Affinizations and R-matrices for quiver Hecke algebras

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 D and construct a tensor functor F^D: Mod_gr(R^D) ? Mod_gr(R) between graded module categories of quiver Hecke algebras R and R^D arising from D. The functor F^D sends finite-dimensional modules to finite-dimensional modules, and is exact when R^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 every simple module to a simple module or zero when R^D is of finite type. We give several examples of the functors F^D from the graded module category of the quiver Hecke algebra of type D, C, B_?1, A_?1 to that of type A, A, B, B, respectively.