Infinite rank module categories over finite dimensional ${\mathfrak{sl}}_2$-modules in Lie-algebraic context

Abstract

We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathcal{C}$ of finite dimensional modules for the complex Lie algebra ${\mathfrak{sl}}_2$. Combinatorics of such realizations is governed by six infinite Coxeter diagrams. We show that five of these are realizable in our setup, while one (type $B_{\infty }$) is not. We also describe the $\mathcal{C}$-module subcategories of ${\mathfrak{sl}}_2$-mod generated by simple modules as well as the $\mathcal{C}$-module categories coming from the natural action of $\mathcal{C}$ on the categories of finite dimensional modules over Lie subalgebras of ${\mathfrak{sl}}_2$.