Crystals, quiver varieties and coboundary categories for Kac-Moody algebras

TL;DR

This paper provides a geometric interpretation of the crystal commutor for Kac-Moody algebras using quiver varieties, establishing the category of crystals as a coboundary category.

Contribution

It introduces a new geometric perspective on the crystal commutor and confirms its coboundary structure for symmetrizable Kac-Moody algebras.

Findings

Geometric interpretation of the crystal commutor via quiver varieties

Confirmation that the category of crystals forms a coboundary category

Extension of previous finite-dimensional results to Kac-Moody algebras

Abstract

Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley.