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

Abstract: Let $U'q(\mathfrak{g})$ 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 $U'q(\mathfrak{g})$-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 ${\mathcal F}{Q}^{(2)}: Rep(R) \rightarrow {\mathcal C}{Q}^{(2)}$, 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 $K(Rep(R)) \simeq K({\mathcal C}_{Q}^{(2)})$.