Lectures on Lie Groups and Lie Algebras 9780521499224, 9780521495790

Table of contents :
Foreword M. J. Taylor page vii
Lie Algebras and Root Systems R. W. Carter 1
Preface 3
1 Introduction to Lie algebras 5
1.1 Basic concepts 5
1.2 Representations and modules 7
1.3 Special kinds of Lie algebra 8
1.4 The Lie algebras s/ n (C) 10
2 Simple Lie algebras over C 12
2.1 Cartan subalgebras 12
2.2 The Cartan decomposition 13
2.3 The Killing form 15
2.4 The Weyl group 16
2.5 The Dynkin diagram 18
3 Representations of simple Lie algebras 25
3.1 The universal enveloping algebra 25
3.2 Verma modules 26
3.3 Finite dimensional irreducible modules 27
3.4 Weyl's character and dimension formulae 29
3.5 Fundamental representations 32
4 Simple groups of Lie type 36
4.1 A Chevalley basis of g 36
4.2 Chevalley groups over an arbitrary field 38
4.3 Finite Chevalley groups 39
4.4 Twisted groups 41
4.5 Suzuki and Ree groups 43
4.6 Classification of finite simple groups 44
Lie Groups Graeme Segal 45
Introduction 47
1 Examples 49
2 SU 2 , SO 3 , and SL 2 R 53
3 Homogeneous spaces 59
4 Some theorems about matrices 63
5 Lie theory 69
6 Representation theory 82
7 Compact groups and integration 85
8 Maximal compact subgroups 89
9 The Peter-Weyl theorem 91
10 Functions on R" and S n ~ { 100
11 Induced representations 104
12 The complexiiication of a compact group 108
13 The unitary and symmetric groups 110
14 The Borel-Weil theorem 115
15 Representations of non-compact groups 120
16 Representations of SL 2 R 124
17 The Heisenberg group 128
Linear Algebraic Groups /. G. Macdonald 133
Preface 135
Introduction 137
1 Affine algebraic varieties 139
2 Linear algebraic groups: definition and elementary properties 146
Interlude 154
3 Projective algebraic varieties 157
4 Tangent spaces. Separability 162
5 The Lie algebra of a linear algebraic group 166
6 Homogeneous spaces and quotients 172
7 Borel subgroups and maximal tori 177
8 The root structure of a linear algebraic group 182
Notes and references 186
Bibliography 187
Index 189

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LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managing editor: Professor C M . Series, Mathematical Institute University of Warwick, Coventry CV4 7AL, United Kingdom

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Local fields, J.W.S. CASSELS An introduction to twistor theory: Second edition, S.A. HUGGETT & K.P. TOD Introduction to general relativity, L.P. HUGHSTON & K.P. TOD The theory of evolution and dynamical systems, J. HOFBAUER & K. SIGMUND Summing and nuclear norms in Banach space theory, G.J.O. JAMESON Automorphisms of surfaces after Nielsen and Thurston, A. CASSON & S. BLEILER Spacetime and singularities, G. NABER Undergraduate algebraic geometry, MILES REID An introduction to Hankel operators, J.R. PARTINGTON Presentations of groups: Second edition, D.L. JOHNSON Aspects of quantum field theory in curved spacetime, S.A. FULLING Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN Steps in commutative algebra, R.Y. SHARP Communication theory, C M . GOLDIE & R.G.E. PINCH Representations of finite groups of Lie type, FRANCOIS DIGNE & JEAN MICHEL Designs, graphs, codes, and their links, P.J. CAMERON & J.H. VAN LINT Complex algebraic curves, FRANCES KIRWAN Lectures on elliptic curves, J.W.S. CASSELS An Introduction to the theory of L-functions and Eisenstein series, H. HIDA Hilbert Space: compact operators and the trace theorem, J.R. RETHERFORD Potential theory in the complex plane, T. RANSFORD Undergraduate commutative algebra, M. REID The Laplacian on a Riemannian manifold, S. ROSENBERG Lectures on Lie Groups and Lie Algebras, R. CARTER, G. SEGAL & I. MACDONALD A primer of algebraic D-modules, S.C COUTINHO Complex algebraic surfaces, A. BEAUVILLE Young tableaux, W FULTON A mathematical introduction to wavelets, P. WOJTASCZYK Harmonic maps, loop groups and integrable systems, M. GUEST Set theory for the working mathematician, K. CIESIELSKI Ergodic theory and dynamical systems, M. POLLICOTT & M. YURI The algorithmic resolution of diophantine equations, N.P. SMART Equilibrium states in ergodic theory, G. KELLER Classical invariant theory, P. OLVER Permutation groups, P.J. CAMERON

