# Modules of quotients of enveloping algebras

I'd like to ask for any resource recommendations whether research literature or pedagogic, on working out the representation theory (i.e. finding all simple modules) for quotients of universal enveloping algebras of Lie algebras. Specifically, quotienting by a non-principal ideal.

There was a book called Methods of Representation Theory by Curtis and Reiner which I considered just based on the title, that seems to use quiver machinery which seems widely applicable. However, before embarking on that journey, I wanted to check if there was some specialised toolkit for quotients of enveloping algebras, since my interest is very specific.

• If $ρ:A→B:=A/I$ is a surjective map of associative unital rings and if $V$ is a left $B$-module and $f:V≅V_1⊕V_2$ an isomorphism of left $B$-modules, it follows $f$ is an isomorphism of left $A$-modules. Hence to classify irreducible $U(\mathfrak{g})/I$-modules is equivalent to classifying irreducible $U(\mathfrak{g})$-modules $V$ with $I⊆ann(V)$. If $V$ is finite dimensional, these modules have been classified if $\mathfrak{g}$ is semi simple. There is a class of infinite dimensional modules (Verma modules) that have been classified. Commented Oct 16, 2021 at 12:49

Finding all simple modules for an arbitrary quotient of the enveloping algebra of a Lie algebra is a hopelessly difficult question. For instance, for the Lie algebra $$\mathfrak{sl}_2$$, it boils down to classifying all irreducible $$D$$-modules on $$\mathbf{P}^1(\mathbf{C})$$.