Lie Groups

This book is devoted to an exposition of the theory of finite-dimensional Lie groups and Lie algebras, which is a beautiful and central topic in modern mathematics. At the end of the nineteenth century this theory came to life in the works of Sophus Lie. It had its origins in Lie's idea of applying Galois theory to differential equations and in Klein's "Erlanger Programm" of treat­ ing symmetry groups as the fundamental objects in geometry. Lie's approach to many problems of analysis and geometry was mainly local, that is, valid in local coordinate systems only. At the beginning of the twentieth century E. Cartan and Weyl began a systematic treatment of the global aspects of Lie's theory. Since then this theory has ramified tremendously and now, as the twentieth century is coming to a close, its concepts and methods pervade mathematics and theoretical physics. Despite the plethora of books devoted to Lie groups and Lie algebras we feel there is justification for a text that puts emphasis on Lie's principal idea, namely, geometry treated by a blend of algebra and analysis. Lie groups are geometrical objects whose structure can be described conveniently in terms of group actions and fiber bundles. Therefore our point of view is mainly differential geometrical. We have made no attempt to discuss systematically the theory of infinite-dimensional Lie groups and Lie algebras, which is cur­ rently an active area of research. We now give a short description of the contents of each chapter.

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.