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.
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.