Lie Groups

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.- 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.- 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.- 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.- 4.15Notes.- References for Chapter Four.- Appendices and Index.- A Appendix: Some Notions from Differential Geometry.- B Appendix: Ordinary Differential Equations.- References for Appendix.