Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University (Lecture Notes in Mathematics (1500))

The main general theorems on Lie Algebras are covered, roughly the content of Bourbaki's Chapter I. I have added some results on free Lie algebras, which are useful, both for Lie's theory itself (Campbell-Hausdorff formula) and for applications to pro-Jrgroups. of time prevented me from including the more precise theory of Lack semisimple Lie algebras (roots, weights, etc.); but, at least, I have given, as a last Chapter, the typical case ofal,.. This part has been written with the help of F.Raggi and J.Tate. I want to thank them, and also Sue Golan, who did the typing for both parts. Jean-Pierre Serre Harvard, Fall 1964 Chapter I. Lie Definition and Examples Let Ie be a commutativering with unit element, and let A be a k-module, then A is said to be a Ie-algebra if there is given a k-bilinear map A x A~ A (i.e., a k-homomorphism A0" A -+ A). As usual we may define left, right and two-sided ideals and therefore quo tients. Definition 1. A Lie algebra over Ie isan algebrawith the following 1). The map A0i A -+ A admits a factorization A i A -+ A2A -+ A i.e., ifwe denote the imageof(x,y) under this map by [x,y) then the condition becomes for all x e k. [x,x)=0 2). (lx,II], z]+ny, z), x) + ([z,xl, til = 0 (Jacobi's identity) The condition 1) implies [x,1/]=-[1/,x). 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 This Book Reproduces J-p. Serre's 1964 Harvard Lectures. The Aim Is To Introduce The Reader To The Lie Dictionary: Lie Algebras And Lie Groups. Special Features Of The Presentation Are Its Emphasis On Formal Groups (in The Lie Group Part) And The Use Of Analytic Manifolds On P-adic Fields. Some Knowledge Of Algebra And Calculus Is Required Of The Reader, But The Text Is Easily Accessible To Graduate Students, And To Mathematicians At Large. Jean-pierre Serre. Originally Published: New York : W.a. Benjamin, 1965. Includes Bibliographical References. Annotation This book reproduces J-P. Serre's 1964 Harvard lectures. The aim is to introduce the reader to the "Lie dictionary": Lie algebras and Lie groups. Special features of the presentation are its emphasis on formal groups (in the Lie group part) and the use of analytic manifolds on p-adic fields. Some knowledge of algebra and calculus is required of the reader, but the text is easily accessible to graduate students, and to mathematicians at large Reproduces J-P Serre's 1964 Harvard lectures. This book aims to introduce the reader to the "Lie dictionary: Lie algebras and Lie groups". It provides emphasis on formal groups (in the Lie group part) and the use of analytic manifolds on p-adic fields. Intended for graduate students, it assumes some knowledge of algebra and calculus.