Face Functors for KLR Algebras

TL;DR

This paper introduces a functorial framework connecting representations of KLR algebras for sub Kac-Moody algebras with cuspidal representations of the original algebra, strengthening the categorical understanding of crystal realizations.

Contribution

It establishes a functorial relationship that provides a firmer categorical foundation for the realization of crystals via cuspidal representations in finite and affine types.

Findings

Constructs a functor between representation categories.

Strengthens the categorical understanding of crystal realizations.

Applies to finite and affine types.

Abstract

Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of cuspidal representations realize crystals for sub Kac-Moody algebras. Here we put that observation an a firmer categorical footing by exhibiting a functor between the category of representations of the KLR algebra for the sub Kac-Moody algebra and the category of cuspidal representations of the original KLR algebra.