Table Of ContentA Tour of Representation Theory
March12,2018
Martin Lorenz
2010MathematicsSubjectClassification. Primary16Gxx,16Txx,17Bxx,
20Cxx,20Gxx
For Maria
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 142
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 254
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,BialgebrasandHopfAlgebras 427
9.1. Coalgebras 427
9.2. Comodules 441
9.3. BialgebrasandHopfAlgebras 446
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 558
12.5. Frobenius-SchurIndicators 566
Appendices
AppendixA. TheLanguageofCategoriesandFunctors 575
A.1. Categories 575
x Contents
A.2. Functors 578
A.3. Naturality 579
A.4. Adjointness 583
AppendixB. BackgroundfromLinearAlgebra 587
B.1. TensorProducts 587
B.2. Hom-⊗ Relations 593
B.3. VectorSpaces 594
AppendixC. SomeCommutativeAlgebra 599
C.1. TheNullstellensatz 599
C.2. TheGenericFlatnessLemma 601
C.3. TheZariskiTopologyonaVectorSpace 602
AppendixD. TheDiamondLemma 605
D.1. TheGoal 605
D.2. TheMethod 606
D.3. FirstApplications 608
D.4. ASimplification 611
D.5. ThePoincaré-Birkhoff-WittTheorem 612
AppendixE. TheSymmetricRingofQuotients 615
E.1. DefinitionandBasicProperties 615
E.2. TheExtendedCenter 617
E.3. ComparisonwithOtherRingsofQuotients 619
Bibliography 623
SubjectIndex 635
IndexofNames 647
Notation 651
Preface
In brief, the objective of representation theory is to investigate the different ways
inwhichagivenalgebraicobject—suchasanalgebra,agroup,oraLiealgebra—
can act on a vector space. The benefits of such an action are at least twofold:
the structure of the acting object gives rise to symmetries of the vector space on
whichitacts;and,intheotherdirection,thehighlydevelopedmachineryoflinear
algebracanbebroughttobearontheactingobjectitselftohelpuncoversomeofits
hiddenproperties. Besidesbeingasubjectofgreatintrinsicbeauty,representation
enjoys the additional benefit of having applications in myriad contexts other than
algebra, ranging from number theory, geometry and combinatorics to probability
and statistics [58], general physics [198], quantum field theory [210], the study of
moleculesinchemistry[49]and,morerecently,machinelearning[126].
Thisbookhasevolvedfrommylecturenotesforatwo-semestergraduatecourse
titledRepresentationTheorythatIgaveatTempleUniversityduringtheacademic
years 2012/13 and 2015/16. Some traces of the informality of my original notes
andthestyleofmylectureshaveremainedintact: thetextmakesrathercopioususe
of pictures and expansively displayed formulae; definitions are not numbered and
neither are certain key results, such as Schur’s Lemma or Wedderburn’s Structure
Theorem, which are referred to by name rather than number throughout the book.
However,duetotherestrictionsimposedbyhavingtosetforththematerialonthe
page in a linear fashion, the general format of this book does not in fact duplicate
my actual lectures and it only locally reflects their content. I will comment more
onthisbelow.
The title A Tour of Representation Theory (ToR) is meant to convey the
panoramic view of the subject that I have aimed for.1 Rather than offering an
1ThechoiceoftitleisalsoanodtotheTourdeFrance,and“Tor”inGermanis“gate”aswellas“goal”
(scored)and“fool”.
xi
xii Preface
in-depthtreatmentofoneparticulararea,ToRgivesanintroductiontothreedistinct
flavorsofrepresentationtheory—representationsofgroups,Liealgebras,andHopf
algebras—and all three are presented as incarnations of algebra representations.
Thebookloopsrepeatedlythroughthesetopics,emphasizingsimilaritiesandcon-
nections. Group representations, in particular, are revisited frequently after their
initial treatment in Part II. For example, Schur-Weyl duality is first discussed in
Section 4.7 and later again in 8.8; Frobenius-Schur indicators are introduced in
§3.6.3inconnectionwiththeBrauer-FowlerTheoremandtheyaretreatedintheir
propergeneralityinSection12.5;andChapter11,onaffinealgebraicgroups,brings
together groups, Lie algebras, and Hopf algebras. This mode of exposition owes
muchtothe“holistic”viewpointofthemonograph[72]byEtingofetal.,although
ToR forgoes the delightful historical intermezzos that punctuate [72] and it omits
quiversinfavorofHopfalgebras. Ourtourdoesnotventureveryfarintoanyofthe
areasitpassesthrough,butIhopethatToRwillengenderinsomereadersthedesire
topursuethesubjectandthatitwillprovideaplatformforfurtherexplorations.
OverviewoftheContents. ThetopicscoveredinToRandthemethodsemployed
∗
areresolutelyalgebraic. Liegroups,C -algebrasandotherareasofrepresentation
theory requiring analysis are not covered. On the other hand, in keeping with
the widely acknowledged truth that algebraic representation theory benefits from
a geometric perspective, the discourse involves a modicum of algebraic geometry
on occasion and I have also tried my hand at depicting various noncommutative
geometricspacesthroughoutthebook. Nopriorknowledgeofalgebraicgeometry
isassumed,however.
Representations of algebras form the unifying thread running through ToR.
Therefore, Part I is entirely written in the setting of associative algebras. Chap-
ter 1 develops the basic themes of representation theory: irreducibility, complete
reducibility, spaces of primitive ideals, characters, ...; the chapter establishes
notationtobeusedthroughouttheremainderofthebook;anditfurnishesthefun-
damental general results of representation theory, such as Wedderburn’s Structure
Theorem. Chapter 2 covers topics that are somewhat more peripheral to the main
thrustofToR:projectivemodules(Section2.1)andFrobeniusalgebras(Section2.2).
Readers whose main interest is in groups or Lie algebras may skip this chapter at
afirstreading. However,Section2.2deployssometoolsthatareindispensablefor
thediscussionoffinite-dimensionalHopfalgebrasinChapter12.
Parts II and III are respectively devoted to representations of groups and Lie
algebras. To some degree, these two parts can be tackled in any order. However,
Ihavemadeadeliberateeffortatpresentingthematerialongrouprepresentations
inapalatablemanner,offeringitasanentrywaytorepresentationtheory,whilethe
part on Lie algebras is written in a slightly terser style demanding greater math-
ematical maturity from the reader. Most of Part II deals with finite-dimensional
representationsoffinitegroups,usuallyoverabasefieldwhosecharacteristicdoes
