p ⊗ Iq ) is both a quasi-coproduct (Def.2.11.1) and a pseudo-coproduct for

p+q=n

In (Def.2.10.1). Definition 5.1.1 (Descent operators) Let μ |= n. The descent operator associated to the composition μ, denoted Dμ , is defined by Dμ := Iμ1 ∗ · · · ∗ Iμl(μ) , (resp., D∅ =: D0 = I0 = ν). By extension, we also set Dμ := Iμ1 ∗ · · · ∗ Iμl(μ) when μ is a generalized composition. Remark 5.1.1 The name “descent operator” and the use of the word “descent” in this chapter is grounded historically in the study of the particular case where A is the tensor Hopf algebra over a countable alphabet. In this case the operators Dμ can be viewed as elements in the symmetric group algebras having descents in a given subset of [n], see Sect. 5.4. Notice that Dμ ∗ Dβ = Dμ+β and that Dμ ∈ End(An ), where n := s(μ) and End(An ) is viewed as a subset of End(A). Since in the cocommutative (resp., commutative) case quasi-coproducts (resp., pseudo-coproducts) are compatible with convolution products, we have, in this case, that, given μ = (μ1 , . . . , μk ),  βi +γi =μi , i=1,...,k 0≤βi ,γi

Dβ ⊗ Dγ ,

(5.1)

5.1 Descent Gebras of Graded Bialgebras

95

is a quasi-coproduct (resp., a pseudo-coproduct) for Dμ . That is, for a, b ∈ A, 

 ◦ Dμ (a) =

Dβ (a (1) ) ⊗ Dγ (a (2) ),

βi +γi =μi , i=1,...,k 0≤βi ,γi

resp., Dμ (a · b) =



Dβ (a)Dγ (b).

βi +γi =μi , i=1,...,k 0≤βi ,γi

Definition 5.1.2 (Descent gebra) The descent gebra of A, Desc(A), is the linear span of the Dμ (μ |= n, n ≥ 0). Proposition 5.1.1 Equipped with the convolution product, Desc(A) is the convolu∞  tion subalgebra of End(Ai ) ⊂ End(A) generated by the graded projections Ii , i=0

i ≥ 0. As such it is called the descent algebra of A. The algebra (Desc(A), ∗) is a graded algebra (setting |Ii | := i). Its comple  Elements of Desc(A)  are infinite sums tion is written Desc(A). dn , where n≥0

dn ∈ End(An ) ∩ Desc(A).  Recall from the previous Let us give an example of computations in Desc(A). ∞  chapter the notation J := I d − I0 , J n := J ∗n , k := I d ∗k . Expanding I d = Ii i=0

we get



Jnk := In ◦ J k =

Dμ ,

μ|=n,l(μ)=k

nk := In ◦ k =

 k  Dμ . l(μ) μ|=n

Definition 5.1.3 (Ribbon operators) Let us set, for μ |= n, Dμ :=



Rβ .

(5.2)

β≤μ

The operator Rμ is called the ribbon operator associated to the composition μ. By Möbius inversion the Rμ can be expressed as linear combinations of the Dβ , β ≤ μ. The computation will be done later in this chapter exploiting the combinatorics of set partitions.

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5 Graded Hopf Algebras and the Descent Gebra

By construction, the Rμ span linearly Desc(A). Ribbon operators1 are useful to expand elements in the descent gebra but their convolution product rules are slightly more complex than the ones for descent operators: Exercise 5.1.1 Show that for μ |= n, β |= m we have Rμ ∗ Rβ = Rμ+β + Rμ β ,

(5.3)

where μ β := (μ1 , . . . , μl(μ) + β1 , β2 , . . . , βl(β) ). Example 5.1.1 (Ribbon formula for the antipode) It follows from the identities S = I d ∗−1 = (



In )−1 =

n≥0

  I0  = I0 + (−1)k ( In )∗k I0 + In n≥1 k≥1 n≥1

 that the antipode  S belongs to Desc(A). Let us write ζn for the graded components ζn ). We have: of S (S =: n≥0

ζn = (−1)n R1n .

Let us show (using that Im = R(m) and the product formula (5.3)) that (−1)n R1n is indeed a convolution inverse to I d. We have     Im ) ∗ ( (−1)n R1n ) = (−1)n Im ∗ R1n I d ∗ ( (−1)n R1n ) = ( n≥0

m≥0

= I0 +



R(m) +

m≥1

= I0 +



R(m) +

m≥1

= I0 +

n≥0

m,n≥0

  (−1)n R1n + (−1)n R(m) ∗ R1n n≥1

m,n≥1

  (−1)n R1n + (−1)n (R(m)+1n + R(m+1)+1n−1 ) n≥1

k 

m,n≥1

((−1)k−m + (−1)k−m+1 )R(m)+1k−m = I0 .

k≥1 m=1

Exercise 5.1.2 Show that for the polynomial Hopf algebra A = k[x1 , . . . , xn , . . . ] over a set of graded primitive generators (|xi | := i) one has (Desc(A), ∗) ∼ = (k[I1 , . . . , In , . . . ], ·). 1

The terminology “ribbon operators” originates in the theory of symmetric group representations and symmetric functions: irreducible symmetric group representations are encoded by certain (Ferrers or Young) diagrams and “ribbon diagrams” appear naturally in the theory. See I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford university press, 1998. See also Remark 5.4.1.

5.1 Descent Gebras of Graded Bialgebras

97

  is a quasiShow that the coproduct making 1 + i≥1 Ii group-like in Desc(A) coproduct and a pseudo-coproduct (for the action of Desc(A) on A). This coproduct makes Desc(A) a Hopf algebra. Show that A ∼ = Desc(A) as a Hopf algebra (hint:  use, for example, that exp( i≥1 xi ) is a group-like element whose graded components generate A as an algebra). Exercise 5.1.3 Show that for the Hopf algebra of noncommutative polynomials A = k < x1 , . . . , xn , · · · > over a set of graded primitive generators (|xi | := i) one has (Desc(A), ∗) ∼ = (k < I1 , . . . , In , · · · >, ·).  Show that, through this isomorphism, the coproduct making 1 + i≥1 Ii group-like  is a quasi-coproduct (for the action of Desc(A) on A). This coproduct in Desc(A) makes Desc(A) a Hopf algebra; show that A ∼ = Desc(A) as a Hopf algebra. j

Let us introduce some notation. If ν = (νi )1≤ j≤m,1≤i≤n is a m × n matrix of m  j νi and (possibly equal to 0) integers, we will write c(ν) for the sequence ci (ν) := r (ν) for the sequence r j (ν) :=

n  i=1

j=1

j νi .

