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CHAPTER Ⅰ Lie Groups and Lie Algebras 1

1.The Concept of a Lie Group and the Classical Examples 1

2.Left-Invariant Vector Fields and One-Parameter Groups 11

3.The Exponential Map 22

4.Homogeneous Spaces and Quotient Groups 30

5.Invariant Integration 40

6.Clifford Algebras and Spinor Groups 54

CHAPTERⅡ Elementary Representation Theory 64

1.Representations 65

2.Semisimple Modules 72

3.Linear Algebra and Representations 74

4.Characters and Orthogonality Relations 77

5.Representations of SU（2），SO（3），U（2），and O（3）. 84

6.Real and Quaternionic Representations 93

7.The Character Ring and the Representation Ring 102

8.Representations of Abelian Groups 107

9.Representations of Lie Algebras 111

10.The Lie Algebra sl（2，？） 115

CHAPTERⅢ Representative Functions 123

1.Algebras of Representative Functions 123

2.Some Analysis on Compact Groups 129

3.The Theorem of Peter and Weyl 133

4.Applications of the Theorem of Peter and Weyl 136

5.Generalizations of the Theorem of Peter and Weyl 138

6.Induced Representations 143

7.Tannaka-Krein Duality 146

8.The Complexification of Compact Lie Groups 151

CHAPTER Ⅳ The Maximal Torus of a Compact Lie Group 157

1.Maximal Tori 157

2.Consequences of the Conjugation Theorem 164

3.The Maximal Tori and Weyl Groups of the Classical Groups 169

4.Cartan Subgroups of Nonconnected Compact Groups 176

CHAPTER Ⅴ Root Systems 183

1.The Adjoint Representation and Groups of Rank 1 183

2.Roots and Weyl Chambers 189

3.Root Systems 197

4.Bases and Weyl Chambers 202

5.Dynkin Diagrams 209

6.The Roots of the Classical Groups 216

7.The Fundamental Group，the Center and the Stiefel Diagram 223

8.The Structure of the Compact Groups 232

CHAPTER Ⅵ Irreducible Characters and Weights 239

1.The Weyl Character Formula 239

2.The Dominant Weight and the Structure of the Representation Ring 249

3.The Multiplicities of the Weights of an Irreducible Representation 257

4.Representations of Real or Quaternionic Type 261

5.Representations of the Classical Groups 265

6.Representations of the Spinor Groups 278

7.Representations of the Orthogonal Groups 292

Bibliography 299

Symbol Index 305

Subject Index 307