Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain University of Dundee K. Erdmann Oxford University A.MacIntyre Queen Mary, University of London L.C.G. Rogers University of Cambridge E. Süli Oxford University J.F. Toland University of Bath Other books in this series A First Course in Discrete Mathematics I. Anderson Analytic Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD, Second Edition D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z. Brze´zniak and T. Zastawniak Calculus of One Variable K.E. Hirst Complex Analysis J.M. Howie Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and J.M. Jones Elements of Abstract Analysis M. Ó Searcóid Elements of Logic via Numbers and Sets D.L. Johnson Essential Mathematical Biology N.F. Britton Essential Topology M.D. Crossley Fields and Galois Theory J.M. Howie Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker Further Linear Algebra T.S. Blyth and E.F. Robertson Geometry R. Fenn Groups, Rings and Fields D.A.R. Wallace Hyperbolic Geometry, Second Edition J.W. Anderson Information and Coding Theory G.A. Jones and J.M. Jones Introduction to Laplace Transforms and Fourier Series P.P.G. Dyke Introduction to Lie Algebras K. Erdmann and M.J. Wildon Introduction to Ring Theory P.M. Cohn Introductory Mathematics: Algebra and Analysis G. Smith Linear Functional Analysis B.P. Rynne and M.A. Youngson Mathematics for Finance: An Introduction to Financial Engineering M. Capin´ksi andT.Zastawniak Matrix Groups: An Introduction to Lie Group Theory A. Baker Measure, Integral and Probability, Second Edition M. Capin´ksi and E. Kopp Multivariate Calculus and Geometry, Second Edition S. Dineen Numerical Methods for Partial Differential Equations G. Evans, J. Blackledge, P.Yardley Probability Models J.Haigh Real Analysis J.M. Howie Sets, Logic and Categories P. Cameron Special Relativity N.M.J. Woodhouse Symmetries D.L. Johnson Topics in Group Theory G. Smith and O. Tabachnikova Vector Calculus P.C. Matthews Karin Erdmann and Mark J. Wildon Introduction to Lie Algebras Karin Erdmann Mark J. Wildon Mathematical Institute Mathematical Institute 24–29 St Giles’ 24–29 St Giles’ Oxford OX1 3LB Oxford OX1 3LB UK UK [email protected] [email protected] British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2005937687 Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN 978-1-84628-040-5 ISBN 978-1-84628-490-8 (eBook) DOI 10.1007/978-1-84628-490-8 Printed on acid-free paper © Springer-VVVerlag London Limited 2006 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be r eproduced, stored or t ransmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specififi c statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. 9 8 7 6 5 4 3 2 (corrected as of 2nd printing, 2007) Springer Science+Business Media springer.com Preface LietheoryhasitsrootsintheworkofSophusLie,whostudiedcertaintrans- formation groups that are now called Lie groups. His work led to the discovery of Lie algebras. By now, both Lie groups and Lie algebras have become essen- tial to many parts of mathematics and theoretical physics. In the meantime, Lie algebras have become a central object of interest in their own right, not least because of their description by the Serre relations, whose generalisations have been very important. This text aims to give a very basic algebraic introduction to Lie algebras. We begin here by mentioning that “Lie” should be pronounced “lee”. The only prerequisite is some linear algebra; we try throughout to be as simple as possible, and make no attempt at full generality. We start with fundamental concepts, including ideals and homomorphisms. A section on Lie algebras of small dimension provides a useful source of examples. We then define solvable and simple Lie algebras and give a rough strategy towards the classification of the finite-dimensional complex Lie algebras. The next chapters discuss Engel’s Theorem, Lie’s Theorem, and Cartan’s Criteria and introduce some represen- tation theory. We then describe the root space decomposition of a semisimple Lie alge- bra and introduce Dynkin diagrams to classify the possible root systems. To practice these ideas, we find the root space decompositions of the classical Lie algebras.Wethenoutlinetheremarkableclassificationofthefinite-dimensional simple Lie algebras over the complex numbers. The final chapter is a survey on further directions. In the first part, we introduce the universal enveloping algebra of a Lie algebra and look in more vi Preface detail at representations of Lie algebras. We then look at the Serre relations andtheirgeneralisationstoKac–MoodyLiealgebrasandquantumgroupsand describe the Lie ring associated to a group. In fact, Dynkin diagrams and the classification of the finite-dimensional complex semisimple Lie algebras have had a far-reaching influence on modern mathematics; we end by giving an illustration of this. In Appendix A, we give a summary of the basic linear and bilinear alge- bra we need. Some technical proofs are deferred to Appendices B, C, and D. In Appendix E, we give answers to some selected exercises. We do, however, encourage the reader to make a thorough unaided attempt at these exercises: it is only when treated in this way that they will be of any benefit. Exercises aremarked†ifananswermaybefoundinAppendixEand(cid:2)iftheyareeither somewhat harder than average or go beyond the usual scope of the text. University of Oxford Karin Erdmann January 2006 Mark Wildon Contents Preface ..................................................... v 1. Introduction................................................ 1 1.1 Definition of Lie Algebras ................................. 1 1.2 Some Examples .......................................... 2 1.3 Subalgebras and Ideals .................................... 