# Lie Groups

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

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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/