We write ω(ν) for the composition obtained

by removing all zero from the sequence ν11 , ν21 , . . . , νn1 , . . . , ν1m , . . . , νnm . For ⎛ entries ⎞ 34 example, if ν = ⎝0 5⎠, c(ν) = (5, 9), r (ν) = (7, 5, 2) and ω(ν) = (3, 4, 5, 2). 20 Theorem 5.1.1 When A is commutative (resp., cocommutative) we have Dμ ◦ Dβ =



Dω(ν) ,

c(ν)=μ r (ν)=β

(resp., Dβ ◦ Dμ =



Dω(ν) .)

c(ν)=μ r (ν)=β

In both cases Dμ ◦ Dβ = 0 if s(μ) = s(β). Proof Let us prove the theorem in the commutative case, the cocommutative one will follow by duality. Recall that we write m k for the iterated product (from A⊗k to A) and k for the iterated coproduct (from A to A⊗k ). The natural product (resp., coproduct) on A⊗k is written m (k) (resp., (k) ). Since A is a Hopf algebra, the coproduct (resp., the iterated coproduct k ) is a morphism of algebras from A to A ⊗ A (resp., from A to A⊗k ) and we have  ◦ m = m (2) ◦ ⊗2

98

5 Graded Hopf Algebras and the Descent Gebra

and more generally

l ◦ m k = m k(l) ◦ l⊗k .

(5.4)

We get, for μ and β two compositions of n: Dμ ◦ Dβ = (Iμ1 ∗ · · · ∗ Iμl(μ) ) ◦ (Iβ1 ∗ · · · ∗ Iβl(β) ) = m l(μ) ◦ (Iμ1 ⊗ · · · ⊗ Iμl(μ) ) ◦ l(μ) ◦ m l(β) ◦ (Iβ1 ⊗ · · · ⊗ Iβl(β) ) ◦ l(β) (l(μ))

= m l(μ) ◦ (Iμ1 ⊗ · · · ⊗ Iμl(μ) ) ◦ m l(β) ◦ (l(μ) )⊗l(β) ◦ (Iβ1 ⊗ · · · ⊗ Iβl(β) ) ◦ l(β) . Since i ◦ In =

 ν|=0 n, l(ν)=i

(Iν1 ⊗ · · · ⊗ Iνi ) ◦ i , we obtain

(l(μ) )⊗l(β) ◦ (Iβ1 ⊗ · · · ⊗ Iβl(β) ) =



(Iν 1 ⊗ · · · ⊗ Iν 1

ν j |=0 β j

1

l(μ)

⊗ · · · ⊗ Iν l(β) ) ◦ (l(μ) )⊗l(β) . l(μ)

l(ν j )=l(μ)

The coassociativity of the coproduct implies (l(μ) )⊗l(β) ◦ l(β) = l(β)·l(μ) whereas (l(μ))

1 ⊗ · · · ⊗ Iν l(β) ⊗ · · · ⊗ Iν l(β) ) (Iμ1 ⊗ · · · ⊗ Iμl(μ) ) ◦ m l(β) ◦ (Iν11 ⊗ · · · ⊗ Iνl(μ) 1

l(μ)

is equal to 0 excepted if l(β)

ν11 + · · · + ν1

l(β)

1 = μ1 , . . . , νl(μ) + · · · + νl(μ) = μl(μ)

(l(μ)) 1 and is equal to m l(β) ◦ (Iν11 ⊗ · · · ⊗ Iνl(μ) ⊗ · · · ⊗ Iν l(β) ⊗ · · · ⊗ Iν l(β) ) in that case. (l(μ))

1

l(μ)

Finally, since m is commutative, m l(μ) ◦ m l(β) = m l(μ)·l(β) , 

Dμ ◦ Dβ =

1 Iν11 ∗ · · · ∗ Iνl(μ) ∗ · · · ∗ Iν l(β) ∗ · · · ∗ Iν l(β) , 1

l(μ)

c(ν)=μ r (ν)=β

and the theorem follows.

5.2 Lie Idempotents The present section is mainly devoted to the study of projections from a graded connected cocommutative Hopf algebra H to Prim(H ), its Lie algebra of primitives. When H is the tensor algebra T (X ) over an infinite set X , the projections that

5.2 Lie Idempotents

99

belong to the corresponding descent gebra (that will be studied in detail in the next section) are classically known as Lie idempotents. We use the same name for their generalization to arbitrary descent gebras. They play an important role in the theory of free Lie algebras and connected topics. Let first A be an arbitrary graded connected Hopf algebra and recall from the 1 ∗i previous chapter the notation e1 := log∗ (I d), ei := (ei!) . When A is commutative or cocommutative, the ei are the canonical projections on the eigenspaces associated ∞  to the dilations k . Expanding I d = Ii we get i=0

en1 := In ◦ e1 =





(−1)k−1 (

μ|=n,l(μ)=k

1≤k≤n

Dμ ). k

(5.5)

Corollary 5.2.1 The canonical idempotents (en1 )n∈N∗ generate Desc(A) as an algebra for the convolution product. Proof Indeed, the In n ≥ 1, generate Desc(A) as an algebra (with unit I0 = η) and en1 = In + h.o.t., where h.o.t. (higher order terms) stands for a linear combination of nontrivial convolution products of the Ii , i < n. Exercise 5.2.1 Show that, more generally, we have eni := In ◦ ei = (



s(l(μ), i) )Dμ . l(μ)! μ|=n,i≤l(μ)

Exercise 5.2.2Recall the definition of the derivation D: D(a) := n · a for a ∈ An , i · Ii . Recall also the definition of the Dynkin operator (that we that is, D := i>0

defined in the cocommutative case, the definition is now extended to arbitrary graded connected Hopf algebras) Dyn := S ∗ D so that Dyn =



Dyn n with Dyn n := In ◦ Dyn. Show that the family (Dyn n )n≥1

n≥0

generates Desc(A) as a convolution algebra. Remark 5.2.1 An element f in End(A) is said to be quasi-primitive if  ◦ f = ( f ⊗ ν + ν ⊗ f ) ◦ . An endomorphism f is a quasi-primitive if and only if it maps A to Prim(A). For example, when A is cocommutative the Dyn n and the en1 map to Prim(A) and are quasi-primitive. Proposition 5.2.1 (Ribbon formula for the Dynkin operator) The Dynkin operator can be expanded in the descent gebra as

100

5 Graded Hopf Algebras and the Descent Gebra

Dyn n =

n−1 

(−1)i R1i +(n−i) .

