$\mathfrak{sl}_2$-Harish-Chandra modules for $\mathfrak{sl}_2 \ltimes L(4)$

TL;DR

This paper classifies a specific category of modules over a Lie algebra extension involving sl_2 and a 4-dimensional module, using functor techniques, and discusses challenges in generalizing to higher dimensions.

Contribution

It determines the quiver and relations for the category of sl_2  L(4)-modules, introducing novel functor applications and highlighting obstacles for broader cases.

Findings

Explicit quiver and relations for sl_2  L(4)-modules

Use of analogues of Enright's and Arkhipov's functors

Identification of obstacles in extending results to higher k

Abstract

We use analogues of Enright's and Arkhipov's functors to determine the quiver and relations for a category of $\mathfrak{sl}_2 \ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $\mathfrak{sl}_2$. We also outline serious obstacles to extend our result to $\mathfrak{sl}_2 \ltimes L(k)$, for $k>4$.