Localizations for quiver Hecke algebras

TL;DR

This paper develops a localization method for monoidal categories related to quiver Hecke algebras, enabling the inversion of certain modules and advancing the categorification of quantum unipotent coordinate algebras.

Contribution

It introduces a localization procedure for monoidal categories using braiders, specifically applied to modules over quiver Hecke algebras, to invert modules corresponding to frozen variables.

Findings

Constructed the localization $ ilde{ ext{C}_w}$ of $ ext{C}_w$ by inverting simple modules.

Proved that the localized category is left rigid.

Expected that the localized category is rigid.

Abstract

We provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, $\mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent coordinate algebra $A_q[\mathfrak{n}(w)]$. We construct the localization $\widetilde{\mathscr{C}_w}$ of $\mathscr{C}_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q[\mathfrak{n}(w)]$. The localization $\widetilde{\mathscr{C}_w}$ is left rigid and we expect that it is rigid.