i=0

Proof Indeed, according to the ribbon formula for the antipode (ζn = (−1)n R1n ), we have Dyn n =

n−1 n−1   (−1)i R1i ∗ ((n − i)In−i ) = (−1)i R1i ∗ ((n − i)R(n−i) ) i=0

i=0

= R(n) +

n−1 

(−1)i (n − i)(R1i +(n−i) + R1i−1 +(n+1−i) )

i=1 n−1 n−1   i i+1 = R(n) + ((−1) (n − i) + (−1) (n − i − 1))R1i +(n−i) = (−1)i R1i +(n−i) . i=1

i=0

Theorem 5.2.1 Let us assume that f ∈ Desc(A) is quasi-primitive. Then, f n := In ◦ f is a quasi-idempotent (that is, f n2 = αn f n for some element of the ground field αn ). Moreover, if f n = 0 and αn = 0, f n /αn is a projector onto Prim(A)n . Proof Let us show first that f n2 = αn f n for some αn . Since f ∈ Desc(A), f n can be written f n = αn In + h.o.t., where h.o.t. stands for a linear combination of nontrivial convolution products of the Ii , i > 0 (this decomposition is nonunique, in general, but the proof will imply that αn is uniquely defined). By definition, whenever a ∈ Prim(A) and k, l > 0 we have Ik ∗ Il (a) = Ik (a)Il (1) + Ik (1)Il (a) = 0 since Ik (a) = Il (a) = 0. The same calculation shows that in general Ik1 ∗ . . . Ikn vanishes on Prim(A) when the ki are strictly positive. We get, since for a ∈ A, f n (a) is primitive: f n ◦ f n (a) = (αn In + h.o.t.)( f n (a)) = αn f n (a). The same argument also implies that for a ∈ Prim(A)n f n (a) = αn · a, hence the surjectivity property when αn = 0. Theorem 5.2.2 (Baker–Campbell–Hausdorff ribbon formula). We have en1 =

 (−1)l(β)−1  n − 1 −1 Rβ . n l(β) − 1 β|=n

5.2 Lie Idempotents

101

Proof We have en1 = Therefore, since Dμ =

aβ =

 β≤μ

 (−1)l(μ)−1 Dμ . l(μ) μ|=n

Rβ , we can reexpand en1 as

 β|=n

aβ Rβ with

n−l(β)  (−1)l(μ)−1  (−1) j+l(β)−1 n − l(β) = , l(μ) j + l(β) j β≤μ|=n j=0

where we used implicitly enumerative properties of compositions, namely, the fact that compositions (μ1 , . . . , μk ) of n are in bijection with subsets {μ1 , μ1 + μ2 , . . . , μ1 + · · · + μk−1 } of [n − 1]. Therefore, aβ = (−1)l(β)−1

n−l(β) 1  j=0



1

= (−1)l(β)−1

0

(−1) j

  n − l(β) j+l(β)−1 x dx j

(1 − x)n−l(β) x l(β)−1 d x =

0

  (−1)l(β)−1 n − 1 −1 . n l(β) − 1

Remark 5.2.2 The classical discrete Baker–Campbell–Hausdorff (BCH) formula2 computes the logarithm of a product of exponentials and its continuous version computes the logarithm of the solution of a first-order matrix or operator-valued linear differential equation. The discrete case can be shown to be a particular case of the continuous one. Let us explain the connection of the BCH ribbon formula with logarithms of products of exponentials. In the tensor gebra over a set X , equipped with the concatenation product and the unshuffling coproduct, exponentials of elements x, y in X are group-like elements since x and y are primitive in T (X ), and products of such exponentials are also group-like. However, for a group-like element g we have in general in a Hopf algebra (since ν(g) = 1) log(g) =

 (−1)n−1  (−1)n−1 (I d − ν)⊗n mn ◦ (g − 1)n = ◦ n (g) = log∗ (I d)(g). n n n≥1

n≥1

It follows, for example, that the degree n component of log(exp(x) exp(y)) in T (X ) can be obtained by expanding  (−1)l(β)−1  n − 1 −1 Rβ (exp(x) exp(y)). n l(β) − 1 β|=n

2

We refer to Reutenauer’s Free Lie Algebras (vol. 7, London Mathematical Society Monographs, New Series, 1993) for details on the formula and its various variants.

102

5 Graded Hopf Algebras and the Descent Gebra

5.3 Logarithmic Derivatives The techniques we have developed can be applied beyond the framework of descent algebras. We feature here, for example, the notion of logarithmic derivatives (of which the Dynkin operator will appear to be a canonical example). Most results below do not require that the underlying Hopf algebra be graded; however, it was natural to include these developments in this section. From now on in the section, A denotes a graded connected cocommutative Hopf algebra and L its graded Lie algebra of primitive elements. Recall that A is an enveloping algebra for L and we identify A with U (L), the quotient of the tensor algebra over L, T (L), by the ideal I generated by the elements ll  − l l − [l, l  ], l, l  ∈ L. Let δ be an arbitrary graded Lie algebra derivation of L, that is, ∀l, l  ∈ L , δ([l, l  ]) = [δ(l), l  ] + [l, δ(l  )].

(5.6)

The map δ on L has a unique extension to an algebra derivation of A that we also write abusively δ. This follows from the general theory of enveloping algebras but can be checked directly. Indeed, the derivation δ extends uniquely to a derivation of T (L) (by requiring δ(ab) = δ(a)b + aδ(b) for a, b ∈ T (L)) and goes over to the quotient A = T (L)/I since I is clearly stable under the action of δ. Definition 5.3.1 We write Dδ := S ∗ δ and call Dδ the logarithmic derivative corresponding to the Lie algebra derivation δ. When δ = D, the graduation operator, we recover the Dynkin operator D D = Dyn. The terminology is motivated by the following observation: for an element l ∈ L,  exp(l) is group-like and we have in U (L): Dδ (exp(l)) = S(exp(l))δ(exp(l)) = exp(−l)δ(exp(l)), expression where we recognize the (noncommutative) logarithmic derivative of exp(l) with respect to δ. Proposition 5.3.1 The logarithmic derivative Dδ is quasi-primitive and maps therefore A = U (L) to L. Proof Recall that U (L) is spanned by products l1 ...ln of elements of L. Since δ is a derivation and maps L to L, we get  ◦ δ(l1 . . . ln ) = (

n 

l1 . . . δ(li ) . . . ln ) = (δ ⊗ I d + I d ⊗ δ) ◦ (l1 . . . ln ),

i=1

where the last identity follows from the fact that the li and the δ(li ) are primitive. In particular, δ ⊗ I d + I d ⊗ δ is a quasi-coproduct for δ. We get finally, using that, since  is cocommutative, S ⊗ S is a quasi-coproduct for S,

5.3 Logarithmic Derivatives

103

 ◦ Dδ =  ◦ (S ∗ δ) = (S ⊗ S) ∗ (δ ⊗ I d + I d ⊗ δ) ◦  = (Dδ ⊗ ν + ν ⊗ Dδ ) ◦ . Proposition 5.3.2 For l ∈ L, we have δ(l) = Dδ (l). In particular, when δ is invertible on L, Dδ is a surjection from U (L) onto L. Proof We have indeed Dδ (l) = (S ∗ δ)(l) = π ◦ (S ⊗ δ) ◦ (l) = π ◦ (S ⊗ δ)(l ⊗ 1 + 1 ⊗ l) = π(S(l) ⊗ δ(1) + S(1) ⊗ δ(l)) = δ(l), since δ(1) = 0 and S(1) = 1. We conclude this section by showing how the classical Magnus formula generalizes in the present context. The formula relates logarithms to logarithmic derivatives in the context of linear differential equations. That is, it relates explicitly the logarithm log(X (t)) =: (t) of the solution to an arbitrary matrix (or operator) differential equation X  (t) = A(t)X (t), X (0) = 1 to the infinitesimal generator A(t) of the differential equation:  (t) =

