Trace as an alternative decategorification functor

Abstract

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called “decategorification” functor. Algebras are typically categorified to additive categories with additional structure, and decategorification is usually given by the (split) Grothendieck group. In this article, we study an alternative decategorification functor given by the trace or the zeroth Hochschild–Mitchell homology. We show that this form of decategorification endows any two representations of the categorified quantum $\mathfrak {sl}_{n}$ with an action of the current algebra $\mathbf {U}(\mathfrak {sl}_{n}[t])$ on its center.