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

TL;DR

This paper develops a new categorification framework for tensor product representations of sl(k) using subquotients of category O, establishing equivalences with Webster's tensor categories and extending to superalgebra cases.

Contribution

It constructs tensor product categorifications of sl(k) representations within category O, generalizing prior work and proving their properties using Lie theoretical methods.

Findings

Established tensor product categorifications in category O.

Proved equivalence between category O variants and Webster's tensor categories.

Extended framework to include superalgebra representations.

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.