论文标题
局部量子坐标环上的分类晶体结构
Categorified Crystal Structure on Localized Quantum Coordinate Rings
论文作者
论文摘要
对于与简单的Lie代数相关的Quiver Hecke代数$ r $，让$ r $ -gmod是有限维度分级$ r $ r $ modules的类别。众所周知，它分类了一级量子坐标环。 $ r $ -gmod的本地化已在[12]中定义。它的Grothendieck环定义了局部（单位）量子坐标环。我们将通过局部$ r $ -GMOD中的一组自偶联对象，在局部量子坐标环上提供一定的晶体结构。我们还将晶体的同构为细胞晶体，以使最长的Weyl组元素的任意简化单词。该结果可以看作是Lauda和Vazirani的Nilpotent Farmpotent晶体晶体的局部版本。
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.
