Implicit structure in 2-representations of quantum groups

TL;DR

This paper demonstrates how to extend strong 2-representations of Kac-Moody Lie algebras to categorified quantum groups by verifying additional 2-relations, revealing implicit structures in Rouquier's framework.

Contribution

It provides a method to upgrade 2-representations from Rouquier's to Khovanov-Lauda's setting by explicitly checking implicit relations.

Findings

Explicit 2-relations are necessary for extension.

Rouquier's definitions implicitly contain these relations.

Applications of the extended 2-representations are discussed.

Abstract

Given a strong 2-representation of a Kac-Moody Lie algebra (in the sense of Rouquier) we show how to extend it to a 2-representation of categorified quantum groups (in the sense of Khovanov-Lauda). This involves checking certain extra 2-relations which are explicit in the definition by Khovanov-Lauda and, as it turns out, implicit in Rouquier's definition. Some applications are also discussed.