Lie Groups

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Springer Science & Business Media, Dec 15, 1999 - Mathematics - 344 pages
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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.
 

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Contents

Lie Groups and Lie Algebras
1
12 Examples
6
13 The Exponential Map
16
14 The Exponential Map for a Vector Space
20
15 The Tangent Map of Exp
23
16 The Product in Logarithmic Coordinates
26
17 Dynkins Formula
29
18 Lies Fundamental Theorems
31
36 Compact Lie Algebras
147
37 Maximal Tori
152
38 Orbit Structure in the Lie Algebra
155
39 The Fundamental Group
161
310 The Weyl Group as a Reflection Group
168
311 The Stiefel Diagram
172
312 Unitary Groups
175
313 Integration
179

19 The Component of the Identity
36
110 Lie Subgroups and Homomorphisms
40
111 Quotients
49
112 Connected Commutative Lie Groups
58
113 Simply Connected Lie Groups
62
114 Lies Third Fundamental Theorem in Global Form
72
115 Exercises
81
116 Notes
86
References for Chapter One
90
Proper Actions
93
22 Bochners Linearization Theorem
96
23 Slices
98
24 Associated Fiber Bundles
100
25 Smooth Functions on the Orbit Space
103
26 Orbit Types and Local Action Types
107
27 The Stratification by Orbit Types
111
28 Principal and Regular Orbits
115
29 Blowing Up
122
210 Exercises
126
211 Notes
129
References for Chapter Two
130
Compact Lie Groups
131
31 Centralizers
132
32 The Adjoint Action
139
33 Connectedness of Centralizers
141
34 The Group of Rotations and its Covering Group
143
35 Roots and Root Spaces
144
314 The Weyl Integration Theorem
184
315 Nonconnected Groups
192
316 Exercises
199
317 Notes
202
References for Chapter Three
206
Representations of Compact Groups
209
41 Schurs Lemma
212
42 Averaging
215
43 Matrix Coefficients and Characters
219
44 Gtypes
225
45 Finite Groups
232
46 The PeterWeyl Theorem
233
47 Induced Representations
242
48 Reality
245
49 Weyls Character Formula
252
410 Weight Exercises
263
411 Highest Weight Vectors
285
412 The BorelWeil Theorem
290
413 The Nonconnected Case
306
414 Exercises
318
415 Notes
322
References for Chapter Four
326
Appendix
329
Ordinary Differential Equations
331
References for Appendix
338
Subject Index
339
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About the author (1999)

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/

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