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Part Ⅰ-Lie Algebras 1

Introduction 1

Chapter Ⅰ.Lie Algebras：Definition and Examples 2

Chapter Ⅱ.Filtered Groups and Lie Algebras 6

1.Formulae on commutators 6

2.Filtration on a group 7

3.Integral filtrations of a group 8

4.Filtrations in GL（n） 9

Exercises 10

Chapter Ⅲ.Universal Algebra of a Lie Algebra 11

1.Definition 11

2.Functorial properties 12

3.Symmetric algebra of a module 12

4.Filtration of U？ 13

5.Diagonal map 16

Exercises 17

Chapter Ⅳ.Free Lie Algebras 18

1.Free magmas 18

2.Free algebra on X 18

3.Free Lie algebra on X 19

4.Relation with the free associative algebra on X 20

5.P.Hall families 22

6.Free groups 24

7.The Campbell-Hausdorff formula 26

8.Explicit formula 28

Exercises 29

Chapter Ⅴ.Nilpotent and Solvable Lie Algebras 31

1.Complements on ？-modules 31

2.Nilpotent Lie algebras 32

3.Main theorems 33

3.The group-theoretic analog of Engel’s theorem 35

4.Solvable Lie algebras 35

5.Main theorem 36

5.The group theoretic analog of Lie’s theorem 38

6.Lemmas on endomorphisms 40

7.Cartan’s criterion 42

Exercises 43

Chapter Ⅵ.Semisimple Lie Algebras 44

1.The radical 44

2.Semisimple Lie algebras 44

3.Complete reducibility 45

4.Levi’s theorem 48

5.Complete reducibility continued 50

6.Connection with compact Lie groups over R and C 53

Exercises 54

Chapter Ⅶ.Representations of s？n 56

1.Notations 56

2.Weights and primitive elements 57

3.Irreducible ？-modules 58

4.Determination of the highest weights 59

Exercises 61

Part Ⅱ-Lie Groups 63

Introduction 63

Chapter Ⅰ.Complete Fields 64

Chapter Ⅱ.Analytic Functions 67

“Tournants dangereux” 75

Chapter Ⅲ.Analytic Manifolds 76

1.Charts and atlases 76

2.Definition of analytic manifolds 77

3.Topological properties of manifolds 77

4.Elementary examples of manifolds 78

5.Morphisms 78

6.Products and sums 79

7.Germs of analytic functions 80

8.Tangent and cotangent spaces 81

9.Inverse function theorem 83

10.Immersions，submersions，and subimmersions 83

11.Construction of manifolds：inverse images 87

12.Construction of manifolds：quotients 92

Exercises 95

Appendix 1.A non-regular Hausdorff manifold 96

Appendix 2.Structure of p-adic manifolds 97

Appendix 3.The transfinite p-adic line 101

Chapter Ⅳ.Analytic Groups 102

1.Definition of analytic groups 102

2.Elementary examples of analytic groups 103

3.Group chunks 105

4.Prolongation of subgroup chunks 106

5.Homogeneous spaces and orbits 108

6.Formal groups：definition and elementary examples 111

7.Formal groups：formulae 113

8.Formal groups over a complete valuation ring 116

9.Filtrations on standard groups 117

Exercises 120

Appendix 1.Maximal compact subgroups of GL（n，k） 121

Appendix 2.Some convergence lemmas 122

Appendix 3.Applications of §9：“Filtrations on standard groups” 124

Chapter Ⅴ.Lie Theory 129

1.The Lie algebra of an analytic group chunk 129

2.Elementary examples and properties 130

3.Linear representations 131

4.The convergence of the Campbell-Hausdorff formula 136

5.Point distributions 141

6.The bialgebra associated to a formal group 143

7.The convergence of formal homomorphisms 149

8.The third theorem of Lie 152

9.Cartan’s theorems 155

Exercises 157

Appendix.Existence theorem for ordinary differential equations 158

Bibliography 161

Problem 163

Index 165