Graduate Texts in Mathematics Brian Hall Lie Groups, Lie Algebras, and Representations An Elementary Introduction Second Edition Graduate Texts in Mathematics 222 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 study. More information about this series at http://www.springer.com/series/136 Brian Hall Lie Groups, Lie Algebras, and Representations An Elementary Introduction Second Edition 123 Brian Hall Department of Mathematics University of Notre Dame Notre Dame, IN, USA ISSN 0072-5285 ISSN 2197-5612 (electronic) Graduate Texts in Mathematics ISBN 978-3-319-13466-6 ISBN 978-3-319-13467-3 (eBook) DOI 10.1007/978-3-319-13467-3 Library of Congress Control Number: 2015935277 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2003, 2015 This work is subject to copyright. 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Printed on acid-free paper 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.......................................................... 3 1.1 Definitions ............................................................ 3 1.2 Examples ............................................................. 5 1.3 Topological Properties ............................................... 16 1.4 Homomorphisms ..................................................... 21 1.5 Lie Groups ............................................................ 25 1.6 Exercises .............................................................. 26 2 The Matrix Exponential ................................................... 31 2.1 The Exponential of a Matrix ......................................... 31 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 ............................................ 42 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 ............................. 55 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 ..................................... 70 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 vii viii Contents 4.4 Complete Reducibility ............................................... 90 4.5 Schur’s Lemma ....................................................... 94 4.6 Representations of sl.2I 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 5.1 The “Hard” Questions................................................ 109 5.2 An Illustrative Example .............................................. 110 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 .............................. 118 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.3I 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................................................... 169 7.1 Semisimple and Reductive Lie Algebras............................ 169 7.2 Cartan Subalgebras ................................................... 174 7.3 Roots and Root Spaces ............................................... 176 7.4 The Weyl Group ...................................................... 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 8.1 Abstract Root Systems ............................................... 197 8.2 Examples in Rank Two............................................... 201 8.3 Duality ................................................................ 204 Contents ix 8.4 Bases and Weyl Chambers ........................................... 206 8.5 Weyl Chambers and the Weyl Group ................................ 212 8.6 Dynkin Diagrams..................................................... 216 8.7 Integral and Dominant Integral Elements ........................... 218 8.8 The Partial Ordering ................................................. 221 8.9 Examples in Rank Three ............................................. 228 8.10 The Classical Root Systems ......................................... 232 8.11 The Classification .................................................... 236 8.12 Exercises .............................................................. 238 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