Categorification of Tensor products of representations for current algebras and quantum groups

Motivated by the categorified quantum group of Khovanov-Lauda and Rouquier, Webster defined diagrammatic categories whose split Grothendieck groups are isomorphic to tensor products of integrable highest weight modules for the quantum group. Losev and Webster have proposed an axiomatic definition for a tensor product categorification (TPC) of integrable highest weight modules and shown that these TPCs are unique up to a strong form of equivalence.

In this dissertation, we study the categorification of tensor products of different classes of modules. We show that in ADE type, Webster's category can be regarded as a categorification of a tensor product of Weyl modules for the current algebra by considering the trace decategorification functor.

We also establish a new uniqueness theorem for TPCs of modules over sl_Z motivated by work of Brundan, Losev, and Webster. Using this, we lift the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras of Cheng, Lam, and Wang to a graded equivalence between Koszul graded lifts.