Hall代数理论是联系代数表示论和李理论的重要桥梁。本项目的研究对象是Bridgeland定义的复形范畴的Hall代数和To?n引入的导出Hall代数。我们将使用箭图表示和同调代数的结论和方法，尤其关注与李理论相关联的结构和性质。首先，我们将研究并比较复形范畴的Hall代数和导出范畴的导出Hall代数，然后将Hall代数构造扩展到n-周期复形范畴，与奇周期三角范畴的导出Hall代数作比较，并且试图用后者给出李超代数的实现。其次，我们希望将Bridgeland的工作推广到2-周期(A,c)-复形范畴上，由于后者给出了箭图簇的同调实现，我们试图用Hall代数方法研究Kac-Moody代数的包络代数及其高权表示的实现模型。再次，我们将研究由Hernandez和Leclerc给出的量子Grothendieck环与导出Hall代数的同构，试图建立辫子群作用的对应。

The theory of Hall algebras is an important bridge between representation theory of algebras and Lie theory. The main objects of our research are the Hall algebras of complexes in the sense of Bridgeland and the derived Hall algebras introduced by To?n. We will use the results and methods from the representation theory of quivers and homological algebra freely, while paying special attention to the structures and properties associated with Lie theory. Firstly, we will study and compare the Hall algebras of complexes and derived Hall algebras of derived categories, then extend the construction of Hall algebras to the category of n-periodic complexes，which can also be compared to the derived Hall algebras for odd periodic triangulated categories. Maybe the latter algebras can give a realization of Lie super algebras. Secondly, we hope to extend the work of Bridgeland to the 2-periodic (A,c)-complexes, which were used to give a homological realization of quiver varieties. We will try to realize the enveloping algebras of Kac-Moody algebras and their highest weight representations. Thirdly， we will study the isomorphism between quantum Grothendieck rings and derived Hall algebras given by Hernandez and Leclerc, and try to establish a correspondence of braid group actions.