Algebras, Rings and Modules: Lie Algebras and Hopf Algebras

425 Pages·2010·2.42 MB·English

by  Michiel Hazewinkel

Mathematical
Surveys
and
Monographs
Volume 168
Algebras, Rings
and Modules
Lie Algebras
and Hopf Algebras
Michiel Hazewinkel
Nadiya Gubareni
V. V. Kirichenko
American Mathematical Society
Algebras, Rings
and Modules
Lie Algebras
and Hopf Algebras

Mathematical
Surveys
and
Monographs
Volume 168
Algebras, Rings
and Modules
Lie Algebras
and Hopf Algebras
Michiel Hazewinkel
Nadiya Gubareni
V. V. Kirichenko
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Ralph L. Cohen, Chair Michael A. Singer
Eric M. Friedlander Benjamin Sudakov
Michael I. Weinstein
2010 Mathematics Subject Classification. Primary 16W30, 05E05, 20C30, 16G20,17B37,
20G42, 14L05, 81R50, 16W35, 17Bxx; Secondary 05E10, 58B34, 81R60, 81R12, 82C20.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-168
Library of Congress Cataloging-in-Publication Data
Hazewinkel,Michiel.
Algebras,rings,andmodules: LiealgebrasandHopfalgebras/M.Hazewinkel,N.Gubareni,
V.V.Kirichenko.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-5262-0(alk.paper)
1. Hopf algebras. 2. Lie algebras. I. Gubareni, Nadezhda Mikhailovna. II. Kirichenko,
VladimirV. III.Title.
QA613.8.H39 2010
512(cid:2).55—dc22
2010030618
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Contents
Preface ix
References xi
Chapter 1. Lie algebras and Dynkin diagrams 1
1.1. Lie algebras. Definitions and examples 2
1.2. Ideals, homomorphisms and representations 7
1.3. Solvable and nilpotent Lie algebras 11
1.4. Radical of a Lie algebra. Simple and semisimple Lie algebras 15
1.5. Modules for Lie algebras. Weyl’s theorem. Ado’s theorem 22
1.6. Lie’s theorem 29
1.7. The Lie algebra sl(2;k). Representation of sl(2;k) 32
1.8. The universal enveloping algebra of A lie algebra 35
1.9. Poincar´e-Birkhoff-Witt theorem 38
1.10. Free Lie algebras 43
1.11. Examples of simple Lie algebras 45
1.12. Abstract root systems and the Weyl group 47
1.13. Cartan matrices and Dynkin diagrams 57
1.14. Coxeter groups and Coxeter diagrams 61
1.15. Root systems of semisimple Lie algebras 67
1.16. The Weyl group of a quiver 73
1.17. Reflection functors 78
1.18. Coxeter functors and Coxeter transformations 85
1.19. The Gabriel theorem 87
1.20. Generalized Cartan matrices and Kac-Moody Lie algebras 88
1.21. Historical notes 91
References 94
Chapter 2. Coalgebras: motivation, definitions, and examples 99
2.1. Coalgebras and ’addition formulae’ 100
2.2. Coalgebras and decompositions 102
2.3. Dualizing the idea of an algebra 103
2.4. Some examples of coalgebras 105
2.5. Sub coalgebras and quotient coalgebras 108
2.6. The main theorem of coalgebras 108
2.7. Cofree coalgebras 109
2.8. Algebra - coalgebra duality 112
2.9. Comodules and representations 121
2.10. Graded coalgebras 123
2.11. Reflexive modules 125
2.12. Measuring 126
v
vi CONTENTS
2.13. Addition formulae and duality 128
2.14. Coradical and coradical filtration 128
2.15. Coda to chapter 2 129
References 129
Chapter 3. Bialgebras and Hopf algebras. Motivation, definitions, and
examples 131
3.1. Products and representations 131
3.2. Bialgebras 133
3.3. Hopf algebras 138
3.4. Some more examples of Hopf algebras 140
3.5. Primitive elements 146
3.6. Group-like elements 149
3.7. Bialgebra and Hopf algebra duality 152
3.8. Graded bialgebras and Hopf algebras 153
3.9. Crossed products 159
3.10. Integrals for Hopf algebras 162
3.11. Formal groups 167
3.12. Hopf modules 169
3.13. Historical remarks 170
3.14. The Hopf algebra of an algebra 171
References 172
Chapter 4. The Hopf algebra of symmetric functions 175
4.1. The algebra of symmetric functions 175
4.2. The Hopf algebra structure 184
4.3. PSH algebras 185
4.4. Automorphisms of Symm 193
4.5. The functor of the Witt vectors 194
4.6. Ghost components 197
4.7. Frobenius and Verschiebung endomorphisms 199
4.8. The second multiplication of Symm 202
4.9. Lambda algebras 203
4.10. Exp algebras 210
4.11. Plethysm 213
4.12. The many incarnations of Symm 214
References 215
Chapter 5. The representations of the symmetric groups from the Hopf
algebra point of view 217
5.1. A little bit of finite group representation theory 217
5.2. Double cosets of Young subgroups 221
5.3. The Hopf algebra 223
5.4. Symm as a PSH algebra 226
5.5. The second multiplication on RS 227
5.6. Remarks and acknowledgements 229
References 229
Chapter 6. The Hopf algebra of noncommutative symmetric functions and
the Hopf algebra of quasisymmetric functions 231
CONTENTS vii
6.1. The Hopf algebra NSymm 232
6.2. NSymm over the rationals 234
6.3. The Hopf algebra QSymm 235
6.4. Symm as a quotient of NSymm 238
6.5. More on Shuffle and LieHopf 241
6.6. The autoduality of Symm 248
6.7. Polynomial freeness of QSymm over the integers 250
6.8. Hopf endomorphisms of QSymm and NSymm 254
6.9. Verschiebung and Frobenius on NSymm and QSymm 255
References 260
Chapter 7. The Hopf algebra of permutations 263
7.1. The Hopf algebra of permutations of Malvenuto, Poirier and
Reutenauer 263
7.2. The imbedding of NSymm into MPR 267
7.3. LSD permutations 271
7.4. Second multiplication and second comultiplication 274
7.5. Rigidity and uniqueness of MPR 275
References 276
Chapter 8. Hopf algebras: Applications in and interrelations with other parts
of mathematics and physics 277
8.1. Actions and coactions of bialgebras and Hopf algebras 277
8.2. The quantum groups GL (n,C) and multiparameter generalizations 285
q
8.3. The quantum groups U (sl(n;k)) 294
q
8.4. R-matrices and QIST: Bethe Ansatz, FCR construction and the FRT
theorem 296
8.5. Knot invariants from quantum groups. Yang-Baxter operators 311
8.6. Quiver Hopf algebras 324
8.7. Ringel-Hall Hopf algebras 333
8.8. More interactions of Hopf algebras with other parts of mathematics
and theoretical physics 347
8.8.1. Capelli identities and other formulas for matrices and determinants
with mildly noncommuting entries 347
8.8.2. Quantum symmetry 355
8.8.3. Hopf algebra symmetries in noncommutative geometry 369
8.8.4. Hopf algebras in Galois theory 371
8.8.5. Hopf algebras and renormalization 372
8.8.6. Quantum calculi 372
8.8.7. Umbral calculus and Baxter algebras 376
8.8.8. q-special functions 380
References 381
Index 407

