论文标题
无限集群类别的进度与（刺穿）磁盘的三角剖分有关
Progress on Infinite Cluster Categories Related to Triangulations of the (Punctured) Disk
论文作者
论文摘要
在这篇主要的说明性论文中，我们介绍了与磁盘的三角形相关的无限（弱）集群类别的最新进展，有或没有穿刺。首先，我们回想起集群类别的概念。然后，我们转到无限设置，并调查了无限集群类型的最新工作，类型$ \ mathbb {a} $和$ \ mathbb {d} $。我们以我们的贡献结束了，两个无限（弱）类别的无限族类别$ \ mathbb {d} $。我们首先提出了带有明显点的穿刺磁盘的Schiffler组合模型的离散版本。然后，我们从线程震动的表示，获取派生类别，然后采取适当的轨道类别开始，从而产生每个（弱）群集类别。我们表明，（弱）群集类别中的组合学与带有次数标记的点的穿刺磁盘的相应组合物匹配。我们还陈述了有关（弱）集群类别中弱集群结构的两个猜想。
In this mostly expository paper, we present recent progress on infinite (weak) cluster categories that are related to triangulations of the disk, with and without a puncture. First we recall the notion of a cluster category. Then we move to the infinite setting and survey recent work on infinite cluster categories of types $\mathbb{A}$ and $\mathbb{D}$. We conclude with our contributions, two infinite families of infinite (weak) cluster categories of type $\mathbb{D}$. We first present a discrete, infinite version of Schiffler's combinatorial model of the punctured disk with marked points. We then produce each (weak) cluster category starting with representations of thread quivers, taking the derived category, and then taking the appropriate orbit category. We show that the combinatorics in the (weak) cluster categories match with the corresponding combinatorics of the punctured disk with countably-many marked points. We also state two conjectures concerning weak cluster structures inside our (weak) cluster categories.
