Analysis on Lie Groups: An Introduction
by Jacques FarautEdition: 1st
Format: Hardcover
Pub. Date: 2008-06-09
Publisher(s): Cambridge University Press
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Summary
This self-contained text concentrates on the perspective of analysis, assuming only elementary knowledge of linear algebra and basic differential calculus. The author describes, in detail, many interesting examples, including formulas which have not previously appeared in book form. Topics covered include the Haar measure and invariant integration, spherical harmonics, Fourier analysis and the heat equation, Poisson kernel, the Laplace equation and harmonic functions. Perfect for advanced undergraduates and graduates in geometric analysis, harmonic analysis and representation theory, the tools developed will also be useful for specialists in stochastic calculation and the statisticians. With numerous exercises and worked examples, the text is ideal for a graduate course on analysis on Lie groups.
Table of Contents
| Preface | p. ix |
| The linear group | p. 1 |
| Topological groups | p. 1 |
| The group GL (n, R) | p. 2 |
| Examples of subgroups of GL (n, R) | p. 5 |
| Polar decomposition in GL (n, R) | p. 7 |
| The orthogonal group | p. 11 |
| Gram decomposition | p. 13 |
| Exercises | p. 14 |
| The exponential map | p. 18 |
| Exponential of a matrix | p. 18 |
| Logarithm of a matrix | p. 25 |
| Exercises | p. 29 |
| Linear Lie groups | p. 36 |
| One parameter subgroups | p. 36 |
| Lie algebra of a linear Lie group | p. 38 |
| Linear Lie groups are submanifolds | p. 41 |
| Campbell-Hausdorff formula | p. 44 |
| Exercises | p. 47 |
| Lie algebras | p. 50 |
| Definitions and examples | p. 50 |
| Nilpotent and solvable Lie algebras | p. 56 |
| Semi-simple Lie algebras | p. 62 |
| Exercises | p. 69 |
| Haar measure | p. 74 |
| Haar measure | p. 74 |
| Case of a group which is an open set in R[superscript n] | p. 76 |
| Haar measure on a product | p. 78 |
| Some facts about differential calculus | p. 81 |
| Invariant vector fields and Haar measure on a linear Lie group | p. 86 |
| Exercises | p. 90 |
| Representations of compact groups | p. 95 |
| Unitary representations | p. 95 |
| Compact self-adjoint operators | p. 98 |
| Schur orthogonality relations | p. 103 |
| Peter-Weyl Theorem | p. 107 |
| Characters and central functions | p. 115 |
| Absolute convergence of Fourier series | p. 117 |
| Casimir operator | p. 119 |
| Exercises | p. 123 |
| The groups SU (2) and SO(3), Haar measures and irreducible representations | p. 127 |
| Adjoint representation of SU(2) | p. 127 |
| Haar measure on SU(2) | p. 130 |
| The group SO(3) | p. 133 |
| Euler angles | p. 134 |
| Irreducible representations of SU(2) | p. 136 |
| Irreducible representations of SO(3) | p. 142 |
| Exercises | p. 149 |
| Analysis on the group SU(2) | p. 158 |
| Fourier series on SO(2) | p. 158 |
| Functions of class C[superscript k] | p. 160 |
| Laplace operator on the group SU(2) | p. 163 |
| Uniform convergence of Fourier series on the group SU(2) | p. 167 |
| Heat equation on SO(2) | p. 172 |
| Heat equation on SU(2) | p. 176 |
| Exercises | p. 182 |
| Analysis on the sphere and the Euclidean space | p. 186 |
| Integration formulae | p. 186 |
| Laplace operator | p. 191 |
| Spherical harmonics | p. 194 |
| Spherical polynomials | p. 200 |
| Funk-Hecke Theorem | p. 204 |
| Fourier transform and Bochner-Hecke relations | p. 208 |
| Dirichlet problem and Poisson kernel | p. 212 |
| An integral transform | p. 220 |
| Heat equation | p. 225 |
| Exercises | p. 227 |
| Analysis on the spaces of symmetric and Hermitian matrices | p. 231 |
| Integration formulae | p. 231 |
| Radial part of the Laplace operator | p. 238 |
| Heat equation and orbital integrals | p. 242 |
| Fourier transforms of invariant functions | p. 245 |
| Exercises | p. 246 |
| Irreducible representations of the unitary group | p. 249 |
| Highest weight theorem | p. 249 |
| Weyl formulae | p. 253 |
| Holomorphic representations | p. 260 |
| Polynomial representations | p. 264 |
| Exercises | p. 269 |
| Analysis on the unitary group | p. 274 |
| Laplace operator | p. 274 |
| Uniform convergence of Fourier series on the unitary group | p. 276 |
| Series expansions of central functions | p. 278 |
| Generalised Taylor series | p. 284 |
| Radial part of the Laplace operator on the unitary group | p. 288 |
| Heat equation on the unitary group | p. 292 |
| Exercises | p. 297 |
| Bibliography | p. 299 |
| Index | p. 301 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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