Crystals from categorified quantum groups

TL;DR

This paper explores the crystal structure on categories of graded modules over categorified quantum groups, linking it to Kashiwara's crystals and revealing the highest weight crystal structure of simple modules, thus computing Grothendieck group ranks.

Contribution

It identifies the crystal structure on categorified quantum group modules with Kashiwara's crystal, and shows simple modules form highest weight crystals, enabling rank computations.

Findings

Crystal structure matches Kashiwara's crystal for quantum Kac-Moody algebra

Simple graded modules form highest weight crystals

Grothendieck group ranks are computed for cyclotomic quotients

Abstract

We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendieck group.