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Concept# Weyl's theorem on complete reducibility

Summary

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.

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