Categorification of tensor product representations of sl(k) and category O

Abstract

We construct categorifications of tensor products of arbitrary finite-dimensional irreducible representations of ${\mathfrak{sl}}_k$ with subquotient categories of the BGG category $\mathcal{O}$, generalizing previous work of Sussan and Mazorchuk-Stroppel. Using Lie theoretical methods, we prove in detail that they are tensor product categorifications according to the recent definition of Losev and Webster. As an application we deduce an equivalence of categories between certain versions of category $\mathcal{O}$ and Webster's tensor product categories. Finally we indicate how the categorifications of tensor products of the natural representation of $\mathfrak{gl}(1|1)$ fit into this framework.

arXiv commentAdded statement and proof of the exactness of the standardization functor restricted to subquotient categories of O (see Proposition 5.19)

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Cited by in corpus (2)

• Isotypic faithful $2$-representations of $\mathcal{J}$-simple fiat $2$-categories
• Ozsvath-Szabo bordered algebras and subquotients of category O