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Springer Undergraduate Mathematics Series AdvisoryBoard P.I.Cameron QueenMaryandWestfieldCollege M.A.I.Chaplain UniversityofDundee K.Erdmann OxfordUniversity L.C.G.Rogers CambridgeUniversity E.Silli OxfordUniversity I.F.Toland UniversityofBath Otherbooksinthisseries AFirstCourseinDiscreteMathematics 1.Anderson AnalyticMethodsforPartialDifferentialEquationsG.Evans,J.Blackledge,P. Yardley AppliedGeometryforComputerGraphicsandCAD,SecondEdition D.Marsh BasicLinearAlgebra.SecondEdition T.S.BlythandE.F.Robertson BasicStochasticProcesses Z.BrzeiniakandT.Zastawniak CalculusofOneVariable K.E.Hirst ComplexAnalysis J.M.Howie ElementaryDifferentialGeometry A.Pressley ElementaryNumberTheory G.A.Jonesand].M.Jones ElementsofAbstractAnalysis M.0Searcoid ElementsofLogicviaNumbersandSets D.L.Johnson EssentialMathematicalBiology N.F.Britton EssentialTopology M.D.Crossley FieldsandGaloisTheoryJ.M.Howie Fields,FlowsandWaves:AnIntroductiontoContinuumModels D.F.Parker FurtherLinearAlgebra T.S.BlythandE.F.Robertson Geometry R.Fenn Groups,RingsandFields D.A.R.Wallace HyperbolicGeometry,SecondEdition J.W.Anderson InformationandCodingTheory G.A.Jonesand].M.Jones IntroductiontoLaplaceTransformsandFourierSeries P.P.G.Dyke IntroductiontoRingTheory P.M.Cohn IntroductoryMathematics:AlgebraandAnalysis G.Smith LinearFunctionalAnalysis B.P.RynneandM.A. Youngson MathematicsforFinance:AnIntroductiontoFinancialEngineering M.Capiflksiand T.Zastawniak MatrixGroups:AnIntroductiontoLieGroupTheory A.Baker Measure,IntegralandProbability,SecondEdition M.CapiflksiandE.Kopp MultivariateCalculusandGeometry,SecondEdition S.Dineen NumericalMethodsforPartialDifferentialEquationsG.Evans,].Blackledge,P.Yardley ProbabilityModels J.Haigh RealAnalysis J.M.Howie Sets.LogicandCategories P.Cameron SpecialRelativity N.M.]. Woodhouse Symmetries D.L.Johnson TopicsinGroupTheory G.SmithandO.Tabachnikova VectorCalculus P.C.Matthews Andrew Baker Matrix Groups An Introdudionto LieGroupTheory With16Figures , Springer Andrew Baker, BSc, PhD Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK Cover illustration elements reproduced by kind permission oft Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E. Kent-Kangley Road, MapIe VaIley, WA 98038, USA. Te!: (206) 432 -7855 Fax (206) 432 -7832 .email: info@aptecb-com UR!.: www.aptecb.com American Statistical Association: Chance V0 18 No 1, 1995 article by KS and KW Helner 'Iree Rings of the Northem Shawangunks' page 32 fig 2 Springer-Verlag: Mathematica in Education and Research V0 1 4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor 'Illustrated Mathematics: Visualization of Mathematical Objects' page 9 fig 11, originally published as a CD ROM 'Illustrated Mathematics' by TELOS: ISBN 0-387-14222-3, German edition by Birkhauser: ISBN 3-7643-5100-4. Mathematica in Education and Research Vol 4 Issue 3 1995 article by Richard J Gaylord and Kazurne Nishidate 'Iraffic Engineering with Cellular Automata' page 35 fig 2. Mathematica in Education and Research V0 1 5 Issue 2 1996 article by Michael Trott 'Tbe Implicitization of a Trefoil Knot' page 14. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Lee de Cola 'Coins, Trees, Bars and BeIls: Simulation of the BinomiaI Pro cess' page 19 fig 3. Mathematica in Education and Research Vol5 Issue 2 1996 article by Richard Gaylord and Kazume Nishidat. 'Contagious Spreading' page 33 fig 1. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Joe Buhler and Stan Wagon 'Secrets of the Madelung Constant' page 50 fig I. British Library Cataloguing in Publication Data Baker, Andrew Matrix groups : an introduction to Lie group theory. (Springer undergraduate mathematies series) 1. Matrix groups 2. Lie groups 1. Title 512.5'5 ISBN 978-1-85233-470-3 Library of Congress Cataloging-in-Publication Data Baker, Andrew, 1953- Matrix groups : an introduction to Lie group theory / Andrew Baker. p. cm. -- (Springer undergraduate mathematies series) Indudes bibliographieal references and index. ISBN 978-1-85233-470-3 ISBN 978-1-4471-0183-3 (eBook) DOI 10.1007/978-1-4471-0183-3 1. Matrix groups. J. Title. II. Series. QAI74.2.B35 2001 512'.2-dc21 2001049261 Apart from any fair dealing for the purposes of research or private study, or critieism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographie reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries conceming reproduction outside those terms should be sent to the publishers. Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN 978-1-85233-470-3 Additional material to this book can be down loaded from http://extra.springer.com. springeronline.com © Springer-Verlag London 2002 Originally published by Springer-Verlag London Limited in 2002 3rd printing 2006 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific 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. Typesetting: Camera ready by the author 12/3830-5432 Printed on acid-free paper SPIN 11532026 Preface This work provides a first taste of the theory of Lie groups accessible to ad vanced mathematics undergraduates and beginning graduate students, provid ing an appetiser for a more substantial further course. