Concept

Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way. Given a finite-dimensional Lie algebra representation , let be the associative subalgebra of the endomorphism algebra of V generated by . The ring A is called the enveloping algebra of . If is semisimple, then A is semisimple. (Proof: Since A is a finite-dimensional algebra, it is an Artinian ring; in particular, the Jacobson radical J is nilpotent. If V is simple, then implies that . In general, J kills each simple submodule of V; in particular, J kills V and so J is zero.) Conversely, if A is semisimple, then V is a semisimple A-module; i.e., semisimple as a -module. (Note that a module over a semisimple ring is semisimple since a module is a quotient of a free module and "semisimple" is preserved under the free and quotient constructions.) Here is a typical application. Proof: First we prove the special case of (i) and (ii) when is the inclusion; i.e., is a subalgebra of . Let be the Jordan decomposition of the endomorphism , where are semisimple and nilpotent endomorphisms in . Now, also has the Jordan decomposition, which can be shown (see Jordan–Chevalley decomposition#Lie algebras) to respect the above Jordan decomposition; i.e., are the semisimple and nilpotent parts of . Since are polynomials in then, we see . Thus, they are derivations of . Since is semisimple, we can find elements in such that and similarly for . Now, let A be the enveloping algebra of ; i.e., the subalgebra of the endomorphism algebra of V generated by . As noted above, A has zero Jacobson radical.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.