London Mathematical Society Student Texts 32

Lectures on Lie Groups and Lie Algebras

Roger Carter University of Warwick

Graeme Segal University of Cambridge

Ian Macdonald Queen Mary and Westfield College, London

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521499224 © Cambridge University Press 1995 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1995 Fifth printing 2006 A catalogue record for this publication is available from the British Library ISBN 978-0-521-49579-0 hardback ISBN 978-0-521-49922-4 paperback Transferred to digital printing 2009 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

Contents

Foreword M. J. Taylor

Lie Algebras and Root Systems

page vii

R. W. Carter

Preface 1 Introduction to Lie algebras 1.1 Basic concepts 1.2 Representations and modules 1.3 Special kinds of Lie algebra 1.4 The Lie algebras s/n(C) 2 Simple Lie algebras over C 2.1 Cartan subalgebras 2.2 The Cartan decomposition 2.3 The Killing form 2.4 The Weyl group 2.5 The Dynkin diagram 3 Representations of simple Lie algebras 3.1 The universal enveloping algebra 3.2 Verma modules 3.3 Finite dimensional irreducible modules 3.4 Weyl's character and dimension formulae 3.5 Fundamental representations 4 Simple groups of Lie type 4.1 A Chevalley basis of g 4.2 Chevalley groups over an arbitrary 4.3 Finite Chevalley groups 4.4 Twisted groups 4.5 Suzuki and Ree groups 4.6 Classification of finite simple groups

field

1 3 5 5 7 8 10 12 12 13 15 16 18 25 25 26 27 29 32 36 36 38 39 41 43 44

vi

Lie Groups

Contents

Graeme Segal

45

Introduction 1 Examples 2 SU2, SO3, and SL 2 R 3 Homogeneous spaces 4 Some theorems about matrices 5 Lie theory 6 Representation theory 7 Compact groups and integration 8 Maximal compact subgroups 9 The Peter-Weyl theorem 10 Functions on R" and Sn~{ 11 Induced representations 12 The complexiiication of a compact group 13 The unitary and symmetric groups 14 The Borel-Weil theorem 15 Representations of non-compact groups 16 Representations of SL 2 R 17 The Heisenberg group

47 49 53 59 63 69 82 85 89 91 100 104 108 110 115 120 124 128

Linear Algebraic Groups

133

/. G. Macdonald

Preface 135 Introduction 137 1 Affine algebraic varieties 139 2 Linear algebraic groups: definition and elementary properties 146 Interlude 154 3 Projective algebraic varieties 157 4 Tangent spaces. Separability 162 5 The Lie algebra of a linear algebraic group 166 6 Homogeneous spaces and quotients 172 7 Borel subgroups and maximal tori 177 8 The root structure of a linear algebraic group 182 Notes and references 186 Bibliography 187 Index 189

Foreword

This book consists of notes based on the three introductory lecture courses given at the LMS-SERC Instructional Conference on Lie theory and algebraic groups held at Lancaster University in September 1993: Lie Algebras by Roger Carter; Lie Groups by Graeme Segal; algebraic groups by Ian Macdonald. The aim of the course was to provide an introduction to this important area of mathematics for postgraduate students who had no previous specialised knowledge. Discussions with the students at the end of the meeting suggested that the conference had been extremely successful; it then seemed desirable to further impose on the lecturers by asking them to write-up their lectures, in order that future generations of students could also benefit from their efforts. All three lecturers adopted the same approach of providing a crisp, fast-moving, clear introduction, while at the same time taking care to indicate more advanced material, so as to give the full flavour of the subject. It is clear, from both the lectures and the written account, that a substantial effort was made to ensure a coherent and well-harmonised presentation of these three highly interrelated themes. The general intention of the new series of LMS-SERC Instructional Conferences is to provide postgraduate students with the opportunity to learn important mainstream core mathematics, which they might not otherwise meet. Lie theory and algebraic groups seemed to be a natural first choice, since they are a central mathematical crossroads, which relate to a host of important areas such as group theory, number theory, algebraic geometry, differential geometry, topology, particle physics and strings; indeed, a knowledge of algebraic groups and Lie theory can be quite crucial in making significant progress in many aspects of these related areas.

viii

Foreword

Finally, on behalf of the LMS, I should like to express my deepest gratitude to the three authors for not just accepting to give their lectures and then write them up, but also for carrying out their alloted tasks with such infectious enthusiasm; extra special thanks go to Ian Macdonald for sage advice in the initial planning of the meeting. It is also a pleasure to thank both Roger Astley and David Tranah for their help and cooperation in producing these notes, which will be a very valuable contribution to the mathematical community. Martin Taylor UMIST Manchester

