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

TL;DR

This paper constructs a functor linking modules over certain Khovanov-Lauda-Rouquier algebras to modules over quantum affine algebras, revealing new dualities and categorical structures, especially in the affine Schur-Weyl case.

Contribution

It introduces a functor from graded modules over R^J algebras to quantum affine algebra modules, establishing new dualities and categorical frameworks, particularly for affine Schur-Weyl duality.

Findings

The functor F maps convolution products to tensor products.

F is exact for finite type A,D,E R^J algebras.

The Grothendieck rings of categories C_J and T_J are isomorphic at q=1.

Abstract

Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from the category $S_J$ of finite-dimensional graded $R^J$-modules to the category of finite-dimensional integrable $U_q(g)$-modules. The functor $F$ sends convolution products of $R^J$-modules to tensor products of $U_q(g)$-modules. It is exact if $R^J$ is of finite type A,D,E. When $J$ is the vector representation of $A^{(1)}_{n-1}$, we recover the affine Schur-Weyl duality. Focusing on this case, we obtain an abelian rigid graded tensor category $T_J$ by localizing the category $S_J$. The functor $F$ factors through $T_J$. Moreover, the Grothendieck ring of the category $C_J$, the image of $F$, is isomorphic to the Grothendieck ring of $T_J$ at $q=1$.