Affinizations and R-matrices for quiver Hecke algebras

Abstract

We introduce the notion of affinizations and R-matrices for arbitrary quiver Hecke algebras. It is shown that they enjoy similar properties to those for symmetric quiver Hecke algebras. We next define a duality datum $\mathcal{D}$ and construct a tensor functor ${\mathfrak{F}}^{\mathcal{D}}:{\mathrm{Mod}}_{g}r(R^{\mathcal{D}})\to {\mathrm{Mod}}_{g}r(R)$ between graded module categories of quiver Hecke algebras $R$ and $R^{\mathcal{D}}$ arising from $\mathcal{D}$. The functor ${\mathfrak{F}}^{\mathcal{D}}$ sends finite-dimensional modules to finite-dimensional modules, and is exact when $R^{\mathcal{D}}$ is of finite type. It is proved that affinizations of real simple modules and their R-matrices give a duality datum. Moreover, the corresponding duality functor sends a simple module to a simple module or zero when $R^{\mathcal{D}}$ is of finite type. We give several examples of the functors ${\mathfrak{F}}^{\mathcal{D}}$ from the graded module category of the quiver Hecke algebra of type $D_{\ell }$, $C_{\ell }$, $B_{\ell -1}$, $A_{\ell -1}$ to that of type $A_{\ell }$, $A_{\ell }$, $B_{\ell }$, $B_{\ell }$, respectively.