Lie Groups

This (post) graduate text gives a broad introduction to Lie groups and algebras with an emphasis on differential geometrical methods. It analyzes the structure of compact Lie groups in terms of the action of the group on itself by conjugation, culminating in the classification of the representations of compact Lie groups and their realization as sections of holomorphic line bundles over flag manifolds. Appendices provide background reviews.

Auteur

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.

Johan Kolk published about harmonic analysis on semisimple Lie groups, the theory of distributions, and classical analysis. Jointly with Duistermaat he has written four books: besides the present one, on Lie groups, and on multidimensional real analysis. Until his retirement in 2009, he was affiliated to the Mathematical Institute of Utrecht University. For more information, see http://www.staff.science.uu.nl/~kolk0101/

Contenu

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