Lie algebras and Lie groups : 1964 lectures given at Harvard University
著者
書誌事項
Lie algebras and Lie groups : 1964 lectures given at Harvard University
（Lecture notes in mathematics, 1500）
Springer-Verlag, c1992
2nd ed
- : gw
- : us
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注記
Originally published: New York : W.A. Benjamin, 1965
Corr. 5th printing: 2006
Bibliography: p. [161]-162
Includes index
内容説明・目次
内容説明
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 Algebras: 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 properties: 1). The map A0i A -+ A admits a factorization A (R)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).
目次
Lie Algebras.- Lie Algebras: Definition and Examples.- Filtered Groups and Lie Algebras.- Universal Algebra of a Lie Algebra.- Free Lie Algebras.- Nilpotent and Solvable Lie Algebras.- Semisimple Lie Algebras.- Representations of .- Lie Groups.- Complete Fields.- Analytic Functions.- Analytic Manifolds.- Analytic Groups.- Lie Theory.
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