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。

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 t（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