Localizations for quiver Hecke algebras II

Abstract

AbstractWe prove that the localization of the monoidal category is rigid, and the category admits a localization via a real commuting family of central objects. For a quiver Hecke algebra and an element in the Weyl group, the subcategory of the category of finite‐dimensional graded ‐modules categorifies the quantum unipotent coordinate ring . In the previous paper, we constructed a monoidal category such that it contains and the objects corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category and . Together with the already known left‐rigidity of , it follows that the monoidal category is rigid. If in the Bruhat order, there is a subcategory of that categorifies the doubly‐invariant algebra . We prove that the family of simple ‐module forms a real commuting family of graded central objects in the category so that there is a localization of in which are invertible. Since the localization of the algebra by the family of the isomorphism classes of is isomorphic to the coordinate ring of the open Richardson variety associated with and , the localization categorifies the coordinate ring .