李代数和表示论导论

PREFACE.
I.BASICCONCEPTS
1.Definitionsandfirstexamples
1.1ThenotionofLiealgebra
1.2LinearLiealgebras
1.3Liealgebrasofderivations
1.4AbstractLiealgebras
2.Idealsandhomomorphisms
2.1Ideals
2.2Homomorphismsandrepresentations
2.3Automorphisms
3.SolvableandnilpotentLiealgebras
3.1Solvability:
3.2Nilpotency
3.3ProofofEngersTheorem
II.SEMISIMPLELIEALGEBRAS
4.TheoremsofLieandCaftan
4.1Lie'sTheorem
4.2Jordan-Chevalleydecomposition
4.3Cartan'sCriterion.
5.Killingform
5.1Criterionforsemisimplicity
5.2SimpleidealsofL
5.3Innerderivations
5.4AbstractJordandecomposition
6.Completereducibilityofrepresentations
6.1Modules
6.2Casimirelementofarepresentation
6.3Weyl'sTheorem
6.4PreservationofJordandecomposition
7.Representationsofsi(2,F)
7.1Weightsandmaximalvectors
7.2Classificationofirreduciblemodules
8.Rootspacedecomposition
8.1Maximaltotalsubalgebrasandroots
8.2CentralizerofH
8.3Orthogonalityproperties
8.4Integralityproperties
8.5Rationalityproperties.Summary
III.ROOTSYSTEMS
9.Axiomatics
9.1Reflectionsinaeuclideanspace
9.2Rootsystems
9.3Examples
9.4Pairsofroots
10.SimplerootsandWeylgroup
10.1BasesandWeylchambers
10.2Lemmasonsimpleroots
10.3TheWeylgroup
10.4Irreduciblerootsystems
11.Classification
11.1Cartanmatrixof
11.2CoxetergraphsandDynkindiagrams
11.3Irreduciblecomponents
11.4Classificationtheorem
12.Constructionofrootsystemsandautomorphisms
12.1ConstructionoftypesA-G
12.2Automorphismsof/b
13.Abstracttheoryofweights
13.1Weights
13.2Dominantweights
13.3Theweightδ
13.4Saturatedsetsofweights
IV.ISOMORPHISMANDCONJUGACYTHEOREMS..
14.Isomorphismtheorem
14.1Reductiontothesimplecase
14.2Isomorphismtheorem
14.3Automorphisms
15.Cartansubaigebras
15.1DecompositionofLrelativetoadx
15.2Engelsubalgebras
15.3Cartansubalgebras
15.4Functorialproperties
16.Conjugacytheorems
16.1Thegroupξ(L)
16.2ConjugacyofCSA's(solvablecase)
16.3Borelsubalgebras
16.4ConjugacyofBorelsubalgebras
16.5Automorphismgroups
V.EXISTENCETHEOREM
17.Universalenvelopingalgebras
17.1Tensorandsymmetricalgebras
17.2ConstructionofЦ(L)
17.3PBWTheoremandconsequences
17.4ProofofPBWTheorem
TableofContents
17.5FreeLiealgebras
18.Generatorsandrelations
18.1RelationssatisfiedbyL
18.2Consequencesof(S1)-(S3)
18.3Serre'sTheorem
18.4Application:Existenceanduniquenesstheorems
19.Thesimplealgebras
19.1Criterionforsemisimplicity
19.2Theclassicalalgebras
19.3ThealgebraG2
VI.REPRESENTATIONTHEORY
20.Weightsandmaximalvectors
20.1Weightspaces
20.2Standardcyclicmodules
20.3Existenceanduniquenesstheorems
21.Finitedimensionalmodules
21.1Necessaryconditionforfinitedimension
21.2Sufficientconditionforfinitedimension
21.3Weightstringsandweightdiagrams
21.4Generatorsandrelationsforν(λ)
22.Multiplicityformula
22.1AuniversalCasimirelement
22.2Tracesonweightspaces
22.3Freudenthal'sformula
22.4Examples
22.5Formalcharacters.
23.Characters
23.1lnvariantpolynomialfunctions
23.2Standardcyclicmodulesandcharacters
23.3Harish-Chandra'sTheorem
Appendix
24.FormulasofWeyi,Kostant,andSteinberg
24.1SomefunctionsonH*
24.2Kostant'smultiplicityformula
24.3Weyl'sformulas
24.4Steinberg'sformula
Appendix
VII.CHEVALLEYALGEBRASANDGROUPS
25.ChevalleybasisofL
25.1Pairsofroots
25.2ExistenceofaCheva!!eybasis
25.3Uniquenessquestions
25.4Reductionmoduloaprime
25.5ConstructionofCheva!leygroups(adjointtype)
26.Kostant'sTheorem
26.1Acombinatoriallemma
26.2Specialcase:si(2,F)
26.3Lemmasoncommutation
26.4ProofofKostant'sTheorem
27.Admissiblelattices
27.1Existenceofadmissiblelattices
27.2Stabilizerofanadmissiblelattice
27.3Variationofadmissiblelattice
27.4Passagetoanarbitraryfield
27.5Surveyofrelatedresults
References
Afterword(1994)
IndexofTerminology
IndexofSymbols...