Cluster structures for 2-Calabi–Yau categories and unipotent groups
Cluster structures for 2-Calabi–Yau categories and unipotent groups
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DOI：
10.1112/s0010437x09003960
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发表时间：
2007-01
影响因子：
1.8
通讯作者：
A. B. Buan;O. Iyama;I. Reiten;J. Scott
中科院分区：
文献类型：
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作者：
A. B. Buan;O. Iyama;I. Reiten;J. Scott
摘要我们研究了三角形2-Calabi-Yau及相关范畴中的倾斜群对象（及其子范畴）。特别是，我们构建了一类新的此类范畴相关的预投射代数的非Dynkin箭图与元素的Coxeter群。这类2-Calabi-Yau范畴作为特例包含Dynkin图的预投射代数的簇范畴和稳定范畴。对于这些2-Calabi-Yau类别，我们构造与每个简化表达式相关联的集群倾斜对象。相关联的双曲正切描述的减少的表达。受簇代数理论的启发，我们提出了（弱）簇结构和子结构的概念，并给出了几个例子。我们讨论了与幂幺群相关的簇代数和子簇代数的连接，在Dynkin和非Dynkin的情况下。
Abstract We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.