Categorified Crystal Structure on Localized Quantum Coordinate Rings

For the quiver Hecke algebra $R$ associated with a simple Lie algebra, let $R$-gmod be the category of finite-dimensional graded $R$-modules. It is well-known that it categorifies the unipotent quantum coordinate ring. The localization of $R$-gmod has been defined in [12]. Its Grothendieck ring defines the localized (unipotent) quantum coordinate ring. We shall give a certain crystal structure on the localized quantum coordinate ring by regarding the set of self-dual simple objects in localized $R$-gmod. We also give the isomorphism of crystals to the cellular crystal for an arbitrary reduced word of the longest Weyl group element. This result can be seen as a localized version of the categorification for the crystal of the nilpotent half of quantum algebra by Lauda and Vazirani.