Lie groups and lie algebras: chapters 4-6

Introduction to Chapters IV, V and VI

Contents

Chapter IV. Coxeter Groups and Tits Systems

1. Coxeter Groups

1. Length and reduced decompositions

2. Dihedral groups

3. First properties of Coxeter groups

4. Reduced decompositions in a Coxeter group

5. The exchange condition

6. Characterisation of Coxeter groups

7. Families of partitions

8. Subgroups of Coxeter groups

9. Coxeter matrices and Coxeter graphs

2. Tits Systems

1. Definitions and first properties

2. An example

3. Decomposition of G into double cosets

4. Relations with Coxeter systems

5. Subgroups of G containing B

6. Parabolic subgroups

7. The simplicity theorem

Appendix. Graphs

1. Definitions

2. The connected components of a graph

3. Forests and trees

Exercises for [section] 1

Exercises for [section] 2

Chapter V. Groups Generated by Reflections

1. Hyperplanes, chambers and facets

1. Notations

2. Facets

3. Chambers

4. Walls and faces

5. Intersecting hyperplanes

6. Simplicial cones and simplices

2. Reflections

1. Pseudo-reflections

2. Reflections

3. Orthogonal reflections

4. Orthogonal reflections in a euclidean affine space

5. Complements on plane rotations

3. Groups of displacements generated by reflections

1. Preliminary results

2. Relation with Coxeter systems

3. Fundamental domain, stabilisers

4. Coxeter matrix and Coxeter graph of W

5. Systems of vectors with negative scalar products

6. Finiteness theorems

7. Decomposition of the linear representation of W on T

8. Product decomposition of the affine space E

9. The structure of chambers

10. Special points

4. The geometric representation of a Coxeter group

1. The form associated to a Coxeter group

2. The plane E[subscript s,s'] and the group generated by [sigma subscript s] and [sigma subscript s']

3. The group and representation associated to a Coxeter matrix

4. The contragredient representation

5. Proof of lemma 1

6. The fundamental domain of W in the union of the chambers

7. Irreducibility of the geometric representation of a Coxeter group

8. Finiteness criterion

9. The case in which B[subscript M] is positive and degenerate

5. Invariants in the symmetric algebra

1. Poincare series of graded algebras

2. Invariants of a finite linear group: modular properties

3. Invariants of a finite linear group: ring-theoretic properties

4. Anti-invariant elements

5. Complements

6. The Coxeter transformation

1. Definition of Coxeter transformations

2. Eigenvalues of a Coxeter transformation: exponents

Appendix. Complements on linear representations

Exercises for [section] 2

Exercises for [section] 3

Exercises for [section] 4

Exercises for [section] 5

Exercises for [section] 6

Chapter VI. Root Systems

1. Root systems

1. Definition of a root system

2. Direct sum of root systems

3. Relation between two roots

4. Reduced root systems

5. Chambers and bases of root systems

6. Positive roots

7. Closed sets of roots

8. Highest root

9. Weights, radical weights

10. Fundamental weights, dominant weights

11. Coxeter transformations

12. Canonical bilinear form

2. Affine Weyl group

1. Affine Weyl group

2. Weights and special weights

3. The normaliser of W[subscript a]

4. Application: order of the Weyl group

5. Root systems and groups generated by reflections

3. Exponential invariants

1. The group algebra of a free abelian group

2. Case of the group of weights: maximal terms

3. Anti-invariant elements

4. Invariant elements

4. Classification of root systems

1. Finite Coxeter groups

2. Dynkin graphs

3. Affine Weyl group and completed Dynkin graph

4. Preliminaries to the construction of root systems

5. Systems of type B[subscript l] (l [greater than or equal] 2)

6. Systems of type C[subscript l] (l [greater than or equal] 2)

7. Systems of type A[subscript l] (l [greater than or equal] 1)

8. Systems of type D[subscript l] (l [greater than or equal] 3)

9. System of type F[subscript 4]

10. System of type E[subscript 8]

11. System of type E[subscript 7]

12. System of type E[subscript 6]

13. System of type G[subscript 2]

14. Irreducible non-reduced root systems

Exercises for [section] 1

Exercises for [section] 2

Exercises for [section] 3

Exercises for [section] 4

Historical Note (Chapters IV, V and VI)

Bibliography

Index of Notation

Index of Terminology

Plate I. Systems of type A[subscript l] (l [greater than or equal] 1)

Plate II. Systems of type B[subscript l] (l [greater than or equal] 2)

Plate III. Systems of type C[subscript l] (l [greater than or equal] 2)

Plate IV. Systems of type D[subscript l] (l [greater than or equal] 3)

Plate V. System of type E[subscript 6]

Plate VI. System of type E[subscript 7]

Plate VII. System of type E[subscript 8]

Plate VIII. System of type F[subscript 4]

Plate IX. System of type G[subscript 2]

Plate X. Irreducible systems of rank 2

Summary of the principal properties of root systems