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。

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.