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Contents

Foreword

Lie Algebras and Root Systems 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 sln(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 field

4.3 Finite Chevalley groups

4.4 Twisted groups

4.5 Suzuki and Ree groups

4.6 Classification of finite simple groups

Lie Groups Graeme Segal

Introduction

1 Examples

Matrix groups

Low dimensional examples

Local isomorphism

2 SU2, S03, and SL2R

3 Homogeneous spaces

Symmetric spaces

Complex structures on R2n

4 Some theorems about matrices

A The polar decomposition

B The Gram-Schmidt process

C Reduced echelon form: the Bruhat decomposition

D Diagonalization and maximal tori

5 Lie theory

Smooth manifolds

Tangent spaces

One-parameter subgroups and the exponential map

Lie's theorems

6 Fourier series and representation theory

General remarks about representations

7 Compact groups and integration

A formula for integration on Un

8 Maximal compact subgroups

9 The Peter-Weyl theorem

The structure of Calg( G)

10 Functions on Rn and sn-l

The Radon transform

11 Induced representations

12 The complexification of a compact group

13 The unitary groups and the symmetric groups

Weyl's correspondence

Quantum groups

14 The Borel-Weil theorem

15 Representations of non-compact groups

16 Representations of S L2R

17 The Heisenberg group the metaplectic representation, and the spin representation

The spin representation

Linear Algebraic Groups I. G. Macdonald

Preface

Introduction

1 Affine algebraic varieties

Morphisms

Products

The image of a morphism

Dimension

2 Linear algebraic groups: definition and elementary properties

Jordan decomposition

Interlude

3 Projective algebraic varieties

Prevarieties and varieties

Projective Varieties

Complete varieties

4 Tangent spaces. Separability

Separability

5 The Lie algebra of a linear algebraic group

The adjoint representation

6 Homogeneous spaces and quotients

7 Borel subgroups and maximal tori

Borel subgroups

Maximal tori

8 The root structure of a linear algebraic group

Characters and one-parameter subgroups of tori

The root system R(G, T)

The root datum R( G, T)

Notes and references

Bibliography

Index