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Infinite rank module categories over finite dimensional $\mathfrak{sl}_2$-modules in Lie-algebraic context
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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 $\mathfrak{sl}_2$.
Forward citations
Cited by 2 Pith papers
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Combinatorics of monoidal actions in Lie-algebraic context
For any semisimple Lie algebra, the action on categories generated by a simple module with generic central character is semi-simple, simple transitive, and described by weight multiplicities.
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Combinatorics of infinite rank module categories over finite dimensional $\mathfrak{sl}_3$-modules in Lie-algebraic context
The combinatorial shadow of any sl3-generated transitive module category is one of eight infinite graphs.
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