This book is a (post)graduate textbook on Lie groups and Lie algebras.Its aim is to give a broad introduction to the field with an emphasis on using differential-geometrical methods, in the spirit of Lie himself. A review of the required background material is provided in the appendices.

The authors are leading specialists and excellent expositors, who have worked a long time on this book project Includes supplementary material: sn.pub/extras

Autorentext

Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was co-author of eleven books. Duistermaat was affiliated to the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.

Inhalt

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.