Lie Groups

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

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/

Résumé

From the reviews:

"This one is worth to read and to keep on your shelf! It presents the theory of Lie groups not only in the usual way of Lie algebraic treatment, but also from the global point of view. ... Every chapter ends with very useful notes on the origins and connections of the chapter's subject. References are given separately in each chapter. ... It is highly recommended to advanced undergraduate and graduated students in mathematics and physics." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 75, 2009)

Contenu

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.