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