Springer Undergraduate Mathematics Series

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P.I. Cameron Queen Mary and Westfield CollegeM.A.I. Chaplain University ofDundeeK. Erdmann Oxford UniversityL.C.G. Rogers Cambridge UniversityE. Silli Oxford UniversityI.F. Toland University ofBath

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Andrew Baker

Matrix GroupsAn Introdudion to Lie Group Theory

With 16 Figures

, 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 V 018 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 V 01 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 V 01 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 Pro­cess' 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

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© Springer-Verlag London 2002 Originally published by Springer-Verlag London Limited in 2002

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Preface

This work provides a first taste of the theory of Lie groups accessible to ad­vanced mathematics undergraduates and beginning graduate students, provid­ing an appetiser for a more substantial further course. Although the formalprerequisites are kept as low level as possible, the subject matter is sophisti­cated and contains many of the key themes of the fully developed theory. Weconcentrate on matrix groups, i.e., closed subgroups of real and complex gen­eral linear groups. One of the results proved is that every matrix group is infact a Lie group, the proof following that in the expository paper of Howe [12].Indeed, the latter, together with the book of Curtis [7], influenced our choice ofgoals for the present book and the course which it evolved from. As pointed outby Howe, Lie theoretic ideas lie at the heart of much of standard undergradu­ate linear algebra, and exposure to them can inform or motivate the study ofthe latter; we frequently describe such topics in enough detail to provide thenecessary background for the benefit of readers unfamiliar with them.

Outline of the ChaptersEach chapter contains exercises designed to consolidate and deepen readers'understanding of the material covered. We also use these to explore relatedtopics that may not be familiar to all readers but which should be in thetoolkit of every well-educated mathematics graduate. Here is a brief synopsisof the 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 groupsand as topological spaces. Matrix groups are defined and a number of standardexamples discussed, including special linear groups SLn(Ik), orthogonal groupsO(n) and special orthogonal groups SO(n), unitary groups U(n) and specialunitary 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 alsostudied. Along the way we discuss various algebraic, analytic and topologi­cal notions including norms, metric spaces, compactness and continuous groupactions.Chapter 2: The exponential function for matrices is introduced and one­parameter subgroups of matrix groups are studied. We show how these ideascan be used in the solution of certain types of differential equations.Chapter 3: The idea of a Lie algebra is introduced and various algebraicproperties are studied. Tangent spaces and Lie algebras of matrix groups aredefined together with the adjoint action. The important special case of 8U(2)and its relationship to 80(3) is studied in detail.Chapters 4 and 5: Finite dimensional algebras over fields, especially lR or C,are defined and their units viewed as a source of matrix groups using the reducedregular representation. The quaternions and more generally the real Cliffordalgebras are defined and spinor groups constructed and shown to double coverthe special orthogonal groups. The quaternionic symplectic groups 8p(n) arealso defined, completing the list of compact connected classical groups and theiruniversal covers. Automorphism groups of algebras are also shown to providefurther examples of matrix groups.Chapter 6: The geometry and linear algebra of Lorentz groups which areof importance in Relativity are studied. The relationship of 8L2(C) to theLorentz group Lor(3, 1) is discussed, extending the work on 8U(2) and 80(3)in Chapter 3.Chapter 7: The general notion of a Lie group is introduced and we show thatall matrix groups are Lie subgroups of general linear groups. Along the waywe introduce the basic ideas of differentiable manifolds and smooth maps. Weshow that not every Lie group can be realised as a matrix group by consideringthe simplest Heisenberg group.Chapters 8 and 9: Homogeneous spaces of Lie groups are defined and we showhow to recognise them as orbits of smooth actions. We discuss connectivity ofLie groups and use homogeneous spaces to prove that many familiar Lie groupsare path connected. We also describe some important families of homogeneousspaces such as projective spaces and Grassmannians, as well as examples relatedto special factorisations of matrices such as polar form.Chapters 10, 11 and 12: The basic theory of compact connected Lie groupsand their maximal tori is studied and the relationship to some well-known ma­trix diagonalisation results highlighted. We continue this theme by describingthe classification theory of compact connected simple Lie groups, showing howthe families we meet in earlier chapters provide all but a finite number of theisomorphism types predicted. Root systems, Weyl groups and Dynkin diagramsare defined and many examples described.

