2022/08/17 by Nakashima, Toshiki
#16T20 #18D10 #6D90 #81R10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2208.08396
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.