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Preface 1 Lie Groups and Lie Algebras 1.1 Lie Groups and their Lie Algebras 1.2 Examples 1.3 The Exponential Map 1.4 The Exponential Map for a Vector Space 1.5 The Tangent Map of Exp 1.6 The Product in Logarithmic Coordinates 1.7 Dynkin's Formula 1.8 Lie's Fundamental Theorems 1.9 The Component of the Identity 1.10 Lie Subgroups and Homomorphisms 1.11 Quotients 1.12 Connected Commutative Lie Groups 1.13 Simply Connected Lie Groups 1.14 Lie's Third Fundamental Theorem in Global Form 1.15 Exercises 1.16 Notes References for Chapter One 2 Proper Actions 2.1 Review 2.2 Bochner's Linearization Theorem 2.3 Slices 2.4 Associated Fiber Bundles 2.5 Smooth Functions on the Orbit Space 2.6 Orbit Types and Local Action Types 2.7 The Stratification by Orbit Types 2.8 Principal and Regular Orbits 2.9 Blowing Up 2.10 Exercises 2.11 Notes References for Chapter Two 3 Compact Lie Groups 3.0 Introduction 3.1 Centralizers 3.2 The Adjoint Action 3.3 Connectedness of Centralizers 3.4 The Group of Rotations and its Covering Group 3.5 Roots and Root Spaces 3.6 Compact Lie Algebras 3.7 Maximal Tori 3.8 Orbit Structure in the Lie Algebra 3.9 The Fundamental Group 3.10 The Weyl Group as a Reflection Group 3.11 The Stiefel Diagram 3.12 Unitary Groups 3.13 Integration 3.14 The Weyl Integration Theorem 3.15 Nonconnected Groups 3.16 Exercises 3.17 Notes References for Chapter Three 4 Representations of Compact Groups 4.0 Introduction 4.1 Schur's Lemma 4.2 Averaging 4.3 Matrix Coefficients and Characters 4.4 G-types 4.5 Finite Groups 4.6 The Peter-Weyl Theorem 4.7 Induced Representations 4.8 Reality 4.9 Weyl's Character Formula 4.10 Weight Exercises 4.11 Highest Weight Vectors 4.12 The Borel-Weil Theorem 4.13 The Nonconnected Case 4.14 Exercises 4.15 Notes References for Chapter Four Appendix A Appendix B Appendix

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