Categorified Crystal Structure on Localized Quantum Coordinate Rings

TL;DR

This paper constructs a crystal structure on the localized quantum coordinate ring associated with a quiver Hecke algebra, linking it to the categorification of crystals in quantum algebra.

Contribution

It introduces a crystal structure on the localized quantum coordinate ring via self-dual simple objects, extending categorification results to a localized setting.

Findings

Established a crystal structure on the localized quantum coordinate ring.

Proved isomorphism of this crystal with the cellular crystal for any reduced word.

Connected the structure to the categorification of the crystal of the nilpotent half of quantum algebra.

Abstract

For the quiver Hecke algebra $R$ associated with a simple Lie algebra, let $R$-gmod be the category of finite-dimensional graded $R$-modules. It is well-known that it categorifies the unipotent quantum coordinate ring. The localization of $R$-gmod has been defined in [12]. Its Grothendieck ring defines the localized (unipotent) quantum coordinate ring. We shall give a certain crystal structure on the localized quantum coordinate ring by regarding the set of self-dual simple objects in localized $R$-gmod. We also give the isomorphism of crystals to the cellular crystal for an arbitrary reduced word of the longest Weyl group element. This result can be seen as a localized version of the categorification for the crystal of the nilpotent half of quantum algebra by Lauda and Vazirani.