Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types

TL;DR

This paper establishes a broad algebraic equivalence between module categories over quiver Hecke algebras and certain categories of quantum affine algebra modules, extending previous geometric results to all types.

Contribution

It proves a general algebraic equivalence between categories of modules over quiver Hecke algebras and Hernandez-Leclerc's categories, beyond untwisted ADE types.

Findings

Proves algebraic equivalence in all types.

Extends previous geometric results.

Provides a uniform algebraic proof.

Abstract

We prove in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez-Leclerc's category $\mathcal{C}_Q$. This was previously proved in untwisted $ADE$ types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to general types.