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.

So, in general, it follows that what we’re really looking at here is maps for an associative algebra , such that is a Lie-algebra homomorphism, i.e.

Call such a function an **-map.** Note that makes any -module into a representation of , so these are interesting objects to consider.

The universal enveloping algebra has the **universal** -map. In detail, if is fixed, we can consider the category of pairs , where is an associative algebra and an -map. A morphism between and is an algebra-homomorphism such that .

Definition 1In this category, theuniversal enveloping algebrais the initial object: thus for any -map , we have a unique algebra-homomorphism making the usual diagram commutative.

By the uniqueness of universal objects, the enveloping algebra is unique if it exists.

Proposition 2To give a -representation is the same as to give a -module. Alternatively, the categories of -representations and -modules are equivalent.

*Proof:* Indeed, a -module is a vector space with an algebra homomorphism . The set of such algebra-homomorphisms is functorially isomorphic to the set of -mappings , or -representations.

Of course, we have to still show exists.

** Tensor Algebras **

Given a vector space , we can form its tensor powers for all (say if we are working over the complex numbers).

We can thus form the graded vector space

which we can make into a noncommutative **ring**. Indeed, we have a bilinear map

and we can extend by linearity to the direct sum to get a graded ring. Basically, the structure is described simply: if , form a basis for , then multiplication sends

The ordering of the ‘s does matter, which explains the noncommutativity; if we demand that the ‘s do commute, we end up with a polynomial algebra.

Anyway, the importance of the tensor algebra is the universal property:

Proposition 3Given a linear map for a ring , there is a unique algebra-homomorphism such that the map is just .

*Proof:* The proof follows by sending to ; this gives a ring-homomorphism . The rest is straightforward to check.

** The Enveloping Algebra **

It is often useful to quotient out the tensor algebra by some relations to still get a useful object. For instance, if one abelianizes, one gets the *symmetric algebra*; if one quotients out by the ideal generated by , one gets the *exterior algebra*.

In our case, we have a Lie algebra , and I claim that

Proposition 4is isomorphic to where is the two-sided ideal generated by

*Proof:* We can check this by looking at universal properties. So, first, there’s a map (this is true for any vector space) and a quotient map . Thus there’s a map . Now by the relations defining . From this it follows that is an -map.

Conversely, if is an -map, then there is a unique algebra-homomorphism . By the definition of -maps, it must factor through , so it follows that is universal.

So, next we should probably look at some of the properties of the enveloping algebra. For instance, it’s Noetherian. It also has a nice basis by the PBW theorem.

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