ISBN: 3-540-15293-8
TITLE: Lie Groups
AUTHOR: Duistermaat, J.J.; Kolk, J.A.C.
TOC:

Preface V 
1. Lie Groups and Lie Algebras 
1.1 Lie Groups and their Lie Algebras 1 
1.2 Examples 6 
1.3 The Exponential Map 16 
1.4 The Exponential Map for a Vector Space 20 
1.5 The Tangent Map of Exp 23 
1.6 The Product in Logarithmic Coordinates 26 
1.7 Dynkin's Formula 29 
1.8 Lie's Fundamental Theorems 31 
1.9 The Component of the Identity 36 
1.10 Lie Subgroups and Homomorphisms 40 
1.11 Quotients 49 
1.12 Connected Commutative Lie Groups 58 
1.13 Simply Connected Lie Groups 62 
1.14 Lie's Third Fundamental Theorem in Global Form 72 
1.15 Exercises 81 
1.16 Notes 86 
References for Chapter One 90 
2. Proper Actions 
2.1 Review 93 
2.2 Bochner's Linearization Theorem 96 
2.3 Slices 98 
2.4 Associated Fiber Bundles 100 
2.5 Smooth Functions on the Orbit Space 103 
2.6 Orbit Types and Local Action Types 107 
2.7 The Stratification by Orbit Types 111 
2.8 Principal and Regular Orbits 115 
2.9 Blowing Up 122 
2.10 Exercises 126 
2.11 Notes 129 
References for Chapter Two 130 
3. Compact Lie Groups 
3.0 Introduction 131 
3.1 Centralizers 132 
3.2 The Adjoint Action 139 
3.3 Connectedness of Centralizers 141 
3.4 The Group of Rotations and its Covering Group 143 
3.5 Roots and Root Spaces 144 
3.6 Compact Lie Algebras 147 
3.7 Maximal Tori 152 
3.8 Orbit Structure in the Lie Algebra 155 
3.9 The Fundamental Group 161 
3.10 The Weyl Group as a Reflection Group 168 
3.11 The Stiefel Diagram 172 
3.12 Unitary Groups 175 
3.13 Integration 179 
3.14 The Weyl Integration Theorem 184 
3.15 Nonconnected Groups 192 
3.16 Exercises 199 
3.17 Notes 202 
References for Chapter Three 206 
4. Representations of Compact Groups 
4.0 Introduction 209 
4.1 Schur's Lemma 212 
4.2 Averaging 215 
4.3 Matrix Coefficients and Characters 219 
4.4 G-types 225 
4.5 Finite Groups 232 
4.6 The Peter-Weyl Theorem 233 
4.7 Induced Representations 242 
4.8 Reality 245 
4.9 Weyl's Character Formula 252 
4.10 Weight Exercises 263 
4.11 Highest Weight Vectors 285 
4.12 The Borel-Weil Theorem 290 
4.13 The Nonconnected Case 306 
4.14 Exercises 318 
4.15 Notes 322 
References for Chapter Four 326 
Appendices and Index 
A Appendix: Some Notions from Differential Geometry 329 
B Appendix: Ordinary Differential Equations 331 
References for Appendix 338 
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