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

Abstract

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody algebra of type $A_n^{\left(1\right)}$ ( $n\ge 1$) or $D_n^{\left(1\right)}$ ( $n\ge 4$), and let ${\mathfrak{g}}_0$ be the underlying finite-dimensional simple Lie subalgebra of $\mathfrak{g}$. For each Dynkin quiver $Q$ of type ${\mathfrak{g}}_0$, Hernandez and Leclerc introduced a tensor subcategory ${\mathcal{C}}_Q$ of the category of finite-dimensional integrable $U{\text{'}}_q\left(\mathfrak{g}\right)$-modules and proved that the Grothendieck ring of ${\mathcal{C}}_Q$ is isomorphic to $\mathbb{C}\left[N\right]$, the coordinate ring of the unipotent group $N$ associated with ${\mathfrak{g}}_0$. We apply the generalized quantum affine Schur–Weyl duality to construct an exact functor $\mathcal{F}$ from the category of finite-dimensional graded $R$-modules to the category ${\mathcal{C}}_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to ${\mathfrak{g}}_0$. We prove that the homomorphism induced by the functor $\mathcal{F}$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\mathcal{F}$ sends the simple modules to the simple modules.