Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras III

TL;DR

This paper constructs duality functors linking categories of modules over quiver Hecke algebras and quantum affine algebras, revealing relationships between different types of quantum affine modules.

Contribution

It introduces generalized quantum affine Schur-Weyl duality functors connecting categories of graded modules over quiver Hecke algebras to categories of finite-dimensional modules over quantum affine algebras of types A_{N-1}^{(1)} and A_{N-1}^{(2)}.

Findings

Established duality functors between categories

Connected module categories of different quantum affine types

Provided new tools for studying quantum affine algebra representations

Abstract

Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. In this paper, we investigate the relationship between the categories $\CC^0_{A_{N-1}^{(1)}}$ and $\CC^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur-Weyl duality functors $\F^{(t)}$ from $R^{A_\infty}\gmod$ to $\CC^0_{A_{N-1}^{(t)}}$ $(t=1,2)$.