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisti cated and contains many ofthe key themes of the fully developed theory. We concentrate on matrix groups, i.e., closed subgroups of real and complex gen eral linear groups. One of the results proved is that every matrix group is in fact a Lie group, the prooffollowing that in the expository paper ofHowe [12]. Indeed, the latter, togetherwith the book ofCurtis [7], influencedourchoiceof goalsfor the present bookand the coursewhichit evolvedfrom. As pointedout by Howe, Lie theoretic ideas lie at the heart ofmuch ofstandard undergradu ate linear algebra, and exposure to them can inform or motivate the study of the latter; we frequently describe such topics in enough detail to provide the necessary background for the benefit ofreaders unfamiliar with them. Outline of the Chapters Each chapter contains exercises designed to consolidate and deepen readers' understanding of the material covered. We also use these to explore related topics that may not be familiar to all readers but which should be in the toolkit of every well-educated mathematics graduate. Here is a brief synopsis ofthe chapters. Chapter 1: The general linear groups GLn(lk) for Ik = lR (the real numbers) and Ik = C (the complex numbers) are introduced and studied both as groups and as topological spaces. Matrix groups are defined and a numberofstandard examples discussed, including special linear groups SLn(Ik), orthogonal groups O(n) and special orthogonal groups SO(n), unitary groups U(n) and special unitary groups SU(n), as well as more exotic examples such as Lorentz groups v vi Matrix Groups: An Introduction to Lie Group Theory and symplectic groups. The relation of complex to real matrix groups is also studied. Along the way we discuss various algebraic, analytic and topologi cal notions including norms, metric spaces, compactness and continuous group actions. Chapter 2: The exponential function for matrices is introduced and one parameter subgroups of matrix groups are studied. We show how these ideas can be used in the solution ofcertain types ofdifferential equations. Chapter 3: The idea of a Lie algebra is introduced and various algebraic properties are studied. Tangent spaces and Lie algebras of matrix groups are defined together with the adjoint action. The important special case of8U(2) and its relationship to 80(3) is studied in detail. Chapters 4 and 5: Finite dimensional algebras overfields, especiallylRor C, aredefinedandtheirunitsviewedasasourceofmatrixgroupsusingthe reduced regular representation. The quaternions and more generally the real Clifford algebras are defined and spinor groups constructed and shown to double cover the special orthogonal groups. The quaternionic symplectic groups 8p(n) are alsodefined, completingthelistofcompactconnectedclassicalgroupsandtheir universal covers. Automorphism groups of algebras are also shown to provide further examples of matrix groups. Chapter 6: The geometry and linear algebra of Lorentz groups which are of importance in Relativity are studied. The relationship of 8L (C) to the 2 Lorentz group Lor(3,1) is discussed, extending the work on 8U(2) and 80(3) in Chapter 3. Chapter 7: The general notion ofa Lie group is introduced and we show that all matrix groups are Lie subgroups of general linear groups. Along the way we introduce the basic ideas of differentiable manifolds and smooth maps. We show that not everyLiegroup can berealised as a matrixgroup byconsidering the simplest Heisenberg group. Chapters 8 and 9: Homogeneous spaces ofLiegroupsaredefinedandweshow how to recognise them as orbits of smooth actions. We discuss connectivity of Liegroups anduse homogeneousspacestoprovethat manyfamiliar Liegroups are path connected. We also describe some important families ofhomogeneous spacessuchas projectivespaces and Grassmannians, aswellasexamplesrelated to special factorisations ofmatrices such as polarform. Chapters 10, 11 and 12: The basic theory of compact connected Lie groups and their maximal tori is studied and the relationship to some well-known ma trix diagonalisation results highlighted. We continue this theme by describing the classification theory ofcompact connected simple Lie groups, showing how the families we meet in earlier chapters provide all but a finite number ofthe isomorphism types predicted. Root systems, Weyl groups and Dynkin diagrams are defined and many examples described. Preface vii Some suggestions for using this book Foran advanced undergraduate courseofabout 30 lectures tostudentsalready equipped with basic real and complex analysis, metric spaces, linear algebra, group and ring theory, the material