 Bn ad(t) n A(t) = A(t) + ad(t) (A(t)), ad exp (t) −1 n! n>0

where ad stands for the adjoint action (adx (y) := x y − yx) and the Bn are the Bernoulli numbers, that is, the coefficients of the formal power series expansion of x/(exp(x) − 1). Lemma 5.3.1 For l ∈ L, k ≥ 1 and x ∈ U (L), we have k    k k−i l · (−adl )i (x). x ·l = i i=0 k

Proof The proof is by induction. The formula holds for k = 1 since xl = −[l, x] + lx. Let us assume that it holds for an arbitrary n < p. Then we have x · l p = (x · l p−1 ) · l = (

 p−1   p − 1 p−1−i l · (−adl )i (x)) · l i i=0

  p−1  p−1    p − 1 p−1−i p − 1 p−i i+1 l l = · (−adl ) (x) + · (−adl )i (x). i i i=0 i=0 The identity follows then from the properties of binomial coefficients.  Theorem 5.3.1 (Hopf algebraic Magnus formula) For l ∈ L, we have in U (L) Dδ (exp(l)) =

exp(−adl ) − 1 δ(l). −adl

104

5 Graded Hopf Algebras and the Descent Gebra

Proof Indeed, from the previous formula, we get n−1  1  1  n δ(l ) = δ(exp(l)) = l n−1−k δ(l)l k n! n! n≥1 n≥1 k=0 n−1   n−1  1  k n−1−i )l = (−adl )i (δ(l)) ( n! i n≥1 i=0 k=i

=

=

  n−1  n 1 l n−1−i (−adl )i (δ(l)) n! i + 1 n≥1 i=0

  i≥0

1 l n−1−i (−adl )i (δ(l)) (i + 1)!(n − 1 − i)! n≥i+1 

= exp(l)(

i≥0

1 (−adl )i )δ(l). (i + 1)!

Since exp(l) is the exponential of a primitive element, it is group-like, S(exp(l)) = exp(−l), Dδ (exp(l)) = exp(−l)δ(exp(l)) and the theorem follows. Theorem 5.3.2 When δ is invertible on L, the logarithmic derivative Dδ is a bijection  between the set of group-like elements in U (L) and L. Proof Indeed, by induction on the degree, for h ∈ L the equation in L l = δ −1 (

−adl (h)) exp(−adl ) − 1

has a unique recursive solution l ∈ L such that Dδ (ex p(l)) = h.

5.4 The Descent Gebra The descent gebras of connected graded Hopf algebras satisfy certain universal relations: many identities we obtained are common to all of them. When furthermore one assumes that the underlying Hopf algebras are cocommutative (or commutative), extra relations hold for the composition product. It is therefore natural to define a “universal” descent gebra Desc (an “abstract” descent algebra using the original terminology) using these relations as the defining identities. Remarkably, doing so, the resulting gebra has not only a very rich structure but connects also to fundamental mathematical objects such as Solomon’s descent algebras or the Hopf algebra of quasi-symmetric functions.

5.4 The Descent Gebra

105

In what follows, we use abusively the same notation for the elements of the universal descent gebra Desc and the elements of descent gebras of bialgebras, Desc(A). This should not cause particular problems in practice since by Proposition 5.4.1 the action of Desc on a given Hopf algebra A factorizes through Desc(A). We focus here on the definition of universal operations on graded connected cocommutative Hopf algebras. The definitions and results dualize in the commutative case.  Descn , is, Definition 5.4.1 (Descent gebra) The descent gebra, written Desc = n∈N

as a vector space, the one underlying the free graded associative unital algebra over a set of graded generators In , n ≥ 1, whose degrees are given by |In | = n. The (free, graded) product is written ∗ and called the convolution product. Its unit is denoted I0 . As a free associative algebra, Desc equipped with the convolution product is called the descent algebra. This algebra is equipped with a cocommutative Hopf algebra structure by requiring the generators to form a group-like family of elements: (In ) =



Ik ⊗ In−k .

0≤k≤n

For μ |= n (resp., μ |=0 n), we set Dμ := Iμ1 ∗ · · · ∗ Iμl(μ) , and call Dμ the descent operator associated to μ. Its coproduct is given by 

Dβ ⊗ Dγ .

(5.7)

βi +γi =μi , i=1,...,l(μ) 0≤βi ,γi

As a Hopf algebra, Desc, is called the descent Hopf algebra. Another associative product, called the composition product or the internal product, is defined by Dβ ◦ Dμ :=



Dω(ν)

c(ν)=μ r (ν)=β

when β |= m, μ |= n and m = n. When n = m, we set Dβ ◦ Dμ := 0. Notice that ◦ defines an associative and unital algebra structure on Descn . The coproduct is compatible with the composition product: (Dβ ◦ Dμ ) = (Dβ ) ◦ (Dβ ).  is the completion of Desc with respect to the The complete descent gebra, Desc  grading: Desc = Descn ; it inherits from Desc its various structures. n∈N

106

5 Graded Hopf Algebras and the Descent Gebra

The properties stated in this definition (associativity, compatibilities between products and coproduct...) can be checked directly but also follows from the existence of realizations of Desc as the descent gebra of a particular Hopf algebra (for example, as its own descent algebra since Desc(Desc) = Desc, see Exercise 5.1.3). These properties follow thus from the general properties of descent gebras of Hopf algebras as studied in the previous sections of this chapter. It also follows from the results in these sections that Proposition 5.4.1 Given a graded connected cocommutative Hopf algebra A, there is a canonical map from Desc to Desc(A) mapping the descent operators Dμ in Desc to the corresponding elements in Desc(A). This map is compatible with the various structures: it maps the convolution and composition products to the convolution and composition products; the coproduct of a descent operator Dμ ∈ Desc is mapped to a quasi-coproduct of Dμ ∈ Desc(A). Remark 5.4.1 (Hopf algebra of noncommutative symmetric functions) The Hopf algebra (Desc, ∗, ) identifies with the Hopf algebra of noncommutative symmetric functions. The latter is indeed, by definition, a free commutative algebra over a countable and group-like family of elements. As such it is a noncommutative generalization of the Hopf algebra of symmetric functions which is, as a Hopf algebra, a free commutative algebra over a countable and group-like family of elements. Although the descent Hopf algebra and the one of noncommutative symmetric functions are isomorphic, two different languages and systems of notation have been developed to handle them. The first relates to Lie theory, Lie idempotents, symmetric groups, and structural properties of Hopf algebras, whereas the one of noncommutative symmetric functions insists on operations on alphabets, realizations of the objects, and computations in algebras of noncommutative formal power series. Remark 5.4.2 The definition of the descent gebra is fine-tuned to the properties of graded connected cocommutative Hopf algebras. Dually, one could define a notion of descent gebra fine-tuned to the commutative case by requiring Dμ ◦ Dβ =



Dω(ν) ,

c(ν)=μ r (ν)=β

all the other definitions and properties being unchanged. Remark 5.4.3 The descent gebra carries in fact other structures, for example, it can be given a free noncommutative λ-ring or a free noncommutative -ring structure, and is also a universal algebra of operations from this point of view of noncommutative λ-rings or -rings.3

3

See F. Patras, Lambda-anneaux non commutatifs. Comm. in Algebra 23, 6 (1995), 2067–2078.