Preface
This volume, the third in our series, is about Hopf algebras (mostly) and Lie
algebras. It is independent of the first two volumes; though, to be sure, there are
some references to them, just as there are references to other books and journals.
The first chapter is devoted to Lie algebras. It is a fairly standard concise
treatment of the more established parts of the theory with the exceptions that
there is a bit more emphasis on Dynkin diagrams (also pictorially) and that the
chapter includes a complete treatment of the correspondence, initiated by Peter
Gabriel, between representations of quivers whose underlying graph is a Dynkin
diagramandrepresentationsoftheLiealgebrawiththatDynkindiagram, Gabriel,
[4], [3]; Bernstein-Gel’fand-Ponomarev, [1]; Dlab-Ringel, [2]. The treatment is via
the very elegant approach by Coxeter reflection functors of [1].
The remaining seven chapters are on Hopf algebras.
The first two of these seven are devoted to the basic theory of coalgebras and
Hopf algebras paying special attention to motivation, history, intuition, and prove-
nance. In a way these two chapters are primers1 on their respective subjects. The
remaining five chapters are quite different.
ChapternumberfourisonthesymmetricfunctionsfromtheHopfalgebrapoint
ofview. ThisHopfalgebraispossibly thericheststructuredobjectinmathematics
andamostbeautifulone. OneaspectthatreceivesspecialattentionistheZelevin-
sky theorem on PSH algebras. The acronym PSH stands for ’positive selfadjoint
Hopf’. What one is really dealing with here is a graded, connected, Hopf algebra
with a distinguished basis that is declared orthonormal, such that each component
of the grading is of finite rank, and such that multiplication and comultiplication
are adjoint to each other and positive. If then, moreover, there is only one distin-
guished basis element that is primitive, the Hopf algebra is isomorphic to Symm,
the Hopf (and more) algebra of symmetric functions with the Schur functions as
distinguished basis. Quite surprisingly the second (co)multiplication on Symm,
whichmakeseachgradedsummandaringinitsownrightandwhichisdistributive
over the first one (in the Hopf algebra sense), turns up during the proof of the
theorem, the Bernstein morphism. This certainly calls for more investigations.
TheenormouslyrichstructureofSymmisdiscussedextensively(variouslambda
ringstructures, Frobenius, and Verschiebung morphims, Adams operations, ... etc.
Correspondingly a fair amount of space is given to the big Witt vectors.
1Asaruleaprofessionalmathematicianiswelladvisedtostayawayfrom’primers’;wehope
andbelievethattheseareanexception.
ix