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 𝒜q, that is, the Grothendieck ring 𝒦(R-gmod) is isomorphic to 𝒜q. For the localization of R-gmod, denoted by R̃-gmod, its Grothendieck ring 𝒦(R̃-gmod) defines the localized (unipotent) quantum coordinate ring 𝒜̃q. We shall give a certain crystal structure on the localized quantum coordinate ring 𝒜̃q by regarding the set of self-dual simple objects 𝔹(R̃-gmod) in R̃-gmod.
We also give the isomorphism of crystals from 𝔹(R̃-gmod) to the cellular crystal 𝔹i=Bi1⊗ . . . ⊗BiN for an arbitrary reduced word i=i1 . . . iN of the longest Weyl group element. This result can be seen as a localized version for the categorification of the crystal B(∞) by Lauda-Vazirani since the crystal B(∞) is realized as a subset of the cellular crystal 𝔹i.
This video was part of the Southeastern Lie Theory Workshop XIII.
