Part Ⅰ-Lie Algebras

Introduction

Chapter Ⅰ.Lie Algebras：Definition and Examples

Chapter Ⅱ.Filtered Groups and Lie Algebras

1.Formulae on commutators

2.Filtration on a group

3.Integral filtrations of a group

4.Filtrations in GL（n）

Exercises

Chapter Ⅲ.Universal Algebra of a Lie Algebra

1.Definition

2.Functorial properties

3.Symmetric algebra of a module

4.Filtration of U？

5.Diagonal map

Exercises

Chapter Ⅳ.Free Lie Algebras

1.Free magmas

2.Free algebra on X

3.Free Lie algebra on X

4.Relation with the free associative algebra on X

5.P.Hall families

6.Free groups

7.The Campbell-Hausdorff formula

8.Explicit formula

Exercises

Chapter Ⅴ.Nilpotent and Solvable Lie Algebras

1.Complements on ？-modules

2.Nilpotent Lie algebras

3.Main theorems

3.The group-theoretic analog of Engel’s theorem

4.Solvable Lie algebras

5.Main theorem

5.The group theoretic analog of Lie’s theorem

6.Lemmas on endomorphisms

7.Cartan’s criterion

Exercises

Chapter Ⅵ.Semisimple Lie Algebras

1.The radical

2.Semisimple Lie algebras

3.Complete reducibility

4.Levi’s theorem

5.Complete reducibility continued

6.Connection with compact Lie groups over R and C

Exercises

Chapter Ⅶ.Representations of s？n

1.Notations

2.Weights and primitive elements

3.Irreducible ？-modules

4.Determination of the highest weights

Exercises

Part Ⅱ-Lie Groups

Introduction

Chapter Ⅰ.Complete Fields

Chapter Ⅱ.Analytic Functions

“Tournants dangereux”

Chapter Ⅲ.Analytic Manifolds

1.Charts and atlases

2.Definition of analytic manifolds

3.Topological properties of manifolds

4.Elementary examples of manifolds

5.Morphisms

6.Products and sums

7.Germs of analytic functions

8.Tangent and cotangent spaces

9.Inverse function theorem

10.Immersions，submersions，and subimmersions

11.Construction of manifolds：inverse images

12.Construction of manifolds：quotients

Exercises

Appendix 1.A non-regular Hausdorff manifold

Appendix 2.Structure of p-adic manifolds

Appendix 3.The transfinite p-adic line

Chapter Ⅳ.Analytic Groups

1.Definition of analytic groups

2.Elementary examples of analytic groups

3.Group chunks

4.Prolongation of subgroup chunks

5.Homogeneous spaces and orbits

6.Formal groups：definition and elementary examples

7.Formal groups：formulae

8.Formal groups over a complete valuation ring

9.Filtrations on standard groups

Exercises

Appendix 1.Maximal compact subgroups of GL（n，k）

Appendix 2.Some convergence lemmas

Appendix 3.Applications of §9：“Filtrations on standard groups”

Chapter Ⅴ.Lie Theory

1.The Lie algebra of an analytic group chunk

2.Elementary examples and properties

3.Linear representations

4.The convergence of the Campbell-Hausdorff formula

5.Point distributions

6.The bialgebra associated to a formal group

7.The convergence of formal homomorphisms

8.The third theorem of Lie

9.Cartan’s theorems

Exercises

Appendix.Existence theorem for ordinary differential equations

Bibliography

Problem

Index