LIE GROUPSPDF电子书下载

Chapter 1 Differentiable and Analytic Manifolds 1

1.1 Differentiable Manifolds 1

1.2 Analytic Manifolds 20

1.3 The Frobenius Theorem 25

1.4 Appendix 31

Exercises 35

Chapter 2 Lie Groups and Lie Algebras 41

2.1 Definition and Examples of Lie Groups 41

2.2 Lie Algebras 46

2.3 The Lie Algebra of a Lie Group 51

2.4 The Enveloping Algebra of a Lie Group 55

2.5 Subgroups and Subalgebras 57

2.6 Locally isomorphic Groups 61

2.7 Homomorphisms 67

2.8 The Fundamental Theorem of Lie 72

2.9 Closed Lie Subgroups and Homogeneous Spaces.Orbits and Spaces of Orbits 74

2.10 The Exponential Map 84

2.11 The Uniqueness of the Real Analytic Structure of a Real Lie Group 92

2.12 Taylor Series Expansions on a Lie Group 94

2.13 The Adjoint Representations of ？and G 101

2.14 The Differential of the Exponential Map 107

2.15 The Baker-Campbell-Hausdorff Formula 114

2.16 Lie’s Theory of Transformation Groups 121

Exercises 133

Chapter 3 Structure Theory 149

3.1 Review of Linear Algebra 149

3.2 The Universal Enveloping Algebra of a Lie Algebra 166

3.3 The Universal Enveloping Algebra as a Filtered Algebra 176

3.4 The Enveloping Algebra of a Lie Group 184

3.5 Nilpotent Lie Algebras 189

3.6 Nilpotent Analytic Groups 195

3.7 Solvable Lie Algebras 200

3.8 The Radical and the Nil Radical 204

3.9 Cartan’s Criteria for Solvability and Semisimplicity 207

3.10 Semisimple Lie Algebras 213

3.11 The Casimir Element 216

3.12 Some Cohomology 219

3.13 The Theorem of Weyl 222

3.14 The Levi Decomposition 224

3.15 The Analytic Group of a Lie Algebra 228

3.16 Reductive Lie Algebras 230

3.17 The Theorem of Ado 233

3.18 Some Global Results 238

Exercises 247

Chapter 4 Complex Semisimple Lie Algebras And Lie Groups：Structure and Representation 260

4.1 Cartan Subalgebras 260

4.2 The Representations of ？？（2，C）. 267

4.3 Structure Theory 273

4.4 The Classical Lie Algebras 293

4.5 Determination of the Simple Lie Algebras over C 305

4.6 Representations with a Highest Weight 313

4.7 Representations of Semisimple Lie Algebras 324

4.8 Construction of a Semisimple Lie Algebra from its Cartan Matrix 329

4.9 The Algebra of Invariant Polynomials on a Semisimple Lie Algebra 333

4.10 Infinitesimal Characters 337

4.11 Compact and Complex Semisimple Lie Groups 342

4.12 Maximal Tori of Compact Semisimple Groups 351

4.13 An Integral Formula 356

4.14 The Character Formula of H.Weyl 364

4.15 Appendix.Finite Reflection Groups 369

Exercises 387

Bibliography 417

Index 423