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The enveloping algebra July 25, 2009

Posted by Akhil Mathew in algebra.
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As we saw in the first post, a representation of a finite group {G} can be thought of simply as a module over a certain ring: the group ring. The analog for Lie algebras is the enveloping algebra. That’s the topic of this post.

Definition

The basic idea is as follows. Just as a representation of a finite group {G} was a group-homomorphism {G \rightarrow Aut(V)} for a vector space, a representation of a Lie algebra {\mathfrak{g}} is a Lie-algebra homomorphism {\mathfrak{g} \rightarrow \mathfrak{g}l(V)}. Now, {\mathfrak{g}l(V)} is the Lie algebra constructed from an associative algebra, {End(V)}—just as {Aut(V)} is the group constructed from {End(V)} taking invertible elements.

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