# Trace decategorification of tensor product algebras

## TL;DR

This paper establishes a connection between the trace of Webster's categorification of tensor products of irreducible quantum group modules and tensor products of Weyl modules for current algebras in ADE type, extending previous results.

## Contribution

It extends prior work by showing the trace of Webster's categorification for tensor products aligns with tensor products of Weyl modules, using a deformation argument.

## Key findings

- Trace of Webster's categorification is isomorphic to tensor products of Weyl modules.
- Extension of previous results from single irreducibles to tensor products.
- Uses deformation techniques based on unfurling 2-representations.

## Abstract

We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra $\dot{U}(\mathfrak{g}[t])$. This extends a result of Beliakova, Habiro, Lauda, and Webster who showed that the trace of the categorified quantum group $\dot{\mathcal{U}}^*(\mathfrak{g})$ is isomorphic to $\dot{U}(\mathfrak{g}[t])$, and the trace of a cyclotomic quotient of $\dot{\mathcal{U}}^*(\mathfrak{g})$, which categorifies a single irreducible for the quantum group, is isomorphic to a Weyl module for $\dot{U}(\mathfrak{g}[t])$. We use a deformation argument based on Webster's technique of unfurling 2-representations.