Current algebras and categorified quantum groups

TL;DR

This paper establishes a connection between the trace of categorified quantum groups of type ADE and their current algebras, linking 2-representations to Weyl modules and exploring category centers.

Contribution

It identifies the trace of categorified quantum groups with current algebras and relates 2-representations to Weyl modules, advancing understanding of categorification in quantum algebra.

Findings

Trace of categorified quantum groups matches current algebra of same type

2-representations correspond to local and global Weyl modules

Implications for centers of categories in 2-representations

Abstract

We identify the trace, or 0th Hochschild homology, of type ADE categorified quantum groups with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.