Leavitt路代数及其应用

Leavitt路代数起源于环的不变基数理论和图C*-代数的研究，是代数学发展的重要组成部分。Leavitt代数研究涉及到代数学的众多分支领域，如环理论、同调代数、C*-代数、代数K-理论、代数表示论和非交换代数几何等。. Leavitt路代数的研究已产生一系列丰富而深刻的结果，而从交换环、同调代数和K-群角度对Leavitt路代数进行研究是一个年轻且富有活力的课题。本项目主要研究：（1）Leavitt路代数的单边理想、局部clean性和自同态环；（2）交换环上的Leavitt路代数；（3）Leavitt路代数的同调性质、导出范畴及K-群；（4）Leavitt路代数的分类问题及其它应用。本项目的研究成果将为Leavitt路代数理论提供新的思路与方法，并将促进Leavitt路代数与图C*-代数、代数表示论和代数K-理论等数学分支的交叉融合。

Leavitt path algebra originated from the research of the Invariant Basis Number property of rings and graph C*-algebra, which is an important part in the development of algebra. Leavitt path algebras connect many branches of algebra, such as ring theory, homological algebra, C*-algebra, algebraic K-theory, representation theory of algebras and noncommutative algebraic geometry.. The research of Leavitt path algebra has produced a series of meaningful and profound results. The study of Leavitt path algebra from the viewpoint of commutative rings, homological algebras and K-groups is a relatively new but viable topic. This project will investigate: (1) one-sided ideals, local cleanness, and endomorphism rings of Leavitt path algebras; (2)Leavitt path algebras over commutative rings; (3)Homological properties, derived categories and K-groups; (4) classification of Leavitt path algebras and other applications. The results of the project will bring new ideas and new methods to Leavitt path algebra, and promote the interaction between Leavitt path algebra and other branches of mathematics, such as graph C*-algebra, representation theory of algebras and algebraic K-theory.