universal enveloping algebra

Definition

Let be a Lie algebra. The universal enveloping algebra is a map where is an associative algebra with unit such that

1. is a Lie algebra homomorphism (taking the commutator Lie bracket on ). That is,

2. If is any associative algebra with unit and is any Lie algebra homomorphism (again with the commutator bracket on ) then there exists a unique homomorphism such that . In other words, the following diagram commutes:

Note, by the last part, if this exists, it is unique up to unique isomorphism (like any universal property).

Existence

Let be a Lie algebra, then it is also a vector space so we can look at the tensor algebra. Let be the (two-sides) ideal generated by the elements . Then

is a universal algebra for .

Proof

Consider the map , this is biliear by definition of the Lie bracket, so by the universal property of tensor product, we get an induced linear map

Then, by construction so by the universal property of quotients we have the unique morphism as seen in the following diagram

Functoriality

If is a Lie algebra homomorphism then the composition

is a Lie algebra homomorphism, so by the definition of this induces a homomorphism

This assignment means that the assignment of a Lie algebra to it’s universal enveloping algebra is a functor.