Lie Algebras and Lie Groups

Cover
Title page
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 Ug
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 g-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 endomorplùsms
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 sl_n
1. Notations
2. Weights and primitive elements
3. Irreducible g-modules
4. Determination of the highest weights
Exercises
Part II - Lie Groups
Introduction
Chapter I. Complete Fields
Chapter II. Analytic Fuctions
"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 funetions
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 homomorplùsms
8. The third theorem of Lie
9. Cartan's theorems
Exercises
Appendix. Existence theorem for ordinary differential equations
Bibliography
Problem
Index