A CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS
A CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS
复制标题
DOI：
10.1017/nmj.2023.6
复制
发表时间：
2017-02
影响因子：
0.8
通讯作者：
Matthew Pressland
中科院分区：
文献类型：
--
作者：
Matthew Pressland
在之前的工作中，作者介绍了一种从内部Calabi-Yau代数的数据出发构造具有冻结变量的聚类代数的Frobenius分类的方法，该分类成为聚类倾斜对象的自同态代数。本文构造了具有极化主系数的聚类代数的合适的内部Calabi-Yau代数（与那些有更多冻结变量的主系数的聚类代数不同），并在无环情况下得到了Frobenius分类。通过部分镇定，我们定义了在Nakaoka和Palu意义上的外三角化范畴，对一般不存在Frobenius范畴的无环主系数聚类代数进行了分类。正如我们将指出的那样，许多用于获得这些分类的中间结果在没有非周期性假设的情况下仍然有效，并且它们本身就很有趣。最值得注意的是，我们提供了Van den Bergh结果的Frobenius版本，即具有势的颤振的Ginzburg g-代数为双模$3$ -Calabi-Yau。
Abstract In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$ -Calabi–Yau.