# Categorical representations and KLR algebras

## TL;DR

This paper establishes a relationship between KLR algebras of cyclic quivers of different lengths, providing geometric insights and implications for categorical representations, with generalizations to broader quiver types.

## Contribution

It proves that KLR algebras of cyclic quivers of length e are subquotients of those of length e+1 and explores their geometric interpretation and categorical implications.

## Key findings

- KLR algebra of cyclic quiver length e is a subquotient of length e+1
- Provides geometric interpretation of the algebraic relationship
- Shows categories with sl_{e+1} actions contain sl_{e} subcategories

## 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 $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. We also give generalizations of these results to more general quivers and Lie types.