Hall代数建立了代数表示论与李理论的深刻联系，而Nakajima的箭图簇从几何的角度深化了这些联系。本项目主要研究1-周期复形的Hall代数和由循环箭图簇构造的结合代数。我们将使用同调代数、箭图表示以及反常层的方法进行研究，关注与李理论联系紧密的结构和性质。1）在1-周期复形的幂零Hall李代数的工作基础上，我们希望刻画D、E型Hall李代数的生成关系，并进一步将工作推广到仿射型。我们还将研究1-周期复形的（退化）Hall代数的结构，建立它们与Hall李代数的关系，实现对它们的生成元和生成关系的刻画。2）我们将在两个层面构造结合代数。其一是循环箭图簇上的反常层在卷积乘法下生成的代数，我们希望将覃的结果推广到到n-周期复形。其二是投射模复形的仿射簇上的可构函数生成的代数，我们希望将肖-徐-张的结果推广到1-周期复形。3）我们希望比较由代数方向得到的Hall代数和由几何方向得到的结合代数。

Hall algebras established deep connections between representation theory of algebras and Lie theory. Then the quiver varieties introduced by Nakajima greatly strengthened the connections in the geometric way. The main objects of this project are the Hall algebras of 1-cyclic complexes and the associative algebras constructed from cyclic quiver varieties. We will use the methods from homological algebra, representation of quivers and the theory of perverse sheaves, and pay special attentions to the structures and properties connected with Lie theory..Firstly, based on our former work on nilpotent Hall Lie algebras associated to 1-cyclic complexes, we will determine the generating relations for type D and E, then try to study the affine case. At the same time, we will investigate the structures of degenerate Hall algebras and Hall algebras associated to 1-cyclic complexes and establish their relations with Hall Lie algebras. Finally, we hope to characterize their generators and generating relations..Secondly, we want to construct associative algebras on two stages. The first one is generated by perverse sheaves on cyclic quiver varieties defined by Nakajima under convolution multiplications. We hope to generalize Qin’s results to n-cyclic complexes. The second one is generated by constructible functions on affine varieties of complexes of projective modules. We want to generalize Xiao-Xu-Zhang’s results to 1-cyclic complexes..Lastly, we will compare the Hall algebras from the algebraic side and the associative algebras from the geometric side.