Symmetric quiver Hecke algebras and R-matrices of Quantum affine algebras IV

Abstract

Let $U_q^{\prime }(\mathfrak{g})$ be a twisted affine quantum group of type $A_N^{(2)}$ or $D_N^{(2)}$ and let ${\mathfrak{g}}_0$ be the finite-dimensional simple Lie algebra of type $A_N$ or $D_N$. For a Dynkin quiver of type ${\mathfrak{g}}_0$, we define a full subcategory ${\mathcal{C}}_Q^{(2)}$ of the category of finite-dimensional integrable $U_q^{\prime }(\mathfrak{g})$-modules, a twisted version of the category ${\mathcal{C}}_Q$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${\mathcal{F}}_Q^{(2)}:Rep(R)\to {\mathcal{C}}_Q^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type ${\mathfrak{g}}_0$ with nilpotent actions of the generators $x_k$. We show that ${\mathcal{F}}_Q^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R))\simeq K({\mathcal{C}}_Q^{(2)})$.

30pages

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