5.5 Combinatorial Descents

107

5.5 Combinatorial Descents Let us turn now to a combinatorial model for Desc, useful in many situations relevant to group theory, Lie theory, and their relationships. We start by recalling various fundamental definitions. Definition 5.5.1 (Descent sets of permutations) A permutation σ ∈ Sn is said to have a descent in position i < n if and only if σ (i) > σ (i + 1). The set of descents of a permutation is denoted desc(σ ): desc(σ ) := {i < n, σ (i) > σ (i + 1)}. For example, with the notation σ = (σ (1), . . . , σ (n)): desc((31425)) = {1, 3}. Definition 5.5.2 To each subset S = S   {n} of [n] containing n are associated two elements in the group algebra Q[Sn ]: De=S :=



σ,

De S :=

σ ∈Sn desc(σ )=S 



σ.

σ ∈Sn desc(σ )⊂S 

Lemma 5.5.1 By inclusion/exclusion, the two elements are related by the following formulas:   De S = De=T , De=S = (−1)|S|−|T | DeT . T ⊂S {n}∈T

T ⊂S {n}∈T

Moreover, the De S (resp., the De=S ), S = S   {n} ⊂ [n] are linearly independent in Q[Sn ]. Proof The sets {σ ∈ Sn , desc(σ ) = S  } with S  ⊂ [n − 1] are non-empty and define a partition of Sn , linear independence follows. Let now X = {x1 , . . . , xn , . . . } be a countable alphabet and T (X ) the tensor gebra over X equipped with a graded connected cocommutative Hopf algebra structure by the concatenation product and the unshuffle coproduct. That is, we view T (X ) as the tensor Hopf algebra over X . Recall the notation x S = xs1 . . . xsk for S = {s1 , . . . , sk }. Permutations in Sn act on Tn (X ) on the right by permutation of the letters of words of length n (σ (y1 . . . yn ) = yσ (1) . . . yσ (n) for yi ∈ X, i ≤ n). This action is extended by the null action on the components of degree different from n so that Sn embeds in End(T (X )). Notice that since this is a right action, the composition law of permutations viewed as endomorphisms of T (X ) is the opposite of the usual law: β ◦ σ (y1 . . . yn ) = β(σ (y1 . . . yn )) = σ · β(y1 . . . yn ), where ◦ stand for the composition of endomorphisms of T (X ) and · for the usual composition product of permutations. The action of the element De S on a word y1 . . . yn , yi ∈ X is then given by

108

5 Graded Hopf Algebras and the Descent Gebra

Lemma 5.5.2 Let S = S   {n} = {s1 , . . . , sk } ⊂ [n], then 

De S (y1 . . . yn ) = I1

... Ik =[n],|Ii |=si −si−1

y I1 · . . . · y Ik ,

where s0 := 0. Proof The lemma follows from the observation that a permutation σ has a descent set included in {s1 , . . . , sk−1 } if and only if σ (1) < · · · < σ (s1 ), . . . , σ (sk−1 + 1) < · · · < σ (n). Setting I1 := {σ (1), . . . , σ (s1 )}, . . . , Ik : ={σ (sk−1 + 1), . . . , σ (n)} gives then a bijection between the permutations with descent set included in S  and the index set of the sum in the right-hand side of the identity. Theorem 5.5.1 The canonical map can from Desc to Desc(T (X )) is an isomorphism preserving the products ∗, ◦ and the coproduct . It maps Dμ , μ |= n to De S(μ) , with S(μ) := {μ1 , μ1 + μ2 , . . . , μ1 + · · · + μl(μ) }, where μ1 + · · · + μl(μ) = n. Equivalently, it maps the ribbon operation Rμ to De=S(μ) . In particular the various ribbon formulas translate into formulas in the symmetric group algebras. Proof By Prop. 5.4.1, the map is compatible with the various products and the coproduct. It is therefore enough to prove that Dμ = De S(μ) in Desc(T (X )): the fact that the canonical map from Desc to Desc(T (X )) is an isomorphism will then follow from the linear independence of the De S . Let us prove the property by induction on the length l(μ) of compositions. First, can(In ) = I dT (X )n = De{n} and, by induction, can(Dμ+(k) ) = can(Dμ ∗ Dk ) = De S(μ) ∗ De{k} . Since a permutation σ ∈ Sn is completely characterized by its action on the word x1 . . . xn (and similarly for a linear combination of permutations), let us compute De S(μ) ∗ De{k} (x1 . . . xn+k ), where n := l(μ). By definition of the convolution product of endomorphisms of T (X ), we get can(Dμ+(k) )(x1 . . . xn+k ) = De S(μ) ∗ De{k} (x1 . . . xn+k ) =



I J =[n+k] |I |=n,|J |=k



= I1



... Il(μ) J =[n+k] |Ii |=μi ,|J |=k

De S(μ) (x I )De{k} (x J )

y I1 · . . . · y Il(μ) · y J = De S(μ+(k)) ,

5.5 Combinatorial Descents

109

by Lemma 5.5.2.The statement on ribbon operators follows from their definition and the one of the De=S , both related by Möbius inversion to the descent operators and the De S . Notice that the proof implies that—as implicit in the statement of the theorem—the quasi-coproduct of formula (5.1) defines a Hopf algebra structure on Desc(T (X )). Corollary 5.5.1 The descent algebra Desc(T (X )) is a graded cocommutative Hopf algebra, freely generated as a unital associative algebra by any of the following families: • The identity permutations in the groups Sn , n ≥ 1, that form a group-like family. • The Dynkin operators Dyn n =

n−1 

(−1)i De={1,...,i,n} ,

i=0

which are primitive elements. • The canonical projections (also called in that case Solomon or Eulerian idempotents)  (−1)l(β)−1  n − 1 −1 soln := en1 = De=S(β) , n l(β) − 1 β|=n which are primitive elements. The graded components Descn (T (X )) are subalgebras of the opposite algebras to the symmetric group algebras. The composition law is given by De S(β) ◦ De S(μ) = De S(μ) · De S(β) =



De S(ω(ν))

c(ν)=μ r (ν)=β

when β |= n, μ |= n, where · denotes the product in Q[Sn ], the group algebra of the symmetric group. Remark 5.5.1 One can dualize the graded cocommutative Hopf algebra structure of Desc, viewing the basis of descent operators as an orthonormal basis. This defines another, commutative, Hopf algebra structure that identifies with a Hopf algebra known as the Hopf algebra of quasi-symmetric functions.

