Actions of ${\mathfrak{sl}}_2$ on algebras appearing in 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.