A Tour of Representation Theory (Graduate Studies in Mathematics)

674 Pages·2018·9.411 MB·English

by  Martin Lorenz

GRADUATE STUDIES 193
IN MATHEMATICS
A Tour of
Representation
Theory
Martin Lorenz
A Tour of
Representation
Theory

GRADUATE STUDIES 193
IN MATHEMATICS
A Tour of
Representation
Theory
Martin Lorenz
EDITORIAL COMMITTEE
Daniel S. Freed (Chair)
Bjorn Poonen
Gigliola Staffilani
Jeff A. Viaclovsky
2010 Mathematics Subject Classification. Primary 16Gxx, 16Txx, 17Bxx, 20Cxx, 20Gxx.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-193
Library of Congress Cataloging-in-Publication Data
Names: Lorenz,Martin,1951-author.
Title: Atourofrepresentationtheory/MartinLorenz.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2018]|Series: Gradu-
atestudiesinmathematics;volume193|Includesbibliographicalreferencesandindexes.
Identifiers: LCCN2018016461|ISBN9781470436803(alk. paper)
Subjects: LCSH: Representations of groups. | Representations of algebras. | Representations
of Lie algebras. | Vector spaces. | Categories (Mathematics) | AMS: Associative rings and
algebras – Representation theory of rings and algebras – Representation theory of rings and
algebras. msc | Associative rings and algebras – Hopf algebras, quantum groups and related
topics – Hopf algebras, quantum groups and related topics. msc | Nonassociative rings and
algebras – Lie algebras and Lie superalgebras – Lie algebras and Lie superalgebras. msc |
Grouptheoryandgeneralizations–Representationtheoryofgroups –Representationtheory
ofgroups. msc|Grouptheoryandgeneralizations–Linearalgebraicgroupsandrelatedtopics
–Linearalgebraicgroupsandrelatedtopics. msc
Classification: LCCQA176.L672018|DDC515/.7223–dc23
LCrecordavailableathttps://lccn.loc.gov/2018016461
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Contents
Preface xi
Conventions xvii
PartI. Algebras
Chapter1. RepresentationsofAlgebras 3
1.1. Algebras 3
1.2. Representations 24
1.3. PrimitiveIdeals 41
1.4. Semisimplicity 50
1.5. Characters 65
Chapter2. FurtherTopicsonAlgebras 79
2.1. Projectives 79
2.2. FrobeniusandSymmetricAlgebras 96
PartII. Groups
Chapter3. GroupsandGroupAlgebras 113
3.1. Generalities 113
3.2. FirstExamples 124
3.3. MoreStructure 131
3.4. SemisimpleGroupAlgebras 143
3.5. FurtherExamples 150
3.6. SomeClassicalTheorems 159
vii
viii Contents
3.7. Characters,SymmetricPolynomials,andInvariantTheory 170
3.8. DecomposingTensorPowers 179
Chapter4. SymmetricGroups 187
4.1. Gelfand-ZetlinAlgebras 189
4.2. TheBranchingGraph 192
4.3. TheYoungGraph 197
4.4. ProofoftheGraphIsomorphismTheorem 205
4.5. TheIrreducibleRepresentations 217
4.6. TheMurnaghan-NakayamaRule 222
4.7. Schur-WeylDuality 235
PartIII. LieAlgebras
Chapter5. LieAlgebrasandEnvelopingAlgebras 245
5.1. LieAlgebraBasics 246
5.2. TypesofLieAlgebras 253
5.3. ThreeTheoremsaboutLinearLieAlgebras 257
5.4. EnvelopingAlgebras 266
5.5. GeneralitiesonRepresentationsofLieAlgebras 278
5.6. TheNullstellensatzforEnvelopingAlgebras 287
5.7. Representationsofsl 300
2
Chapter6. SemisimpleLieAlgebras 315
6.1. CharacterizationsofSemisimplicity 316
6.2. CompleteReducibility 320
6.3. CartanSubalgebrasandtheRootSpaceDecomposition 325
6.4. TheClassicalLieAlgebras 334
Chapter7. RootSystems 341
7.1. AbstractRootSystems 342
7.2. BasesofaRootSystem 349
7.3. Classification 356
7.4. LatticesAssociatedtoaRootSystem 361
Chapter8. RepresentationsofSemisimpleLieAlgebras 373
8.1. Reminders 374
8.2. Finite-DimensionalRepresentations 377
8.3. HighestWeightRepresentations 379
Contents ix
8.4. Finite-DimensionalIrreducibleRepresentations 385
8.5. TheRepresentationRing 390
8.6. TheCenteroftheEnvelopingAlgebra 393
8.7. Weyl’sCharacterFormula 408
8.8. SchurFunctorsandRepresentationsofsl(V) 418
PartIV. HopfAlgebras
Chapter9. Coalgebras,Bialgebras,andHopfAlgebras 427
9.1. Coalgebras 427
9.2. Comodules 441
9.3. BialgebrasandHopfAlgebras 447
Chapter10. RepresentationsandActions 465
10.1. RepresentationsofHopfAlgebras 466
10.2. FirstApplications 476
10.3. TheRepresentationRingofaHopfAlgebra 485
10.4. ActionsandCoactionsofHopfAlgebrasonAlgebras 492
Chapter11. AffineAlgebraicGroups 503
11.1. AffineGroupSchemes 503
11.2. AffineAlgebraicGroups 508
11.3. RepresentationsandActions 512
11.4. Linearity 515
11.5. IrreducibilityandConnectedness 520
11.6. TheLieAlgebraofanAffineAlgebraicGroup 526
11.7. AlgebraicGroupActionsonPrimeSpectra 530
Chapter12. Finite-DimensionalHopfAlgebras 541
12.1. FrobeniusStructure 541
12.2. TheAntipode 549
12.3. Semisimplicity 552
12.4. DivisibilityTheorems 559
12.5. Frobenius-SchurIndicators 567
Appendices
AppendixA. TheLanguageofCategoriesandFunctors 575
A.1. Categories 575