Preface vii

Some suggestions for using this bookFor an advanced undergraduate course of about 30 lectures to students alreadyequipped with basic real and complex analysis, metric spaces, linear algebra,group and ring theory, the material of Chapters 1, 2, 3, 7 provide an introduc­tion to matrix groups, while Chapters 4, 5, 6, 8, 9 supply extra material thatmight be quarried for further examples. A more ambitious course aimed at pre­senting the classical compact connected Lie groups might take in Chapters 4, 5and perhaps lead on to some of the theory of compact connected Lie groupsdiscussed in Chapters 10, 11, 12.

A reader (perhaps a graduate student) using the book on their own wouldfind it useful to follow up some of the references [6, 8, 17, 18, 25, 29] to seemore advanced approaches to the topics on differential geometry and topologycovered in Chapters 7, 8, 9 and the classification theory of Chapters 10, 11, 12.

Each chapter has a set of Exercises of varying degrees of difficulty. Hintsand solutions are provided for some of these, the more challenging questionsbeing indicated by the symbols ~ or ~~ with the latter intended forreaders wishing to pursue the material in greater depth.

Prerequisites and assumptionsThe material in Chapters 1, 2, 3, 7 is intended to be accessible to a well­equipped advanced undergraduate, although many topics such as non-metrictopological spaces, normed vector spaces and rings may be unfamiliar so wehave given the relevant definitions. We do not assume much abstract algebrabeyond standard notions of homomorphisms, subobjects, kernels and imagesand quotients; semi-direct products of groups are introduced, as are Lie alge­bras. A course on matrix groups is a good setting to learn algebra, and thereare many significant algebraic topics in Chapters 4, 5, 11, 12. Good sources ofbackground material are [5, 15, 16, 22, 28]

The more advanced parts of the theory which are described in Chap­ters 7, 8, 9, 10, 11, 12 should certainly challenge students and naturally pointto more detailed studies of Differential Geometry and Lie Theory. Occasionallyideas from Algebraic Topology are touched upon (e.g., the fundamental groupand Lefschetz Fixed Point Theorem) and an interested reader might find ithelpful to consult an introductory book on the subject such as [9, 20, 25].

viii Matrix Groups: An Introduction to Lie Group Theory

TypesettingThis book was produced using Iffi.'IE;X and the American Mathematical Society'samsmath package. Diagrams were produced using JW-pic. The symbol ~ wasproduced by my colleague J. Nimmo, who also provided other help with 'IE;Xand Iffi.'IE;X.

Acknowledgements

I would like to thank the following: the Universitiit Bern for inviting me tovisit and teach a course in the Spring of 2000; the students who spotted errorsand obscurities in my notes and Z. Balogh who helped with the problem classes;the mathematicians of Glasgow University, especially I. Gordon, J. Nimmo,R. Odoni and J. Webb; the topologists and other mathematicians of ManchesterUniversity from whom I learnt a great deal over many years. Also thanks toRoger and Marliese Delaquis for providing a temporary home in the wonderfulcity of Bern. Finally, special thanks must go to Carole, Daniel and Laura forputting up with it all.

Remarks on the second printing

I have taken the opportunity to correct various errors found in the first print­ing and would like to thank R. Barraclough, R. Chapman, P. Eccles, S. Hendren,N. Pollock, R. Wickner and M. Yankelevitch for their helpful comments anderror-spotting. The web page

http://www.maths.gla.ac.uk/~ajb/MatrixGroups/

contains an up to date list of known 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 ofBaez [30], while representation theory is covered by Brocker and tom Dieck [31]and Sternberg [27].

