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}}\colon \mathrm {Mod}_gr(R^{\mathcal D}) \to \mathrm {Mod}_gr(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.