Trace as an alternative decategorification functor

TL;DR

This paper explores an alternative decategorification method using trace or Hochschild--Mitchell homology, revealing new algebraic actions on 2-representations of categorified quantum sl(n).

Contribution

It introduces trace-based decategorification as a novel approach, linking 2-representations to current algebra actions on their centers.

Findings

Trace decategorification induces current algebra actions on centers of 2-representations.

Shows equivalence between trace decategorification and traditional Grothendieck group in certain contexts.

Provides new insights into the structure of categorified quantum groups.

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 expository 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 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.