Localizations for quiver Hecke algebras III

TL;DR

This paper studies the localization of categories of modules over quiver Hecke algebras, showing how to obtain a localized category with properties like right rigidity, using determinantial modules.

Contribution

It demonstrates that the localization of the module category can be achieved via right braiders from determinantial modules, revealing new structural properties.

Findings

Localization obtained via right braiders from determinantial modules

The localized category exhibits right rigidity

Provides new insights into the structure of quiver Hecke algebra modules

Abstract

Let $R$ be a quiver Hecke algebra, and let $\mathcal{C}_{w,v}$ be the category of finite-dimensional graded $R$-module categorifying a $q$-deformation of the doubly-invariant algebra $^{N'(w)} \mathbb{C}[N] ^{N(v)} $. In this paper, we prove that the localization $\tilde{\mathcal{C}}_{w,v}$ of the category $\mathcal{C}_{w,v}$ can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category $\tilde{\mathcal{C}}_{w,v} $ including the right rigidity.