A diagrammatic approach to reflection functors

TL;DR

This paper develops a diagrammatic method to construct reflection functors for quiver Hecke algebras related to Kac-Moody algebras, categorifying Lusztig's braid group action and proving their braid relation properties.

Contribution

It introduces a new diagrammatic construction of reflection functors for quiver Hecke algebras from a higher representation-theoretic perspective, extending prior work.

Findings

Constructed reflection functors for arbitrary symmetrizable Kac-Moody algebras.

Provided a categorification of Lusztig's braid group action.

Proved that the functors satisfy braid relations as natural isomorphisms.

Abstract

We construct reflection functors for quiver Hecke algebras associated with arbitrary symmetrizable Kac-Moody algebras, from a higher representation-theoretic viewpoint. These functors provide a categorification of Lusztig's braid group action on the quantum group. Similar functors were recently constructed independently by Kashiwara-Kim-Oh-Park via a different approach. Moreover, we prove that our reflection functors satisfy the braid relations as natural isomorphisms.