5.6 Bibliographical Indications Descent (al)gebras of graded Hopf algebras were introduced and studied by the second author in [Pat3] as a follow up of [Pat1, Pat2], and of the approach to the structure of unipotent Hopf algebras presented in the previous chapter. Their study

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5 Graded Hopf Algebras and the Descent Gebra

was refined and their various structures investigated progressively in a series of articles together with Ch. Reutenaeur emphasizing, in particular, applications related to Lie idempotents [PR]. The results on logarithmic derivatives given as an example of application of these ideas outside the context of descent gebras are motivated by problems in the theory of dynamical systems, they generalize classical results on the Dynkin idempotents and the Magnus formula: see Menous and Patras [MP], that we followed. The combinatorial approach to the descent gebra in terms of symmetric groups appeared much earlier and was not motivated originally by Hopf algebras. It originates with Solomon [Sol2]. Interested in the representation theory of symmetric groups, he introduced and studied algebras that identify with the graded components of Desc(T (X )), X a countable alphabet, equipped with the restriction to Desc(T (X )) of the product of permutations. These algebras are usually referred to as Solomon’s algebras of type An−1 in representation theory. Solomon’s algebras and their links with the combinatorics of tensors were studied by F. Bergeron, N. Bergeron, Garsia, Reutenauer, and others: see in particular [GR]. These ideas gave rise to what is now called the noncommutative representation theory of symmetric groups that was investigated systematically by L. Blessenohl, M. Schocker, and others [BS]. This theory also applies to other finite Coxeter groups than symmetric groups. The structure of the full combinatorial descent gebra Desc(T (X )), that is, when moving beyond the structure of its graded components, was investigated by Malvenuto and Reutenauer who also featured the existence of a coproduct and duality with quasi-symmetric functions [Mal, MR, Reu1]. Another key contribution to the theory came from the development of the theory of noncommutative symmetric functions. As we already mentioned, the gebra of noncommutative symmetric functions, that was introduced in [G+], identifies indeed canonically with the descent gebra. The approach of noncommutative symmetric functions lead to new insights on descents and Lie idempotents. Among others, the role of the coproduct was put in evidence, for example, with the striking result that primitive elements in the descent gebra are quasi-idempotents (Theorem 5.2.1), a result that was first obtained in this context. The theory of noncommutative symmetric functions was developed intensively and systematically by J.-Y. Thibon and his collaborators during the last 25 years. It forms now an important chapter of the combinatorial theory of Hopf algebras that goes largely in another direction than the one we will develop in Chap. 9. Classical Lie idempotents (projections from the tensor algebra to the graded components of the free Lie algebra that belong to the symmetric group algebras) have a long history. Dynkin’s idempotents are among the most important ones. The other main idempotents in the symmetric group algebra are the canonical ones: soln :=

 (−1)|S|−1 D S ∈ Q[Sn ] |S| S⊂[n] n∈S

which are known under various names in the literature: Solomon idempotent of type An−1 , Barr idempotent, n-th Eulerian idempotent (Eulerian numbers enumer-

5.6 Bibliographical Indications

111

ate permutations with a given number of descents)... The family of the soln was discovered by Solomon in 1968 [Sol1]. He showed that they encode the natural projection from the free associative algebra T (X ) over a set X to the corresponding free Lie algebra Lie(X ) induced by the Poincaré–Birkhoff–Witt theorem. Mielnik and Pleba´nski obtained them independently in 1970 [MiP] in relation to the Baker– Campbell–Hausdorff formula. A fundamental discovery by Reutenauer [Reu2] that was generalized later to graded connected or unipotent cocommutative Hopf algebras in [Pat1, Pat2] was the computation of these idempotents using the logarithm of the identity in the convolution algebra of linear endomorphisms of tensor algebras. For more details on classical Lie idempotents, we refer to Reutenauer’s monograph [Reu1].

References [BS] [GR] [G+] [Mal]

[MR] [MP] [MiP] [Pat1] [Pat2] [Pat3] [PR]

[Reu1] [Reu2]

[Sol1] [Sol2]

Blessenohl, D., Schocker, M.: Noncommutative Character theory of symmetric Groups. Imperial College Press, London (2005) Garsia, A., Reutenauer, C.: A decomposition of Solomon’s descent algebra. Adv. Math. 77(2), 189–262 (1989) Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218–348 (1995) Malvenuto, C.: Produits et Coproduits des fonctions quasi-symétriques et de l’algèbre des descentes. Publications du Laboratoire de Combinatoire et d’Informatique Mathématique - UQAM 16, (1994) Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967–982 (1995) Menous, F. , Patras, F.: Logarithmic Derivatives and Generalized Dynkin Operators. Journal of Algebraic Combinatorics 38(4), 901–913 (2013) Mielnik, B., Pleba´nski, J.: Combinatorial approach to Baker-Campbell-Hausdorff exponents. Ann. Inst. H. Poincaré A 12, 215–254 (1970) Patras, F.: Homothéties simpliciales. Janvier 1992. PhD Thesis. Université Paris 7 Patras, F.: La décomposition en poids des algèbres de Hopf. Ann. Inst. Fourier. 43(4), 1067–1087 (1993) Patras, F.: L’algèbre des descentes d’une bigèbre graduée. J. Algebra 170(2), 547–566 (1994) Patras, F., Reutenauer, C.: Higher Lie idempotents. J. Algebra 222, 51–64 (1999); Lie representations and an algebra containing Solomon’s. J. Alg. Comb 16, 301–314 (2002); On Dynkin and Klyachko idempotents in graded bialgebras. Adv. Appl. Math. 28(3), 560– 579 (2002) Reutenauer, Ch.: Free Lie Algebras. Oxford University Press, Oxford (1993) Reutenauer, Ch.: Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers. In Combinatoire énumérative, Springer 267–284 (1986) Solomon, L.: On the Poincaré-Birkhoff-Witt theorem. J. Combin. Theory 4, 363–375 (1968) Solomon, L.: A Mackey formula in the group algebra of a finite Coxeter group, J. Algebra 41, 255–268 (1976). Tits, J.: Two properties of Coxeter complexes, (Appendix to Solomon’s article), J. Algebra 41, 255–268 (1976)

Chapter 6

Pre-lie Algebras

Pre-Lie algebras, also called Vinberg algebras, have become an important tool in combinatorics, differential geometry, the theory of operads and in various application domains such as perturbative quantum field theory or numerical analysis, to quote only a few. They generate a special class of Lie algebras and have enveloping algebras enjoying more properties than usual enveloping algebras of Lie algebras. The notions of pre-Lie products and symmetric braces together with their associated Hopf algebra structures have a nonsymmetric variant important in various contexts, for example, in homological algebra or algebraic topology where it originates. Brace algebras are a far reaching generalization of the notion of shuffle tensor Hopf algebra and of quasi-shuffle tensor Hopf algebra, the latter introduced and accounted for in this chapter. We start with the description of the basic definitions and structures related to pre-Lie algebras, featuring the Hopf algebraic point of view and the link with symmetric braces. Follows an account of the Lie and group-theoretical part of the theory emphasizing the interplay between two logarithms and exponential maps relating a pre-Lie algebra to the associated group under suitable completion assumptions. The purpose of this chapter is also to survey fundamental examples of pre-Lie algebras, mostly those who have a structural meaning for the theory. A second part introduces brace algebras and the associated right-handed tensor Hopf algebras. Among others, it explains why the notion is a natural generalization of the one of shuffle tensor Hopf algebras.

