Lectures on Lie Groups and Lie Algebras

Cover......Page 1
List of Series Publications......Page 3
Title: Lectures on Lie Groups and Lie Algebras......Page 4
ISBN 0 521 49922 4......Page 5
Contents......Page 6
Foreword......Page 8
Lie Algebras and Root Systems by R.W. Carter......Page 10
Contents: Lie Algebras and Root Systems......Page 11
Preface......Page 12
1.1 Basic concepts......Page 14
1.2 Representations and modules......Page 16
1.3 Special kinds of Lie algebra......Page 17
1.4 The Lie algebras sln(C)......Page 19
2.1 Cartan subalgebras......Page 21
2.2 The Cartan decomposition......Page 22
2.3 The Killing fom......Page 24
2.4 The Weyl group......Page 25
2.5 The Dynkin diagram......Page 27
3.1 The universal enveloping algebra......Page 34
3.2 Verma modules......Page 35
3.3 Finite dimensional irreducible modules......Page 36
3.4 Weyl's character and dimension formulae......Page 38
3.5 Fundamental representations......Page 41
4.1 A Chevalley basis of g......Page 45
4.2 Chevalley groups over an arbitrary field......Page 47
4.3 Finite Chevalley groups......Page 48
4.4 Twisted groups......Page 50
4.5 Suzuki and Ree groups......Page 52
4.6 Classification of finite simple groups......Page 53
Lie Groups by Graeme Segal......Page 54
Contents: Lie Groups......Page 55
Introduction......Page 56
1 Examples......Page 58
Matrix groups......Page 59
Low dimensional examples......Page 60
Local isomorphism......Page 61
2 SU2, S03, and SL2R......Page 62
A picture of SL2R.......Page 65
3 Homogeneous spaces......Page 68
Symmetric spaces......Page 69
Complex structures on R^2n......Page 70
B The Gram-Schmidt process......Page 72
C Reduced echelon form: the Bruhat decomposition......Page 73
D Diagonalization and maximal tori......Page 76
Smooth manifolds......Page 78
Tangent spaces......Page 81
One-parameter subgroups and the exponential map......Page 82
Lie's theorems......Page 84
6 Fourier series and representation theory......Page 91
General remarks about representations......Page 93
7 Compact groups and integration......Page 94
A formula for integration on U,.......Page 95
8 Maximal compact subgroups......Page 98
9 The Peter-Weyl theorem......Page 100
The structure of Calg( G)......Page 104
10 Functions on R^n and S^(n-1)......Page 109
The Radon transform......Page 112
11 Induced representations......Page 113
12 The complexification of a compact group......Page 117
Weyl's correspondence......Page 119
Quantum groups......Page 122
14 The Borel-Weil theorem......Page 124
15 Representations of non-compact groups......Page 129
16 Representations of S L2R......Page 133
17 The Heisenberg group, the metaplectic representation, and the spin representation......Page 137
The spin representation......Page 141
Linear Algebraic Groups by I. G. Macdonald......Page 142
Contents: Linear Algebraic Groups......Page 143
Preface......Page 144
Introduction......Page 146
1 Affine algebraic varieties......Page 148
Products......Page 152
The image of a morphism......Page 153
Dimension......Page 154
Examples......Page 155
Jordan decomposition......Page 160
Interlude......Page 163
3 Projective algebraic varieties......Page 166
Prevarieties and varieties......Page 167
Projective Varieties......Page 168
Complete varieties......Page 169
4 Tangent spaces. Separability......Page 171
Separability......Page 173
5 The Lie algebra of a linear algebraic group......Page 175
The adjoint representation......Page 179
6 Homogeneous spaces and quotients......Page 181
7 Borel subgroups and maximal tori......Page 186
Borel subgroups......Page 187
Maximal tori......Page 188
Characters and one-parameter subgroups of tori......Page 191
The root datum B(G, T)......Page 192
Notes and references......Page 195
Bibliography......Page 196
Index......Page 198