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The enveloping algebra
*July 25, 2009*

*Posted by Akhil Mathew in algebra.*

Tags: algebra, enveloping algebras, Lie algebras, tensor algebras, universal properties

2 comments

Tags: algebra, enveloping algebras, Lie algebras, tensor algebras, universal properties

2 comments

As we saw in the first post, a representation of a finite group 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 was a group-homomorphism for a vector space, a representation of a Lie algebra is a Lie-algebra homomorphism . Now, is the Lie algebra constructed from an associative algebra, —just as is the group constructed from taking invertible elements.