Contents

Part I. Basic Ideas and Examples

1. Real and Complex Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Groups of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Groups of Matrices as Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . 51.3 Compactness............................................. 121.4 Matrix Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 151.5 Some Important Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181.6 Complex Matrices as Real Matrices. . . . . . . . . . . . . . . . . . . . . . . .. 291.7 Continuous Homomorphisms of Matrix Groups. . . . . . . . . . . . . .. 311.8 Matrix Groups for Normed Vector Spaces 331.9 Continuous Group Actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37

2. Exponentials, Differential Equations and One-parameter Sub-groups 452.1 The Matrix Exponential and Logarithm 452.2 Calculating Exponentials and Jordan Form. . . . . . . . . . . . . . . . .. 512.3 Differential Equations in Matrices 552.4 One-parameter Subgroups in Matrix Groups. . . . . . . . . . . . . . . .. 562.5 One-parameter Subgroups and Differential Equations 59

3. Tangent Spaces and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . .. 673.1 Lie Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 673.2 Curves, Tangent Spaces and Lie Algebras. . . . . . . . . . . . . . . . . . .. 713.3 The Lie Algebras of Some Matrix Groups. . . . . . . . . . . . . . . . . . .. 76

ix

3.4 Some Observations on the Exponential Function of a MatrixGroup 84

3.5 SO(3) and SU(2) . . . .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. . .. .. 863.6 The Complexification of a Real Lie Algebra. . . . . . . . . . . . . . . . .. 92

4. Algebras, Quaternions and Quaternionic Symplectic Groups 994.1 Algebras 994.2 Real and Complex Normed Algebras , 1114.3 Linear Algebra over a Division Algebra 1134.4 The Quaternions 1164.5 Quaternionic Matrix Groups 1204.6 Automorphism Groups of Algebras 122

5. Clifford Algebras and Spinor Groups 1295.1 Real Clifford Algebras 1305.2 Clifford Groups 1395.3 Pinor and Spinor Groups 1435.4 The Centres of Spinor Groups 1515.5 Finite Subgroups of Spinor Groups 152

6. Lorentz Groups 1576.1 Lorentz Groups 1576.2 A Principal Axis Theorem for Lorentz Groups 1656.3 SL2(C) and the Lorentz Group Lor(3, 1) 171

Part II. Matrix Groups as Lie Groups

7. Lie Groups , 1817.1 Smooth Manifolds 1817.2 Tangent Spaces and Derivatives 1837.3 Lie Groups 1877.4 Some Examples of Lie Groups 1897.5 Some Useful Formulre in Matrix Groups 1937.6 Matrix Groups are Lie Groups 1997.7 Not All Lie Groups are Matrix Groups 203

8. Homogeneous Spaces 2118.1 Homogeneous Spaces as Manifolds 2118.2 Homogeneous Spaces as Orbits 2158.3 Projective Spaces 2178.4 Grassmannians 222

Contents xi

8.5 The Gram-Schmidt Process 2248.6 Reduced Echelon Form 2268.7 Real Inner Products 2278.8 Symplectic Forms 229

9. Connectivity of Matrix Groups 2359.1 Connectivity of Manifolds 2359.2 Examples of Path Connected Matrix Groups 2389.3 The Path Components of a Lie Group 2419.4 Another Connectivity Result 244

Part III. Compact Connected Lie Groups and their Classification

10. Maximal Tori in Compact Connected Lie Groups 25110.1 Tori 25110.2 Maximal Tori in Compact Lie Groups 25510.3 The Normaliser and Weyl Group of a Maximal Torus 25910.4 The Centre of a Compact Connected Lie Group 263

11. Semi-simple Factorisation 26711.1 An Invariant Inner Product 26711.2 The Centre and its Lie Algebra 27011.3 Lie Ideals and the Adjoint Action 27211.4 Semi-simple Decompositions 27611.5 Structure of the Adjoint Representation 278

12. Roots Systems, Weyl Groups and Dynkin Diagrams 28912.1 Inner Products and Duality 28912.2 Roots systems and their Weyl groups 29112.3 Some Examples of Root Systems 29312.4 The Dynkin Diagram of a Root System 29712.5 Irreducible Dynkin Diagrams 29812.6 From Root Systems to Lie Algebras 299

Hints and Solutions to Selected Exercises 303

Bibliography 323

Index 325