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

TL;DR

This paper explores the structure and realizability of infinite rank module categories over $ ext{sl}_2$-modules, revealing combinatorial constraints and classifying subcategories in a Lie-algebraic framework.

Contribution

It characterizes realizable infinite rank module categories over $ ext{sl}_2$, identifies which Coxeter diagrams are realizable, and describes subcategories generated by simple modules.

Findings

Five Coxeter types are realizable, one (type $B_$) is not.

Provides a classification of $ ext{sl}_2$-module subcategories.

Analyzes the action of the monoidal category on subcategories.

Abstract

We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathscr{C}$ 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 $\mathscr{C}$-module subcategories of $\mathfrak{sl}_2$-mod generated by simple modules as well as the $\mathscr{C}$-module categories coming from the natural action of $\mathscr{C}$ on the categories of finite dimensional modules over Lie subalgebras of $\mathfrak{sl}_2$.