Localizations for quiver Hecke algebras

Abstract

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) over tilde (w) of C-w by adding the inverses of simple modules M(w Lambda(i), Lambda(i)) which correspond to the frozen variables in the quantum cluster algebra A(q)(n(w)). The localization (C) over tilde (w) is left rigid and it is conjectured that (C) over tilde (w) is rigid.