${\mathfrak{s}\mathfrak{l}}_2$-Harish-Chandra modules for ${\mathfrak{s}\mathfrak{l}}_2⋉L(4)$

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