Actions of sl_2 on algebras appearing in categorification

TL;DR

This paper demonstrates that various algebras in categorification can be endowed with an $rak{sl}_2$ action by derivations, revealing a structured representation theory linked to tensor products of coverma modules.

Contribution

It introduces an $rak{sl}_2$ action on categorification-related algebras, providing a new perspective on their structure and representation theory.

Findings

Algebras in categorification admit $rak{sl}_2$ actions by derivations.

The $rak{sl}_2$ representations are filtered by tensor products of coverma modules.

Future work will explore the implications of this structure for categorification.

Abstract

We prove that many of the recently-constructed algebras and categories which appear in categorification can be equipped with an action of $\mathfrak{sl}_2$ by derivations. The $\mathfrak{sl}_2$ representations which appear are filtered by tensor products of coverma modules. In a future paper, we will address the implications of the $\mathfrak{sl}_2$ structure for categorification.