Crystal structure of localized quantum unipotent coordinate category

Abstract

A localized quantum unipotent coordinate category associated with a Weyl group element is a rigid monoidal category that is obtained by applying the localization process to the category , a subcategory of the category of finite‐dimensional graded modules of a quiver Hecke algebra . We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in possesses a crystal structure and it is isomorphic to the cellular crystal associated with . As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal.