Lie groups / Johannes J. Duistermaat, Johan A.C. Kolk.
Series UniversitextEditor: Berlin ; New York : Springer, c2000Descripción: viii, 344 p. : ill. ; 24 cmISBN: 3540152938 (softcover)Tema(s): Lie groupsOtra clasificación: 22Exx (22-01 22C05 43-01) Recursos en línea: Publisher description | Table of contents onlyPreface V -- 1. Lie Groups and Lie Algebras -- 1.1 Lie Groups and their Lie Algebras [1] -- 1.2 Examples [6] -- 1.3 The Exponential Map [16] -- 1.4 The Exponential Map for a Vector Space [20] -- 1.5 The Tangent Map of Exp [23] -- 1.6 The Product in Logarithmic Coordinates [26] -- 1.7 Dynkin’s Formula [29] -- 1.8 Lie’s Fundamental Theorems [31] -- 1.9 The Component of the Identity [36] -- 1.10 Lie Subgroups and Homomorphisms [40] -- 1.11 Quotients .. [49] -- 1.12 Connected Commutative Lie Groups [58] -- 1.13 Simply Connected Lie Groups [62] -- 1.14 Lie’s Third Fundamental Theorem in Global Form [72] -- 1.15 Exercises [81] -- 1.16 Notes [86] -- References for Chapter One [90] -- 2. Proper Actions -- 2.1 Review [93] -- 2.2 Bochner’s Linearization Theorem [96] -- 2.3 Slices [98] -- 2.4 Associated Fiber Bundles [100] -- 2.5 Smooth Functions on the Orbit Space [103] -- 2.6 Orbit Types and Local Action Types [107] -- 2.7 The Stratification by Orbit Types [111] -- 2.8 Principal and Regular Orbits [115] -- 2.9 Blowing Up [122] -- 2.10 Exercises [126] -- 2.11 Notes [129] -- References for Chapter Two [130] -- 3. Compact Lie Groups -- 3.0 Introduction [131] -- 3.1 Centralizers [132] -- 3.2 The Adjoint Action [131] -- 3.3 Connectedness of Centralizers [141] -- 3.4 The Group of Rotations and its Covering Group [143] -- 3.5 Roots and Root Spaces [144] -- 3.6 Compact Lie Algebras [147] -- 3.7 Maximal Tori [152] -- 3.8 Orbit Structure in the Lie Algebra [155] -- 3.9 The Fundamental Group [161] -- 3.10 The Weyl Group as a Reflection Group [168] -- 3.11 The Stiefel Diagram [172] -- 3.12 Unitary Groups [175] -- 3.13 Integration [179] -- 3.14 The Weyl Integration Theorem [184] -- 3.15 Nonconnected Groups [192] -- 3.16 Exercises [199] -- 3.17 Notes [202] -- References for Chapter Three [206] -- 4. Representations of Compact Groups -- 4.0 Introduction [209] -- 4.1 Schur’s Lemma [212] -- 4.2 Averaging [215] -- 4.3 Matrix Coefficients and Characters [219] -- 4.4 G-types [225] -- 4.5 Finite Groups [232] -- 4.6 The Peter-Weyl Theorem [233] -- 4.7 Induced Representations [242] -- 4.8 Reality [245] -- 4.9 Weyl’s Character Formula [252] -- 4.10 Weight Exercises [263] -- 4.11 Highest Weight Vectors [285] -- 4.12 The Borel-Weil Theorem [290] -- 4.13 The Nonconnected Case [306] -- 4.14 Exercises [318] -- 4.15 Notes [322] -- References for Chapter Four [326] -- Appendices and Index -- A Appendix: Some Notions from Differential Geometry [329] -- B Appendix: Ordinary Differential Equations [331] -- References for Appendix [338] -- Subject Index [339] --
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Preface V --
1. Lie Groups and Lie Algebras --
1.1 Lie Groups and their Lie Algebras [1] --
1.2 Examples [6] --
1.3 The Exponential Map [16] --
1.4 The Exponential Map for a Vector Space [20] --
1.5 The Tangent Map of Exp [23] --
1.6 The Product in Logarithmic Coordinates [26] --
1.7 Dynkin’s Formula [29] --
1.8 Lie’s Fundamental Theorems [31] --
1.9 The Component of the Identity [36] --
1.10 Lie Subgroups and Homomorphisms [40] --
1.11 Quotients .. [49] --
1.12 Connected Commutative Lie Groups [58] --
1.13 Simply Connected Lie Groups [62] --
1.14 Lie’s Third Fundamental Theorem in Global Form [72] --
1.15 Exercises [81] --
1.16 Notes [86] --
References for Chapter One [90] --
2. Proper Actions --
2.1 Review [93] --
2.2 Bochner’s Linearization Theorem [96] --
2.3 Slices [98] --
2.4 Associated Fiber Bundles [100] --
2.5 Smooth Functions on the Orbit Space [103] --
2.6 Orbit Types and Local Action Types [107] --
2.7 The Stratification by Orbit Types [111] --
2.8 Principal and Regular Orbits [115] --
2.9 Blowing Up [122] --
2.10 Exercises [126] --
2.11 Notes [129] --
References for Chapter Two [130] --
3. Compact Lie Groups --
3.0 Introduction [131] --
3.1 Centralizers [132] --
3.2 The Adjoint Action [131] --
3.3 Connectedness of Centralizers [141] --
3.4 The Group of Rotations and its Covering Group [143] --
3.5 Roots and Root Spaces [144] --
3.6 Compact Lie Algebras [147] --
3.7 Maximal Tori [152] --
3.8 Orbit Structure in the Lie Algebra [155] --
3.9 The Fundamental Group [161] --
3.10 The Weyl Group as a Reflection Group [168] --
3.11 The Stiefel Diagram [172] --
3.12 Unitary Groups [175] --
3.13 Integration [179] --
3.14 The Weyl Integration Theorem [184] --
3.15 Nonconnected Groups [192] --
3.16 Exercises [199] --
3.17 Notes [202] --
References for Chapter Three [206] --
4. Representations of Compact Groups --
4.0 Introduction [209] --
4.1 Schur’s Lemma [212] --
4.2 Averaging [215] --
4.3 Matrix Coefficients and Characters [219] --
4.4 G-types [225] --
4.5 Finite Groups [232] --
4.6 The Peter-Weyl Theorem [233] --
4.7 Induced Representations [242] --
4.8 Reality [245] --
4.9 Weyl’s Character Formula [252] --
4.10 Weight Exercises [263] --
4.11 Highest Weight Vectors [285] --
4.12 The Borel-Weil Theorem [290] --
4.13 The Nonconnected Case [306] --
4.14 Exercises [318] --
4.15 Notes [322] --
References for Chapter Four [326] --
Appendices and Index --
A Appendix: Some Notions from Differential Geometry [329] --
B Appendix: Ordinary Differential Equations [331] --
References for Appendix [338] --
Subject Index [339] --
MR, REVIEW #
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