silting理论上的Bongartz引理研究

倾斜理论以及covering理论在有限维代数的表示理论中都具有很重要的作用，silting理论是tilting理论的推广和完善。本项目主要包含两方面研究：silting理论上的Bongartz引理和赋值箭图的covering理论研究。. 关于第一部分，拟研究以下问题：.1. Bongartz引理对于silting对象是否成立？.2. 研究silting复形的自同态代数与原代数A的导出范畴之间的关系。. 关于第二部分，拟研究以下问题：.1. 给出关于赋值箭图的universal covering。.2. 赋值箭图的covering是否能引导它们的表示范畴的AR-箭图之间的covering，以及其导出范畴AR-箭图之间的covering？.3. 运用covering理论工具研究Dynkin型（例如：B_n，C_n）和无穷Dynkin型的cluster范畴。

Tilting theory and covering theory have been playing an important role in the representation theory of finite dimensional algebras, silting theory is a generalization and complement of tilting theory. This project will focus on the study of the Bongartz lemma for silting objects and the covering theory of valued quivers.. For the first part, we mainly consider the following questions:.1. Is the Bongartz lemma also true for silting objects?.2. Study on the relationship between the derived categories of algebra A and the endomorphism algebra of a silting complex of A.. For the second part, we mainly consider the following questions:.1. How to characterize the universal covering for a valued quiver?.2. Given a covering between two valued quivers, will it induce a covering between the AR-quivers of the categories of representations of given valued quivers? What about derived categories of them? .3. Apply the covering technique to study the cluster categories of the Dynkin type (for instance, B_n and C_n) and the infinite Dynkin type.