Categorification of tensor product representations of slkslk and category OO

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