ofChapters 1, 2, 3, 7provide an introduc tion to matrix groups, while Chapters 4, 5, 6, 8, 9 supply extra material that might bequarriedfor further examples. A more ambitious course aimed at pre sentingthe classical compact connected Lie groups might take in Chapters4, 5 and perhaps lead on to some of the theory of compact connected Lie groups discussed in Chapters 10, 11, 12. A reader (perhaps a graduate student) using the book on their own would find it useful to follow up some of the references [6, 8, 17, 18, 25, 29] to see more advanced approaches to the topics on differential geometry and topology covered in Chapters 7, 8, 9and the classification theory ofChapters 10, 11, 12. Each chapter has a set of Exercises of varying degrees of difficulty. Hints and solutions are provided for some of these, the more challenging questions being indicated by the symbols ~ or ~~ with the latter intended for readers wishing to pursue the material in greater depth. Prerequisites and assumptions The material in Chapters 1, 2, 3, 7 is intended to be accessible to a well equipped advanced undergraduate, although many topics such as non-metric topological spaces, normed vector spaces and rings may be unfamiliar so we have given the relevant definitions. We do not assume much abstract algebra beyond standard notions of homomorphisms, subobjects, kernels and images and quotients; semi-direct products ofgroups are introduced, as are Lie alge bras. A course on matrix groups is a good setting to learn algebra, and there are many significant algebraic topics in Chapters 4, 5, 11, 12. Good sources of background material are [5, 15, 16, 22, 28] The more advanced parts of the theory which are described in Chap ters 7, 8, 9, 10, 11, 12 should certainly challenge students and naturally point to moredetailedstudiesofDifferential Geometry and Lie Theory. Occasionally ideas from Algebraic Topology are touched upon (e.g., the fundamental group and Lefschetz Fixed Point Theorem) and an interested reader might find it helpful to consult an introductory book on the subject such as [9, 20, 25]. viii Matrix Groups: An Introduction to Lie Group Theory Typesetting ThisbookwasproducedusingIffi.'IE;XandtheAmericanMathematicalSociety's amsmath package. Diagrams were produced usingJW-pic. The symbol ~ was produced by my colleague J. Nimmo, who also provided other help with 'IE;X and Iffi.'IE;X. Acknowledgements I would like to thank the following: the Universitiit Bern for inviting me to visit and teach a course in the Spring of2000; the students who spotted errors andobscuritiesinmy notesand Z. Baloghwhohelpedwiththeproblemclasses; the mathematicians of Glasgow University, especially I. Gordon, J. Nimmo, R. OdoniandJ.Webb; thetopologistsandothermathematiciansofManchester University from whom I learnt a great deal over many years. Also thanks to Roger and Marliese Delaquis for providing a temporary home in the wonderful city of Bern. Finally, special thanks must go to Carole, Daniel and Laura for putting up with it all. Remarks on the second printing Ihavetakentheopportunitytocorrectvariouserrorsfoundinthefirstprint ingandwouldliketothankR. Barraclough,R. Chapman,P. Eccles,S. Hendren, N. Pollock, R. Wickner and M. Yankelevitch for their helpful comments and error-spotting. The web page http://www.maths.gla.ac.uk/~ajb/MatrixGroups/ contains an up to date list ofknown errors and corrections. A reader wishing to pursue exceptional Lie groups and connections with Physics will find much of interest in the excellent recent survey paper of Baez [30],whilerepresentation theory iscoveredbyBrockerand tom Dieck [31] and Sternberg [27]. Contents Part I. Basic Ideas and Examples 1. Real and Complex Matrix Groups .......................... 3 1.1 Groups ofMatrices. ...................................... 3 1.2 Groups ofMatrices as Metric Spaces. ....................... 5 1.3 Compactness............................................. 12 1.4 Matrix Groups. .......................................... 15 1.5 Some Important Examples. ................................ 18 1.6 Complex Matrices as Real Matrices. ........................ 29 1.7 Continuous Homomorphisms ofMatrix Groups. .............. 31 1.8 Matrix Groups for Normed Vector Spaces 33 1.9 Continuous Group Actions. ................................ 37 2. Exponentials, Differential Equations and One-parameter Sub- groups 45 2.1 The Matrix Exponential and Logarithm 45 2.2 Calculating Exponentials and Jordan Form. ................. 51 2.3 Differential Equations in Matrices 55 2.4 One-parameter Subgroups in Matrix Groups. ................ 56 2.5 One-parameter Subgroups and Differential Equations 59 3. Tangent Spaces and Lie Algebras ........................... 67 3.1 Lie Algebras. ............................................ 67 3.2 Curves, Tangent Spaces and Lie Algebras. ................... 71 3.3 The Lie Algebras ofSome Matrix Groups. ................... 76 ix