1-循环复形的Hall代数与李理论

代数表示论与李理论的交叉研究一直是国际上的热门课题，李理论中的相关对象通过代数表示论中非遗传正合范畴的范畴化在此交叉研究领域中备受关注。设A是一个有限域上的有限维遗传代数，E是由投射A-模的1-循环复形构成的（非遗传）正合范畴。.本项目拟研究E的Hall代数理论与李理论之间的联系。主要内容如下：研究E中Hall多项式的存在性；研究E的退化Hall代数与A的退化Hall代数之间的关系，进一步，研究它们的Hall代数之间的关系；通过E的退化Hall代数定义李代数L，我们试图刻画L的李代数结构，并给出其某种根系的描述；建立李代数L与某种Kac--Moody李代数的正部分以及某种二步幂零李代数之间的联系，同时研究它们的泛包络代数之间的联系；最后，研究李代数L的泛包络代数与E的退化Hall代数之间的关系。

The cross-over study of representation theory of algebras and Lie theory is always an international hot topic, the categorization of certain research objects in Lie theory via non-hereditary exact categories in representation theory of algebras is paid much attention in the field of interdisciplinary research. Let A be a finite dimensional hereditary algebra over a finite field, and let E be the (non-hereditary) exact category of 1-cyclic complexes over projective A-modules..In this project, we will study the connections between the Hall algebra theory of E and Lie theory. The main contents are as follows: we will study the existence of Hall polynomials in E; we will study the relations between degenerate Hall algebra of E and degenerate Hall algebra of A, furthermore, we will study the relations between their Hall algebras; by the degenerate Hall algebra of E, we define a Lie algebra L and try to give a characterization of its Lie algebra structure, moreover, we will also give a description of its root system in some sense; we will establish certain connections of the Lie algebra L to the positive part of some Kac--Moody Lie algebra and some 2-step nilpotent Lie algebra, besides, we will study the relations between their universal enveloping algebras; in the end, we will study the relations between the universal enveloping algebra of L and the degenerate Hall algebra of E.