Crystal structure of localized quantum unipotent coordinate category

Abstract

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