Applications and usefulness of universal enveloping algebra

I know the definition of the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, and I know the PBW theorem. My question is the following:

Where does the concept of the universal enveloping algebra get used?

Sure, maybe some people are interested in the structure of those types of algebras. Buy my question is more something like: does the universal enveloping algebra get used in any particularly interesting proofs in Lie theory, or something else? Is it used in classifying some type of object or the representations of some object (say, a Lie group or Lie algebra)?

The universal enveloping algebra of a Lie algebra is often used for representation theory of Lie algebras. To give an example, we have used it to construct faithful representations of minimal degree by certain quotients of the universal enveloping algebra, see here. This helped us to study affine structures on nilpotent Lie groups.

You have received some answers concerning the use of the universal enveloping algebra within Lie theory. I know of some relation it has with algebraic topology. The moral of the following story is that it will be useful for distinguishing between (not all!) topological spaces.

Let $\mathfrak{g}$ be a Lie algebra. The algebraic structure of the universal enveloping algebra $U(\mathfrak{g})$ is not only of an associative unital algebra, but it is a Hopf algebra. That is: there is an structure of coalgebra such that the multiplication and comultiplication in $U(\mathfrak{g})$ are compatible. This is a much richer structure.

In algebraic topology, the algebraic objects that are used usually come with a grading (i.e., we assign a degree to each element) and a differential (i.e., we have a linear map $\partial :L_i \to L_{i-1}$ decreasing or raising degree which obbeys Leibniz's rule with respect to the product and $\partial^2=0$). So that, for example a differential graded Lie algebra can be thought of as a chain complex which is at the same time a Lie algebra (this is the topologist's meaning of a Lie algebra):

$$(L,\partial, [,]) \quad \equiv \quad \cdots \rightarrow L_1 \rightarrow L_0 \rightarrow L_{-1} \rightarrow \cdots$$

Some differential graded algebraic structures are for example the exterior forms in a differential manifold or the singular cochains of a topological space endowed with the cup product and the boundary operator.

Well, given a Hopf algebra $H$, the set of its primitive elements $$P(H)=\{x\in H \mid \Delta x = x\otimes 1+1\otimes x \}$$ has the structure of a Lie algebra (its closed w.r.t the bracket operation). Taking primitives defines a functor $P$ from dg Hopf algebras to dg Lie algebras, and we also have that taking the universal enveloping algebra defines a functor $U$ from dg Lie algebras to dg Hopf algebras in such a way that if we restrict the objects in both categories to be positively graded and the dg Hopf algebras are cocommutative, we have an equivalence of categories given by (this is Milnor-Moore theorem):

$$U:DGL \to DGH \quad \text{ and } \quad P:DGH \to DGL$$

Getting back to the topological context, given a pointed topological space $X$ which is simply connected, we build its loop space $\Omega X$, which is s.t. its homology with rational coefficients $H_*(\Omega X;\mathbb{Q})$ is a Hopf algebra (with trivial differential). As we restrict to rational coefficients, we also have that the homotopy groups $\pi_* (\Omega X)\otimes \mathbb{Q}$ have a dg Lie algebra structure via the Samelson product. (some statements do not need rational coefficients, but for simplicity let us assume that) This might be a lot of machinery, but the main result I wanted to write down is the following:

Theorem (Cartan, Serre): The rational Hurewicz homomorphism induces an isomorphism of dg Lie algebras $$\pi_* (\Omega X)\otimes \mathbb{Q} \cong P(H_*(\Omega X;\mathbb{Q}))$$

Applying the above stated equivalence, we also have (and here we use the universal enveloping algebra, at last!):

Corollary: $H_*(\Omega X;\mathbb{Q}) \cong U(\pi_* (\Omega X)\otimes \mathbb{Q})$

This means that we can do interesting computations in algebraic topology taking the universal enveloping algebra of Lie algebras! And of course, conversely, taking primitive elements. So that if the algebraic structures obtained via this procedure are different, we can conclude that our original spaces are not homotopy equivalent.

As you probably know, the universal enveloping algebra is an associative algebra with unity associated to a Lie algebra. This algebra is generated as "freely" as possible from the commutation relations in the Lie algebra. Besides the PBW theorem, it is introduced because it is a helpful tool in representation theory: representations of the Lie algebra coincide with modules over the universal enveloping algebra.

Unrelated, but possibly interesting: from a homological algebra point of view, the left adjoint to the forgetful functor from the category of Lie algebras to the category of associative algebras is given by the functor sending a Lie algebra to its universal enveloping algebra.

• In your last paragraph, those are both functors from LieAlgebras to AssociativeAlgebras. The universal enveloping algebra functor is left adjoint to the functor from AssociativeAlgebras to LieAlgebras that defines the bracket $[xy]$ as a literal commutator $xy-yx$ in the associative algebra. – Mike Pierce Oct 1 '18 at 23:05