Lie Groups, Lie Algebras, and Representations

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Lie Groups, Lie Algebras, and Representations基础理论介绍
Graduate Texts in mathematics Series editors: Sheldon axler San francisco state University san francisco, CA, USA Kenneth ribet University of California, Berkeley, CA, USA Advisory board: Alejandro adem, University of british Columbia David Jerison, University of California Berkeley& msrI Irene M. Gamba, The University of Texas at austin Jeffrey C Lagarias, University of michigan Ken Ono, Emory University Jeremy Quastel, University of Toronto Fadil santosa, University of minnesota Barry Simon, California Institute of Technology Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual stud Moreinformationaboutthisseriesathttp://www.springer.com/series/136 Brian hall Lie groups Lie algebras and representations An Elementary Introduction Second edition ② Springer Brian hall Department of mathematics University of Notre Dame Notre dame IN. USA ISSN0072-5285 isSn 2197-5612(electronic) Graduate Texts in mathematics ISBN978-3-319-13466-6 ISBN978-3-319-13467-3( e Book) DOⅠ10.1007/978-3-319-13467-3 Library of Congress Control Number: 2015935277 Springer Cham Heidelberg New York dordrecht Londo O Springer International Publishing Switzerland 2003, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free pap Springer International Publishing AG Switzerland is part of Springer Science+Business Media(www springer. com) For Carla Contents Part I General Theory 1 Matrix Lie groups 1.1 Definitions 1. 2 Example 1.3 Topological Properties........................ 16 1. 4 Homomorphisms 21 1.5 Lie groups 25 1.6 Exercises 26 2 The Matrix Exponential 2.1 The exponential of a matrix 2.2 Computing the Exponential.…… 34 2. 3 The matrix Logarithm 36 2. 4 Further Properties of the exponential 40 2.5 The Polar decomposition 2.6 Exercises 46 3 Lie algebras 49 3.1 Definitions and First Examples 49 3.2 Simple, Solvable, and Nilpotent Lie Algebras 53 3.3 The Lie Algebra of a Matrix Lie Group 15 3.4 Examples 57 3.5 Lie Group and lie Algebra Homomorphisms . ....................60 3.6 The Complexification of a Real Lie Algebra. ........... 65 3.7 The Exponential Map........... 67 3.8 Consequences of Theorem 3. 42 3.9 Exercises 73 4 Basic Representation Theory 77 4.1 Representations 77 4.2 Examples of representations 81 4.3 New Representations from Old 84 VIl Contents 4.4 Complete Reducibility 4.5 Schur’ s Lemma.. 94 4.6 Representations of sl (2 C) 96 4.7 Group Versus Lie Algebra Representations 101 4.8 A Nonmatrix Lie Group 103 4.9 Exercises 105 5 The Baker-Campbell-Hausdorff Formula and Its Consequences .. 109 5.1The‘Hard” Questions....…...,109 5.2 An Illustrative example 5.3 The Baker-Campbell-Hausdorff formula 113 5.4 The Derivative of the Exponential Map 114 5.5 Proof of the bch formula 117 5.6 The Series Form of the bch formula 5.7 Group Versus Lie Algebra Homomorphisms............ 119 5. 8 Universal Covers 126 5.9 Subgroups and Subalgebras.......... 128 5.10 Lie's Third Theorem 135 5.11 Exercises .135 Part II Semisimple Lie algebras 6 The Representations of sl (3; C) 141 6.1 Preliminaries 141 6.2 Weights and roots ∴.142 6.3 The Theorem of the Highest Weight .146 6.4 Proof of the theorem 148 6.5 An Example: Highest weight(1,1)…… ·;···· 153 6.6 The Weyl Group 154 6.7 Weight Diagrams 158 6.8 Further Properties of the representations ............. 159 6.9 Exercises 165 7 Semisimple Lie Algebras 7.1 Semisimple and reductive Lie algebras 169 7.2 Cartan Subalgebras 174 7.3 Roots and root spaces 176 7.4 The Weyl grou 182 7.5 Root systems∴ 183 7.6 Simple lie algebras 185 7.7 The root Systems of the Classical Lie algebras 188 7. 8 Exercises 193 8 Root systems 197 8. Abstract root systems .197 8.2 Examples in Rank Two 201 8.3 Duality..……… 204 Contents 8.4 Bases and Weyl Chambers 206 8.5 Weyl Chambers and the Weyl group…….212 8.6 Dynkin Diag 216 8.7 Integral and Dominant Integral Elements 218 8. 8 The Partial Ordering 8.9 Examples in Rank Three 228 8.10 The Classical Root Systems 232 8.11 The Classification 236 8.12Eⅹ excises.. 38 9 Representations of Semisimple Lie algebras 241 9.1 Weights of representations..………241 9.2 Introduction to Verma Modules 244 9.3 Universal Enveloping Algebras 246 9.4 Proof of the Pbw Theorem 250 9.5 Construction of Verma Modules 254 9.6 Irreducible Quotient Modules 257 9.7 Finite-Dimensional Quotient Modules 260 9.8 Exercises 263 10 Further Properties of the representations 265 10.1 The Structure of the Weights 265 10.2 The Casimir element 269 10.3 Complete Reducibility 273 10.4 The Weyl character Formula....................275 10.5 The Weyl Dimension formula 281 10.6 The Kostant Multiplicity Formula.................. 287 10.7 The Character Formula for Verma modules 294 10. 8 Proof of the character formula 295 10.9 Exercises 303 Part II Compact Lie Groups 11 Compact Lie Groups and Maximal Tori 307 11.1 Tori 308 11.2 Maximal Tori and the weyl group 312 11.3 Mapping degrees ..315 11. 4 Quotient manifold 321 11.5 Proof of the Torus theorem 326 11.6 The Weyl Integral Formula.……330 11.7 Roots and the Structure of the Weyl group 333 11. 8 Exercises 339 12 The Compact Group Approach to Representation Theory .343 12.1 Representations 343 12.2 Analytically Integral Elements 346 12.3 Orthonormality and Completeness for Characters......... 351 【实例截图】
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