Lie Algebras and Root Systems R.W. Carter

Contents Lie Algebras and Root Systems

Preface 1 Introduction to Lie algebras 1.1 Basic concepts 1.2 Representations and modules 1.3 Special kinds of Lie algebra 1.4 The Lie algebras s/n(C) 2 Simple Lie algebras over C 2.1 Cartan subalgebras 2.2 The Cartan decomposition 2.3 The Killing form 2.4 The Weyl group 2.5 The Dynkin diagram 3 Representations of simple Lie algebras 3.1 The universal enveloping algebra 3.2 Verma modules 3.3 Finite dimensional irreducible modules 3.4 Weyl's character and dimension formulae 3.5 Fundamental representations 4 Simple groups of Lie type 4.1 A Chevalley basis of g 4.2 Chevalley groups over an arbitrary 4.3 Finite Chevalley groups 4.4 Twisted groups 4.5 Suzuki and Ree groups 4.6 Classification of finite simple groups

field

3 5 5 7 8 10 12 12 13 15 16 18 25 25 26 27 29 32 36 36 38 39 41 43 44

Preface

The following notes on Lie Algebras and Root Systems follow fairly closely the lectures I gave on this subject at the Lancaster meeting, although more detail has been included in a number of places. The aim has been to give an outline of the main ideas involved in the structure and representation theory of the simple Lie algebras over C, and the construction of the corresponding groups of Lie type over an arbitrary field. It has not been possible to give all the proofs in detail, and so interested readers are encouraged to consult books in which more complete information is given. The following books are particularly recommended. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9 (1972) Springer. N. Jacobson, Lie Algebras. Interscience Publishers, J. Wiley, New York (1962). R. W. Carter, Simple Groups of Lie Type, Wiley Classics Library Edition (1989), J. Wiley, New York.

1 Introduction to Lie algebras

1.1 Basic concepts A Lie algebra is a vector space g over a field F on which a multiplication g xg

-+ g

x,y

-* [xy]

is defined satisfying the axioms: (i) [xy] is linear in x and in y. (ii) [xx] = 0 for all x e g. (iii) [[xy]z] + [[yz]x] + [[zx]y] = 0 for all x,y,z € g. Property (iii) is called the Jacobi identity. We note that the multiplication is not associative, i.e., it is not true in general that [[xy]z] = [x[yz]]. It is therefore essential to include the Lie brackets in products of elements. For any pair of elements x j G g w e have [x + y, x + y] = [xx] + [xy] + [yx] + \yy]. We also know that [xx] = 0 ,

[yy] = 0 ,

[x + y,x + y] = 0 .

It follows that [yx] = — [xy] for all x,y G g. Thus multiplication in a Lie algebra is anticommutative. Lie algebras can be obtained from associative algebras by the following method. Let A be an associative algebra, i.e., a vector space with a bilinear associative multiplication xy. Then we may obtain a Lie algebra [A] by redefining the multiplication on A. We define [xy] = xy — yx. It is clear

6

/ Lie Algebras

that [xy] is linear in x and in y and that [xx] = 0. We also have [[xy]z]

=

(xy - yx)z - z(xy - yx)

=

xyz — yxz — zxy 4- zyx.

It follows that [[xy]z] + [[yz]x] + [[zx]y] =

xyz — yxz — zxy -f zyx

4-yzx — zyx — xyz -f- xzy -\-zxy — xzy — yzx 4- yxz = 0, so that the Jacobi identity is satisfied. Let gugi be Lie algebras over F. A homomorphism of Lie algebras is a linear map 8 : g\ —> g2 such that 0[xy] = [0x, 0y] for all x,y G gi. 6 is an isomorphism of Lie algebras if 0 is a bijective homomorphism. Let g be a Lie algebra and h,k be subspaces of g. We define the product [hk] to be the subspace spanned by all products [xy] for x € h, y € k. Each element of [hk] is thus a finite sum xiyi 4- • • • 4- x r y r with Xi € h, y\ € k. We note that [We] = [kh], i.e., multiplication of subspaces is commutative. This follows from the fact that multiplication of elements is anticommutative. So if x € h, y € k we have [yx] = — [xy] G [hk]. A subalgebra of g is a subspace h of g such that [hh] c /z. An ideal of g is a subspace /i of g such that [hg] a h. We observe that, since [hg] = [g/z], there is no distinction in the theory of Lie algebras between left ideals and right ideals. Every ideal is two-sided. Now let h be an ideal of the Lie algebra g. Let g/h be the vector space of cosets h -f x for x € g. h 4- x consists of all elements of form y + x for y € h. We claim that g/h can be made into a Lie algebra, the factor algebra of g with respect to h9 by introducing the Lie multiplication [h + x,h + y]

=h+[xy].

We must take care to check that this operation is well defined, i.e., that if h + x = h + x' and h + y = h + y' then /i 4- [xy] = h + [ x y ] . This follows from the fact that h is an ideal of g. We have x' = a + x,

yr = b 4- y

for a,b € h.

