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.
