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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 [...] and [...]. 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 ®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).
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 [...] and [...]. 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 ®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).
Inhaltsverzeichnis
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.
Details
Fachbereich: | Arithmetik & Algebra |
---|---|
Genre: | Mathematik |
Rubrik: | Naturwissenschaften & Technik |
Medium: | Taschenbuch |
ISBN-13: | 9783540550082 |
ISBN-10: | 3540550089 |
Sprache: | Englisch |
Herstellernummer: | 978-3-540-55008-2 |
Autor: | Serre, Jean-Pierre |
Auflage: | 2nd ed. Corr. pr. |
Hersteller: |
Springer
Springer, Berlin Springer Berlin Heidelberg |
Abbildungen: | VII, 173 p. |
Maße: | 241 x 168 x 11 mm |
Von/Mit: | Jean-Pierre Serre |
Gewicht: | 0,328 kg |
Inhaltsverzeichnis
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.
Details
Fachbereich: | Arithmetik & Algebra |
---|---|
Genre: | Mathematik |
Rubrik: | Naturwissenschaften & Technik |
Medium: | Taschenbuch |
ISBN-13: | 9783540550082 |
ISBN-10: | 3540550089 |
Sprache: | Englisch |
Herstellernummer: | 978-3-540-55008-2 |
Autor: | Serre, Jean-Pierre |
Auflage: | 2nd ed. Corr. pr. |
Hersteller: |
Springer
Springer, Berlin Springer Berlin Heidelberg |
Abbildungen: | VII, 173 p. |
Maße: | 241 x 168 x 11 mm |
Von/Mit: | Jean-Pierre Serre |
Gewicht: | 0,328 kg |
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