Preface
Chapter 1 Differentiable and Analytic Manifolds
1.1 Differentiable Manifolds
1.2 Analytic Manifolds
1.3 The Frobcnius Theorem
1.4 Appendix
Exercises
Chapter 2 Lie Groups and Lie Algebras
2.1 Definition and Examples of Lie Groups
2.2 Lie Algebras
2.3 The Lie Algebra of a Lie Group
2.4 The Enveloping Algebra of a Lie Group
2.5 Subgroups and Subalgebras
2.6 Locally isomorphic Groups
2.7 Homomorphisms
2.8 The Fundamental Theorem of Lie
2.9 Closed Lie Subgroups and Homogeneous Spaces. Orbits and Spaces of Orbits
2.10 The Exponential Map
2.11 The Uniqueness of the Real Analytic Structure of a Real Lie Group
2.12 Taylor Series Expansions on a Lie Group
2.13 The Adjoint Representations of g and G
2.14 The Differential of the Exponential Map
2.15 The Baker－Campbell－Hausdorff Formula
2.16 Lie's Theory of Transformation Groups
Exercises
Chapter 3 Structure Theory
3.1 Review of Linear Algebra
3.2 The Universal Enveloping Algebra of a Lie Algebra
3.3 The Universal Enveloping Algebra as a Filtered Algebra
3.4 The Enveloping Algebra of a Lie Group
3.5 Nilpotent Lie Algebras
3.6 Nilpotent Analytic Groups
3.7 Solvable Lie Algebras
3.8 The Radical and the Nil Radical
3.9 Cartan's Criteria for Solvability and Semisimplicity
3.10 Semisimple Lie Algebras
3.11 The Casimir Element
3.12 Some Cohomology
3.13 The Theorem of Weyl
3.14 The Levi Decomposition
3.15 The Analytic Group of a Lie Algebra
3.16 Reductive Lie Algebras
3.17 The Theorem of Ado
3.18 Some Global Results
Exercises
Chapter 4 Complex Semisimple Lie Algebras And Lie Groups: Structure and Representation
4.1 Cartan Subalgebras
4.2 The Representations of □(特殊字符)(2, C)
4.3 Structure Theory
4.4 The Classical Lie Algebras
4.5 Determination of the Simple Lie Algebras over C
4.6 Representations with a Highest Weight
4.7 Representations of Semisimple Lie Algebras
4.8 Construction of a Semisimple Lie Algebra from its Cartan Matrix
4.9 The Algebra of Invariant Polynomials ON a Semisimple Lie Algebra
4.10 Infinitesimal Characters
4.11 Compact and Complex Semisimple Lie Groups
4.12 Maximal Tori of Compact Semisimple Groups
4.13 An Integral Formula
4.14 The Character Formula of H. Weyl
4.15 Appendix. Finite Reflection Groups
Exercises
Bibliography
Index