6.1 The Basic Concept Lie remarked that for commutators [a, b] := ab − ba the Jacobi identity, J (a, b, c) := [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0, © Springer Nature Switzerland AG 2021 P. Cartier and F. Patras, Classical Hopf Algebras and Their Applications, Algebra and Applications 29, https://doi.org/10.1007/978-3-030-77845-3_6

113

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pertaining to the definition of Lie algebras, is implied by associativity. The computation is trivial, but it will be useful to recall it here. For a, b, c elements of any algebra: [a, [b, c]] = a(bc) − a(cb) − (bc)a + (cb)a, [b, [c, a]] = b(ca) − b(ac) − (ca)b + (ac)b, [c, [a, b]] = c(ab) − c(ba) − (ab)c + (ba)c. Define the associator A(a, b, c) of three elements a, b, c to be A(a, b, c) = a(bc) − (ab)c (when a product is denoted by a symbol such as m, its associator will be denoted Am (a, b, c), so that Am (a, b, c) = m(a, m(b, c)) − m(m(a, b), c)). An algebra is associative iff A(a, b, c) vanishes identically. As for J (a, b, c), it is the total skewsymmetrization of A(a, b, c): J (a, b, c) = A(a, b, c) − A(a, c, b) − A(b, a, c) + A(b, c, a) + A(c, a, b) − A(c, b, a).

so that, for J to vanish, it is not necessary that A vanish in turn; it is enough that the right Vinberg identity holds A(a, b, c) − A(a, c, b) = a(bc) − (ab)c − a(cb) + (ac)b = 0,

(6.1)

or, symmetrically, that the left Vinberg identity A(a, b, c) − A(b, a, c) = 0 holds. Definition 6.1.1 A vector space A equipped with a bilinear product  satisfying the right Vinberg identity A (a, b, c) − A (a, c, b) = 0, that is a  (b  c) − (a  b)  c = a  (c  b) − (a  c)  b

(6.2)

is called a pre-Lie algebra (or also right pre-Lie algebra when some ambiguities might arise, see also the next definition). Definition 6.1.2 A vector space A equipped with a bilinear product  satisfying the left Vinberg identity A (a, b, c) − A (b, a, c) = 0, that is a  (b  c) − (a  b)  c = b  (a  c) − (b  a)  c is called a left pre-Lie algebra.

(6.3)

6.1 The Basic Concept

115

The two notions of left and right pre-Lie products are equivalent: if  is a right pre-Lie product, x  y := y  x defines a left pre-Lie product, and conversely. Due to our preceding remarks: Lemma 6.1.1 Let A be a pre-Lie (resp. a left pre-Lie) algebra, then the bracket [a, b] := a  b − b  a (resp. [a, b] := a  b − b  a) is a Lie bracket: it defines a Lie algebra structure on A. The corresponding forgetful functor from pre-Lie (resp. left pre-Lie) algebras to Lie algebras is denoted A −→ A Lie . Lemma 6.1.2 Let A be a pre-Lie algebra. For each b in A, let Rb be the linear operator a → a  b of right multiplication by b in A. The right Vinberg identity can be rewritten as −R[b,c] = Rb Rc − Rc Rb , hence the operators −Rb provide a representation of the Lie algebra ALie in A, the so-called half-adjoint representation. We have a similar property for left preLie algebras and the operators L a : b → a  b that provide also in that case a representation of the Lie algebra ALie in A. Example 6.1.1 (Vector fields) Consider now smooth vector fields X (x) written in local coordinates x = (x 1 , . . . , x n ), with x ∈ Rn . To the functions X α (x), α = 1, . . . , n one can associate the Lie derivative L X , that is the differential operator defined by n  δf LX f = X α α =: X α δα f, (6.4) δx α=1 where we use Einstein’s summation notation. Then, [L X , LY ] = L X LY − LY L X is again a first-order differential operator, hence of the form L[X,Y ] ; the Jacobi identity for the Lie bracket [X, Y ] comes then for free from associativity of the algebra of differential operators. However, things can be done differently. Let us define X  Y = D X Y by (D X Y )β := X α δα Y β .

(6.5)

Notice that this definition is not intrinsic: it depends on a choice of a system of coordinates. Lemma 6.1.3 The product  defines a left pre-Lie algebra structure on the space of vector fields.

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Proof Indeed, X  (Y  Z ) − (X  Y )  Z = D X DY Z − D D X Y Z = X α δα (Y β δβ Z ) − (D X Y )β δβ Z = X α δα (Y β δβ Z ) − X α δα Y β δβ Z = X α Y β δα δβ Z , which is symmetric in X and Y : the left Vinberg identity follows. It is then an easy computation to show that the Lie bracket of vector fields agrees with the Lie bracket inherited from the left pre-Lie algebra structure: [X, Y ] := D X Y − DY X. Example 6.1.2 (Polynomial vector fields) Let us consider now the particular case of polynomial vector fields: this example will pave the way to the use of graphical rules and tree-like structures to handle computations with pre-Lie products. Recall that a polynomial vector field, X (x) is a function from Rn to Rn such that X = X0 + X1 + X2 + · · · + Xk for a certain integer k, where the components of the vector X i are homogeneous polynomial of degree i in the variables x 1 , . . . , x n . Since, for a finite-dimensional vector space V the space of polynomial functions ((V ∗ )⊗i ) Si identifies with the space of symmetric multilinear functions ((V ∗ )⊗i ) Si (covariants and invariants identify over a field of characteristic 0), we can assume that X i (x) = i (x, . . . , x), for i (y1 , . . . , yi ) a uniquely defined symmetric multilinear function. We represent the last identity by means of a graph: X i (x) 0123 7654  R ltltlt i JJRJRJRRR l l l JJ RRRR t l l t JJ RR lll tttt JJ RRR l l l J t RR l l t xl x ... x ... x x Graphically, we have a rooted tree with the root on top and with i unordered leaves. Now we reconsider D X Y with X homogeneous of degree i and Y homogeneous of degree j. Let the symmetric function  j with j entries correspond to Y as i corresponds to X . Then Z = D X Y has degree i + j − 1. Precisely, the Leibniz rule

