Affine highest weight structures on module categories over quiver Hecke algebras

TL;DR

This paper demonstrates that categories of modules over quiver Hecke algebras can be stratified into affine highest weight categories, generalizing previous results and using R-matrices to analyze standard modules.

Contribution

It introduces a new framework for stratifying module categories over quiver Hecke algebras as affine highest weight categories, extending prior work in the field.

Findings

Categories admit numerous stratifications in the sense of Kleshchev.

Full subcategories related to quantum unipotent subgroups are affine highest weight.

Standard modules are realized via determinantial modules and studied using R-matrices.

Abstract

We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to the quantum unipotent subgroup associated with any Weyl group element is an affine highest weight category. Our results significantly generalize earlier works by Kato, Brundan, Kleshchev, McNamara and Muth. The key ingredient is a realization of standard modules via determinantial modules. We utilize the technique of R-matrices to study these standard modules.