Toshiki Nakashima: Categorified crystal structure on quantum coordinate rings and cellular crystals

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 𝒜_q, that is, the Grothendieck ring 𝒦(R-gmod) is isomorphic to 𝒜_q. For the localization of R-gmod, denoted by R̃-gmod, its Grothendieck ring 𝒦(R̃-gmod) defines the localized (unipotent) quantum coordinate ring 𝒜̃_q. We shall give a certain crystal structure on the localized quantum coordinate ring 𝒜̃_q 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 𝔹_i=B_i1⊗ . . . ⊗B_iN for an arbitrary reduced word i=i₁ . . . i_N 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 𝔹_i.

This video was part of the Southeastern Lie Theory Workshop XIII.