Leavitt路代数的Grothendieck群与导出等价

Leavitt路代数是由箭图给出的一类具有自然Z-分次结构的结合代数，是Cuntz-Krieger代数的相似版本，现已成为代数表示论和代数K-理论中的重要研究对象。Leavitt路代数的Grothendieck群与代数表示论及非交换几何密切相关，为当前的热门研究课题。在本项目中我们拟研究以下两方面的内容：1.研究Leavitt路代数的分次Grothendieck群结构，运用分次Grothendieck群分类Leavitt路代数；2.研究Leavitt路代数的导出等价和silting对象，并进一步建立Leavitt路代数与silting理论的联系。本项目的展开有助于我们对Leavitt路代数的代数结构以及其分次Grothendieck群的理解。我们将导出范畴作为工具应用到Leavitt路代数的分类问题，有助于我们找到导出等价下的不变量进而来分类Leavitt路代数。

Leavitt path algebras are a class of associative algebras with natural Z-grading given by quivers. They are analogues of Cuntz-Krieger algebras. Now Leavitt path algebras have become important subjects in representation theory of algebras and algebraic K-theory. The Grothendieck groups of Leavitt path algebras are closely related to representation theory and non-commutative geometry, and have recently attracted strong attention. In this project we plan to investigate the following two parts:..1. We study the structures of Grothendieck groups for Leavitt path algebras and expect to classify Leavitt path algebras via graded Grothendieck groups;..2. We study derived equivalences and silting objects for Leavitt path algebras, and further establish connections between Leavitt path algebras and silting theory...This project will help us to understand the algebraic structures and the graded Grothendieck groups of Leavitt path algebras. We will apply derived categories as tools to investigate the classification of Leavitt path algebras. This will help us to find invariants up to derived equivalences for the classification of Leavitt path algebras.