3 1.4 Homomorphisms ......................................... 4 1.5 Algebras ................................................ 5 1.6 Derivations .............................................. 6 1.7 Structure Constants ...................................... 7 2. Ideals and Homomorphisms ................................ 11 2.1 Constructions with Ideals.................................. 11 2.2 Quotient Algebras ....................................... 12 2.3 Correspondence between Ideals............................. 14 3. Low-Dimensional Lie Algebras.............................. 19 3.1 Dimensions 1 and 2....................................... 20 3.2 Dimension 3 ............................................. 20 4. Solvable Lie Algebras and a Rough Classification ........... 27 4.1 Solvable Lie Algebras ..................................... 27 4.2 Nilpotent Lie Algebras .................................... 31 4.3 A Look Ahead ........................................... 32 viii Contents 5. Subalgebras of gl(V)........................................ 37 5.1 Nilpotent Maps .......................................... 37 5.2 Weights ................................................. 38 5.3 The Invariance Lemma.................................... 39 5.4 An Application of the Invariance Lemma .................... 42 6. Engel’s Theorem and Lie’s Theorem ........................ 45 6.1 Engel’s Theorem ......................................... 46 6.2 Proof of Engel’s Theorem.................................. 48 6.3 Another Point of View .................................... 48 6.4 Lie’s Theorem............................................ 49 7. Some Representation Theory ............................... 53 7.1 Definitions............................................... 53 7.2 Examples of Representations............................... 54 7.3 Modules for Lie Algebras .................................. 55 7.4 Submodules and Factor Modules ........................... 57 7.5 Irreducible and Indecomposable Modules .................... 58 7.6 Homomorphisms ......................................... 60 7.7 Schur’s Lemma........................................... 61 8. Representations of sl(2,C) .................................. 67 8.1 The Modules V .......................................... 67 d 8.2 Classifying the Irreducible sl(2,C)-Modules .................. 71 8.3 Weyl’s Theorem.......................................... 74 9. Cartan’s Criteria ........................................... 77 9.1 Jordan Decomposition .................................... 77 9.2 Testing for Solvability..................................... 78 9.3 The Killing Form......................................... 80 9.4 Testing for Semisimplicity ................................. 81 9.5 Derivations of Semisimple Lie Algebras...................... 84 9.6 Abstract Jordan Decomposition ............................ 85 10. The Root Space Decomposition............................. 91 10.1 Preliminary Results....................................... 92 10.2 Cartan Subalgebras ...................................... 95 10.3 Definition of the Root Space Decomposition ................. 97 10.4 Subalgebras Isomorphic to sl(2,C).......................... 97 10.5 Root Strings and Eigenvalues .............................. 99 10.6 Cartan Subalgebras as Inner-Product Spaces.................102 Contents ix 11. Root Systems...............................................109 11.1 Definition of Root Systems ................................109 11.2 First Steps in the Classification ............................111 11.3 Bases for Root Systems ...................................115 11.4 Cartan Matrices and Dynkin Diagrams......................120 12. The Classical Lie Algebras..................................125 12.1 General Strategy .........................................126 12.2 sl((cid:3)+1,C) ..............................................129 12.3 so(2(cid:3)+1,C).............................................130 12.4 so(2(cid:3),C) ................................................133 12.5 sp(2(cid:3),C) ................................................134 12.6 Killing Forms of the Classical Lie Algebras ..................136 12.7 Root Systems and Isomorphisms ...........................137 13. The Classification of Root Systems .........................141 13.1 Classification of Dynkin Diagrams ..........................142 13.2 Constructions............................................148 14. Simple Lie Algebras ........................................153 14.1 Serre’s Theorem..........................................154 14.2 On the Proof of Serre’s Theorem ...........................158 14.3 Conclusion ..............................................160 15. Further Directions..........................................163 15.1 The Irreducible Representations of a Semisimple Lie Algebra...164 15.2 Universal Enveloping Algebras .............................171 15.3 Groups of Lie Type.......................................177 15.4 Kac–Moody Lie Algebras .................................179 15.5 The Restricted Burnside Problem ..........................180 15.6 Lie Algebras over Fields of Prime Characteristic..............183 15.7 Quivers .................................................184 16. Appendix A: Linear Algebra................................189 16.1 Quotient Spaces..........................................189 16.2 Linear Maps .............................................191 16.3 Matrices and Diagonalisation ..............................192 16.4 Interlude: The Diagonal Fallacy ............................197 16.5 Jordan Canonical Form ...................................198 16.6 Jordan Decomposition ....................................200 16.7 Bilinear Algebra..........................................201