Localizations for quiver hecke algebras

In this paper, we provide a generalization of the localization procedure for monoidal categories developed in [12] by Kang-Kashiwara-Kim by introducing the notions of braiders and a real commuting family of braiders. Let R be a quiver Hecke algebra of arbitrary symmetrizable type and R-gmod the category of finite-dimensional graded R-modules. For an element w of the Weyl group, C_w is the subcategory of R-gmod which categorifies the quantum unipotent coordinate algebra A_q (n(w)). We construct the localization C^˜_w of C_w by adding the inverses of simple modules M(wΛ_i, Λ_i ) which correspond to the frozen variables in the quantum cluster algebra A_q (n(w)). The localization C^˜_w is left rigid and it is conjectured that C^˜_w is rigid.