Lectures on Lie Groups and Lie Algebras

In this excellent introduction to the theory of Lie groups and Lie algebras, three of the leading figures in this area have written up their lectures from an LMS/SERC sponsored short course in 1993. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie groups. And in the final part, Ian Macdonald gives an introduction to special linear groups. Anybody requiring an introduction to the theory of Lie groups and their applications should look no further than this book.

Foreword

vii

M. J. Taylor

Lie Algebras and Root Systems

1

(44)

R. W. Carter

Preface

3

(2)

Introduction to Lie algebras

5

(7)

Basic concepts

5

(2)

Representations and modules

7

(1)

Special kinds of Lie algebra

8

(2)

The Lie algebras sln (C)

10

(2)

Simple Lie algebras over C

12

(13)

Cartan subalgebras

12

(1)

The Cartan decomposition

13

(2)

The Killing form

15

(1)

The Weyl group

16

(2)

The Dynkin diagram

18

(7)

Representations of simple Lie algebras

25

(11)

The universal enveloping algebra

25

(1)

Verma modules

26

(1)

Finite dimensional irreducible modules

27

(2)

Weyl's character and dimension formulae

29

(3)

Fundamental representations

32

(4)

Simple groups of Lie type

36

(9)

A Chevalley basis of g

36

(2)

Chevalley groups over an arbitrary field

38

(1)

Finite Chevalley groups

39

(2)

Twisted groups

41

(2)

Suzuki and Ree groups

43

(1)

Classification of finite simple groups

44

(1)

Lie Groups

45

(88)

Graeme Segal

Introduction

47

(2)

Examples

49

(4)

SU2, SO3, and SL2R

53

(6)

Homogeneous spaces

59

(4)

Some theorems about matrices

63

(6)

Lie theory

69

(13)

Representation theory

82

(3)

Compact groups and integration

85

(4)

Maximal compact subgroups

89

(2)

The Peter-Weyl theorem

91

(9)

Functions on Rn'' and Sn-1

100

(4)

Induced representations

104

(4)

The complexification of a compact group

108

(2)

The unitary and symmetric groups

110

(5)

The Borel-Weil theorem

115

(5)

Representations of non-compact groups

120

(4)

Representations of SL2R

124

(4)

The Heisenberg group

128

(5)

Linear Algebraic Groups

133

(53)

I.G. Macdonald

Preface

135

(2)

Introduction

137

(2)

Affine algebraic varieties

139

(7)

Linear algebraic groups: definition and elementary properties

146

(11)

Interlude

154

(3)

Projective algebraic varieties

157

(5)

Tangent spaces. Separability

162

(4)

The Lie algebra of a linear algebraic group

166

(6)

Homogeneous spaces and quotients

172

(5)

Borel subgroups and maximal tori

177

(5)

The root structure of a linear algebraic group

182

(4)

Notes and references

186

(1)

Bibliography

187

(2)

Index

189

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Lectures on Lie Groups and Lie Algebras

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