Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object
Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object
复制标题
广义饰带和修改后的 Caldero-Chapoton 贴图取决于刚性物体
DOI：
10.1215/00277630-2891495
复制
发表时间：
2013
影响因子：
0.8
通讯作者：
Peter Jørgensen
中科院分区：
文献类型：
--
作者：
T. Holm;Peter Jørgensen
Caldero-Chapoton映射是从范畴的对象集合到整数上的Laurent多项式环的映射。在簇范畴的情况下，它将可达的不可分解对象映射到簇代数中相应的簇变量。这形式化了簇范畴是簇代数的范畴化的思想。Caldero-Chapoton映射的定义要求范畴是2-Calabi-Yau，并且映射依赖于范畴中的星团倾斜对象。我们研究的Caldero-Chapoton地图的修改版本，只需要类别有一个塞尔函子，只依赖于一个刚性对象的类别。众所周知，通常的Caldero-Chapoton地图会产生所谓的中楣，例如Conway-Coxeter中楣。我们表明，修改后的Caldero-Chapoton地图产生了我们所谓的广义friezes，并为集群类别Dynkin类型A，它恢复了广义friezes介绍了作者和Bessenrodt在最近的工作组合手段。
Abstract The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.