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

TL;DR

This paper constructs a functor linking symmetric quiver Hecke algebras to categories of quantum affine algebra modules, confirming a correspondence of simple modules and connecting algebraic structures.

Contribution

It introduces an exact functor from graded R-modules to Hernandez-Leclerc categories, extending quantum affine Schur-Weyl duality and confirming module correspondence.

Findings

The functor coincides with Hernandez-Leclerc's homomorphism.

Simple modules are preserved under the functor.

Establishes a link between quiver Hecke algebras and quantum affine modules.

Abstract

Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cite{KKK13} to construct an exact functor $\F$ from the category of finite-dimensional graded $R$-modules to the category $\CC_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to $\g_0$. We prove that the homomorphism induced by the functor $\F$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\F$ sends the simple modules to the simple modules.