Thus [xV] = [ofc] + [ay] 4- [xb] 4- [xy] G h 4- [xy]

1.2 Representations and modules

1

since [aft], [ay], [xb] all lie in h. This gives h+ [x'yf] = h+ [xy] as required. There is a natural homomorphism g —• g/h relating a Lie algebra with a factor algebra. 0 is defined by d(x) = h -f x. Conversely given any homomorphism 9 : g\ —• gi of Lie algebras which is surjective, the kernel k of 6 is an ideal of g\ and the factor algebra g\/k is isomorphic to gi. The set of all n x n matrices over the field F can be made into a Lie algebra under the Lie multiplication [A, B] = AB — £ A This Lie algebra is called gln(F), the general linear Lie algebra of degree n over the field F.

1.2 Representations and modules

Let g be a Lie algebra over F. A representation of g is a homomorphism p:g->

gln(F)

for some n. Two representations p,pf of g of degree n are called equivalent if there is a non-singular n x n matrix T over F such that p'(x)=T-1p(x)T,

forallx€g.

There is a close connection between representations of g and g-modules. A left g-module is a vector space V over F with a multiplication

X, V

-+

XV

satisfying the axioms (i) xv is linear in x and in v (ii) [xy]t; = x(yv) — y(xv) for all x,y e g, v e V. Every finite dimensional g-module gives a representation of g, as follows. Choose a basis e\,...,en of V. Then xej is a linear combination of e\9...,en.

Let

Let p(x) be the n x n matrix (pij(x)), Then we have p[xy] = p(x)p(y) - p(y)p(x) = and so the map x —• p(x) is a representation of g. If we choose a different basis for the g-module V we shall get an equivalent representation.

8

/ Lie Algebras

Now let U be a subspace of V and h a subspace of g. Let hU be the subspace of V spanned by all elements xu for x € Ji, w € U. U is called a submodule of F if gU a U. A g-module V is called irreducible if V has no submodules other than V and 0. Now g is itself a g-module under the multiplication gxg —• g given by *>y -* [*?]• To see this we must check [[xy]z] = [x[yz]] — \y[xz]] for x, y, z € g. This follows from the Jacobi identity using the anticommutative law. g is called the adjoint g-module, and it gives rise to the adjoint representation of g.

1.3 Special kinds of Lie algebra So far the theory of Lie algebras has been very analogous to the theory of rings, where one has subrings, ideals, factor rings, etc. However there is also a sense in which the theory of Lie algebras can be considered as analogous to the theory of groups, where the Lie product [xy] is regarded as analogous to the commutator x~ly~lxy of two elements in a group. This analogy motivates the following terminology. A Lie algebra g is called abelian if [gg] = 0. This means that all Lie products are zero. We shall now define a sequence of subspaces gKg2,g3,'" of g. We define them inductively by g1-*,

g n + l = b n g].

Now if h,k are ideals of g so is their product [hk]. For let x e h, y € k, z € g. Then we have [[xy]z] = [x[yz]] + [[xz]y] € [hk]. Thus the product of two ideals is an ideal. It follows that all the subspaces g1 defined above are ideals of g. Thus we also have g n+1 = [gng] g2 => g3 => ' •' • The Lie algebra g is called nilpotent if g1 = 0 for some i. Every abelian Lie algebra is nilpotent. Example. The set of all n x n matrices (ay) over F with ay = 0 whenever i ^ j is a nilpotent Lie algebra under Lie multiplication [AB] = AB—BA.

1.3 Special kinds of Lie algebra

9

We now define a different sequence of subspaces g (0) ,g (I) ,g (2) , • • • of g. We again define them inductively by g(0) =

g ;

g (n + .) = [ g (n) g (n, ] _

The g(l) are all ideals of g. Also we have g (n + l) = [ g (n, g (n) ] c g (n)

and so we again have a descending series

The Lie algebra g is called soluble if g(l) = 0 for some I Proposition. Every nilpotent Lie algebra is soluble. Proof We show first that [gmgn] cz g m + n for all m,n. We proceed by induction on n9 the result being clear if n = 1. Assuming inductively that [gmgn] let x 6 g m , y G g n , z e g. Then we have [x\yz]] = [[xy]z] - [[xz]y] G g m + n + 1

by induction. Thus [gmgn+l] j is a soluble Lie algebra. A Lie algebra g is called simple if g has no ideals other than g and 0. A Lie algebra g of dimension 1 is of course simple because g has no proper subspaces at all. We have g = Kx for some x G g. Since [xx] = 0 we have [gg] = 0. Such a 1-dimensional Lie algebra will be called a trivial simple Lie algebra. We shall be mainly interested in non-trivial simple Lie algebras.

10

/ Lie Algebras 1.4 The Lie algebras s/n(C)

We shall now take F = C. Let sln(