2-Calabi-Yau三角范畴里的丛倾斜和倾斜

In this project, we shall use cluster-tilting theory invented as a tool for categorification of cluster algebras to study systematically the representation theoretic properties of the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated categories. Namely, we plan to describe the relation between cluster and classical tilting in 2-Calabi-Yau triangulated categories, to study the relation between exchange relation for tilting modules over the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated category and the exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yau triangulated category, and to investigate the relation between the Auslander-Reiten sequences of the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated category and the Auslander-Reiten triangles in the corresponding 2-Calabi-Yau triangulated category.

本项目拟利用代数表示论范畴化丛代数的丛倾斜理论来系统研究与2-Calabi-Yau三角范畴里的极大rigid对象的自同态代数（包括2-Calabi-Yau倾斜代数）相关的一系列表示论热点问题。具体研究，2-Calabi-Yau三角范畴里丛倾斜与经典倾斜的关系；2-Calabi-Yau三角范畴里的极大rigid对象的自同态代数上倾斜模的交换关系与相应的2-Calabi-Yau三角范畴里的交换三角之间的联系；2-Calabi-Yau三角范畴里的极大rigid对象的自同态代数的模范畴里的Auslander-Reiten序列与2-Calabi-Yau三角范畴里Auslander-Reiten三角的关系。

本项目利用2-Calabi-Yau三角范畴丰富的结果来研究与2-Calabi-Yau三角范畴里的极大rigid对象（maximal rigid object）的自同态代数相关的一系列表示论热点问题，得到的主要结果如下：.1) 证明了自同态代数上的任意一个倾斜模均可提升为2-Calabi-Yau三角范畴里的极大rigid对象；给出了自同态代数上的partial倾斜模的刻画。.2) 给出了自同态代数上的几乎完全倾斜模恰有两个补的充分必要条件。.3) 刻画了自同态代数上的Auslander-Reiten变换、Auslander-Reiten序列与三角范畴上的Auslander-Reiten函子、Auslander-Reiten三角之间的关系。.4) 得到了自同态代数的support tau-倾斜对的集合到2-Calabi-Yau三角范畴上的极大rigid对象的集合之间的一一对应，并且精确地给出了这个对应。.此外，我们从三角范畴的recollement出发，构造出三角范畴的商范畴的recollement，进而构造出加权射影线上的凝聚层范畴、单态射范畴、奇点范畴上的recollement。