Categorical representations and KLR algebras

Abstract

We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of ${\widetilde{\mathfrak{s}\mathfrak{l}}}_{e+1}$ contains a subcategory with an action of ${\widetilde{\mathfrak{s}\mathfrak{l}}}_e$. We also give generalizations of these results to more general quivers and Lie types.