Applications of the PBW theorem on enveloping algebras

Question

What are some nice corollaries or applications of the Poincaré Birkhoff Witt theorem? There's this immediate corollary that a Lie algebra sits inside the universal enveloping algebra so in particular, the Lie algebra structure comes from an associative algebra. Why else is the theorem interesting? Are there some nice papers or expository notes for this?

Answer 1

One immediate corollary is that you get to know the "sizes" of induced representations. If $\mathfrak{h}\subset\mathfrak{g}$ is a Lie subalgebra, and $M$ is an $\mathfrak{h}$-module, then the underlying vector space of the $\mathfrak{g}$-module induced from $M$ is $S(\mathfrak{g}/\mathfrak{h})\otimes M$.

Of course, it also depends what version of the PBW theorem you have in mind. The $\mathfrak{g}$-module isomorphism between $U(\mathfrak{g})$ and $S(\mathfrak{g})$ tells you, for instance, that the centre of the universal enveloping algebra is identified, as a vector space, with the adjoint invariants in the symmetric algebra, leading to deep results like the Duflo theorem. The isomorphism of coalgebras between $U(\mathfrak{g})$ and $S(\mathfrak{g})$ is an insight into geometry (making one think of distributions on the Lie group supported at the unit element). And the list can go on :)

Answer 2

The PBW theorem leads to the construction of a monomial basis for the universal enveloping algebra UEA, revealing thus its structure and giving rise to UEA techniques in the represenation theory of Lie algebras: It is one of the main tools for investigating the structure of highest weight modules, their weight space decomposition, their $H$-diagonalizability (where $H$ is the Cartan subalgebra) etc.
(i think this is related to the first paragraph of Vladimir Dotsenko's answer).

On the other hand, it motivates (or at least facilitates) the construction of Verma modules and the investigation of their structure and properties.

Apart from representations, PBW can also provide insight on the ideas (and can be also used to provide a proof) behind fundamental structure theorems of Lie algebras like Ado's theorem.