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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.

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  • $\begingroup$ 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. $\endgroup$
    – hm2020
    Oct 16, 2021 at 12:49

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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.

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  • $\begingroup$ +1. My algebraic geometry is only up to the level of Hartshorne if that, and I haven't studied D-modules. I will look into this, but are there other viewpoints you could name? Just so I know where to look, even if it's not your field of expertise. $\endgroup$
    – JPhy
    Oct 12, 2021 at 20:22
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    $\begingroup$ @JPhy Here is another viewpoint: each primitive ideal (annihilator of a simple module) in the enveloping algebra of a semi-simple Lie algebra is in fact the annihilator of a simple object in category O. So if you care about simple modules only up to their annihilators, you can just classify those in O, which are given by highest weight theory as in e.g. Humphrey's book here: bookstore.ams.org/gsm-94 $\endgroup$
    – Stephen
    Oct 12, 2021 at 20:44

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