李群，李代数及其表示

Preface

Chapter1DifferentiableandAnalyticManifolds

1.1DifferentiableManifolds

1.2AnalyticManifolds

1.3TheFrobcniusTheorem

1.4Appendix

Exercises

Chapter2LieGroupsandLieAlgebras

2.1DefinitionandExamplesofLieGroups

2.2LieAlgebras

2.3TheLieAlgebraofaLieGroup

2.4TheEnvelopingAlgebraofaLieGroup

2.5SubgroupsandSubalgebras

2.6LocallyisomorphicGroups

2.7Homomorphisms

2.8TheFundamentalTheoremofLie

2.9ClosedLieSubgroupsandHomogeneousSpaces.OrbitsandSpacesofOrbits

2.10TheExponentialMap

2.11TheUniquenessoftheRealAnalyticStructureofaRealLieGroup

2.12TaylorSeriesExpansionsonaLieGroup

2.13TheAdjointRepresentationsofgandG

2.14TheDifferentialoftheExponentialMap

2.15TheBaker－Campbell－HausdorffFormula

2.16LiesTheoryofTransformationGroups

Exercises

Chapter3StructureTheory

3.1ReviewofLinearAlgebra

3.2TheUniversalEnvelopingAlgebraofaLieAlgebra

3.3TheUniversalEnvelopingAlgebraasaFilteredAlgebra

3.4TheEnvelopingAlgebraofaLieGroup

3.5NilpotentLieAlgebras

3.6NilpotentAnalyticGroups

3.7SolvableLieAlgebras

3.8TheRadicalandtheNilRadical

3.9CartansCriteriaforSolvabilityandSemisimplicity

3.10SemisimpleLieAlgebras

3.11TheCasimirElement

3.12SomeCohomology

3.13TheTheoremofWeyl

3.14TheLeviDecomposition

3.15TheAnalyticGroupofaLieAlgebra

3.16ReductiveLieAlgebras

3.17TheTheoremofAdo

3.18SomeGlobalResults

Exercises

Chapter4ComplexSemisimpleLieAlgebrasAndLieGroups:StructureandRepresentation

4.1CartanSubalgebras

4.2TheRepresentationsoft（2,C）

4.3StructureTheory

4.4TheClassicalLieAlgebras

4.5DeterminationoftheSimpleLieAlgebrasoverC

4.6RepresentationswithaHighestWeight

4.7RepresentationsofSemisimpleLieAlgebras

4.8ConstructionofaSemisimpleLieAlgebrafromitsCartanMatrix

4.9TheAlgebraofInvariantPolynomialsONaSemisimpleLieAlgebra

4.10InfinitesimalCharacters

4.11CompactandComplexSemisimpleLieGroups

4.12MaximalToriofCompactSemisimpleGroups

4.13AnIntegralFormula

4.14TheCharacterFormulaofH．Weyl

4.15Appendix．FiniteReflectionGroups

Exercises

　　本书是一部学习李群，李代数及其表示论的优秀的研究生教材。本书分为两部分，第一部分主要介绍了李群与李代数，以及它们之间的相互关系，同时还介绍了基础的表示论。第二部分则阐述了半单李群与李代数理论。本书以一种尽可能少地运用流形知识的方法来研究李群。使读者可以快速地掌握知识的核心内容。本书过详尽地介绍SU(2)和SU(3)的表示理论来引入即将介绍的一般内容，使得读者能够在了解一般理论之前已经有了对根系、权，及Weyl群的简单认识。同时，书中众多的例子和图示可以很好地协助学习并理解一些内容。