6.1 The Basic Concept

117

says that Z is obtained by considering the substitution for i (x, . . . , x) of each variable argument in the symmetric function  j , and summing on all the terms obtained. We define, for r between 1 and j,  j |r i (x1 , . . . , xi+ j−1 ) =  j (x1 , . . . , xr −1 , i (xr , . . . , xi+r −1 ), xi+r , . . . , xi+ j−1 ),

(6.6)

89:; ?>=<  j OWOWWW OOOWWWWW ggogogogo g g g g OOO WWWWWW ggg oooo g g g WWWWW OOO g g o g g o WWWWW OOO ggg oo g g o g WWWW g O o o ggg 0123 7654 ...  . . . Q i QQ m m QQQ mmm QQQ mmm QQQ m m m QQQ m mmm and the sum  j  i =

j 

 j |r i ,

(6.7)

r =1

which is a multilinear function in i + j − 1 entries. Thereby Z is given as a sum of insertions or graftings: Z (x) =  j  i (x, . . . , x). We may look now at the Vinberg identity (which we know to hold from the calculations in the previous example) in the light of this graphical representation. Consider (  )   in relation with   (  ): when grafting  on   , we can choose to do it on a -part or on a -part. Now, the insertions on a -part are totally cancelled in (  )   −   (  ) , and there remain the insertions on -parts. But the latter are symmetric in , : thus we recover that the right Vinberg identity holds (notice that we changed the pre-Lie product from a left to a right one).

6.2 Symmetric Brace Algebras Symmetric brace algebras and the equivalent notion of right-handed polynomial Hopf algebras are another way to understand and study pre-Lie algebras. In practice these approaches are important for several reasons: – Symmetric brace operations capture iterations of the pre-Lie product and are useful in many calculations involving the latter, for example, in applications to group theory.

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–Any pre-Lie algebra can be constructed out of a right-handed polynomial Hopf algebra (as its primitive part). –In general, many constructions involving pre-Lie algebras (such as the construction of their enveloping algebras, or the construction of free pre-Lie algebras) are easier to understand from the point of view of symmetric brace operations. Recall that the algebra of polynomials k[V ] over a vector space is equipped with a unique bicommutative Hopf algebra structure by requiring the elements of V to be primitive elements. Hereafter we use the Sweedler notation and write P (1) ⊗ · · · ⊗ P (k) for k (P), where  =  p stands for the corresponding coproduct on k[V ]. Definition 6.2.1 A symmetric brace algebra is a vector space V equipped with an operation called a symmetric brace operation (or simply brace operation when the context is the one of symmetric brace algebras): V ⊗ k[V ] −→ V v ⊗ P −→ v{P} satisfying, for v, w1 , . . . , wn ∈ V and Q ∈ k[V ] the identities: v{1} = v, (v{w1 , · · · , wn }){Q} = v{w1 {Q (1) } · · · wn {Q (n) }Q (n+1) }.

(6.8)

Notice that we used a multilinear notation v{w1 , · · · , wn } (instead of v{w1 · · · wn }). We will use it later again in particular to emphasize that the entries under consideration (here the wi ) are elements of V . These multilinear operations are of course symmetric (invariant by permutation of the entries): ∀σ ∈ Sn , v{wσ (1) , · · · , wσ (n) } = v{w1 , · · · , wn }. Using the cofreeness of k[V ], the definition of symmetric braces can be rephrased using a Hopf algebraic point of view. Definition 6.2.2 A right-handed polynomial Hopf algebra is a polynomial Hopf algebra (k[V ], ·, ) equipped with a second product ∗ with unit 1 ∈ k[V ]0 = k that makes (k[V ], ∗, ) a cocommutative Hopf algebra and such that furthermore  i≥2

that is such that

 i≥2

k[V ]i ∗ k[V ] ⊂



k[V ]i ,

i≥2

k[V ]i is a right ideal of k[V ] for the product ∗.

6.2 Symmetric Brace Algebras

119

Remark 6.2.1 One can use instead the notion of left non-decreasing product (or  k[V ]i , n ≥ 2 be right right sided product), namely, the requirement that all the i≥n

ideals. The terminology right handed was introduced initially by Turaev1 to account for the dual notion: in our context the property (V ) ⊂ V ⊗ k[V ]+ of the coproduct. We prefer to call such a coproduct a left-linear coproduct and the associated Hopf algebra a left-linear polynomial Hopf algebra. Lemma 6.2.1 Equivalently to Definition 6.2.1, a symmetric brace algebra is a vector space V such that k[V ] is equipped with a right-handed polynomial Hopf algebra structure—the categories of symmetric brace algebras and right-handed polynomial Hopf algebras are equivalent. The brace operations are then defined by the corestriction of the product ∗ to V (that is by composition of the action of ∗ on V ⊗ k[V ] with the projection on V along the k[V ]i , i = 1): ∗|V

V ⊗ k[V ] −→ V. Proof Recall from Example 2.13.4 that T S(V ) and k[V ] are cofree in the category of conilpotent cocommutative coalgebras. It follows that a coalgebra map from a coalgebra in this category to k[V ] is entirely determined by its corestriction to V . In particular, when k[V ] is equipped with a product ∗ that is a coalgebra map, this product is entirely determined by its corestriction to a map from k[V ] ⊗ k[V ] to V . When k[V ] is furthermore right handed, the product is, by definition of right handedness, determined by its corestriction to V : a map, denoted π from V ⊗ k[V ] ⊕ k ⊗ V to V , whose restriction to V ⊗ k = k ⊗ V = V is the identity map. The associativity of the product is then expressed by the identity of the two maps from V ⊗ k[V ] ⊗ k[V ] to V : π(π ⊗ I dk[V ] ) = π(I dV ⊗ ∗). According to Eq. (2.5), this identity is equivalent to the fact that π satisfies Eq. (6.8) and defines a symmetric brace algebra structure on V . Indeed, given w1 . . . wn and Q = v1 . . . vm , we have w1 . . . wn ∗ v1 . . . vm =

 1 π ⊗ p ◦  p (w1 . . . wn ⊗ v1 . . . vm ). p! p≥1

Since π is the null map excepted on V ⊗ k[V ] ⊕ k ⊗ V , we get w1 . . . wn ∗ v1 . . . vm =

1

 1 π ⊗ p ◦  p (w1 . . . wn ⊗ v1 . . . vm ), p! p≥n

V. Turaev, Coalgebras of words and phrases, Journal of Algebra 314.1 (2007) 303–323.

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and for such a p, π ⊗ p ◦  p (w1 . . . wn ⊗ v1 . . . vm ) = 





vl1 ⊗ · · · ⊗ vl j1 −1 ⊗ π(wσ (1) ⊗ Q (1) L ) ⊗ vl j1 +1 · · · ⊗

σ ∈Sn 0< j1