Table Of ContentSpringer 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
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Andrew Baker
Matrix Groups
An Introdudionto LieGroupTheory
With16Figures
, Springer
Andrew Baker, BSc, PhD
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
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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
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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.
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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 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