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.
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 $\mathbf{P}^1(\mathbf{C})$.
You might see e.g. the note
https://yisun.io/notes/dmod.pdf
for more information and references on this point of view. Given your question, I worry that it may seem too geometric for your tastes, but geometry is perhaps the correct way to organize the overwhelming amount of information needed for the classification problem you pose.