Bridgeland-Hall代数与李代数的实现

The categorization of infinite dimensional Lie algebras has been paid much attention in the study of Lie theory and algebraic representation theory. It is closely related to many mathematical areas, such as quantum group, algebraic geometry and cluster category. As the first intrinsic global model of the quantum group, the Bridgeland-Hall algebra reveals more profound relationship between the representations of quivers and Lie theory. .In this project, we will study an intrinsic realization of infinite dimensional Lie algebras via Bridgeland-Hall algebras. We mainly focus on a realization of affine Kac-Moody algebras via Bridgeland-Hall algebras, then desribe Bridgeland-Hall Lie algebras in case of quivers with relations, and establish a geometric version of affine Bridgeland-Hall Lie algebras..We have given a realization of the semisimple Lie algebra by using Auslander-Reiten theory and Galois covering theory. Based on this, combined with other methods, we expect to study intrinsic symmetries in Bridgeland-Hall algebras, and will get some new results in these directions, which can provide a new perspective and a tool for the further study of infinite dimensional Lie algebras and their representations.

无限维李代数的范畴化在李理论和代数表示论交叉研究中备受关注。它与量子群、代数几何、丛范畴等数学领域紧密相关。Bridgeland-Hall代数作为量子群首个内蕴的实现模型，揭示了箭图表示范畴与李理论的更深刻联系。.本项目拟研究无限维李代数的内蕴的Bridgeland-Hall代数实现，主要探讨仿射Kac-Moody代数的Bridgeland-Hall代数实现；刻画带关系箭图的Bridgeland-Hall李代数的结构；给出仿射Bridgeland-Hall李代数的几何化。.申请人与合作者利用Auslander-Reiten箭图及Galois覆盖理论等方法，已解决了半单李代数的Bridgeland-Hall代数实现问题。在此基础上结合其他方法我们希望深入挖掘Bridgeland-Hall代数内蕴的对称性，将在相关方向取得新成果，对进一步研究无限维李代数的结构及表示提供新的视野和工具。

Bridgeland-Hall代数作为量子群的一个内蕴的实现，与李理论有深刻联系。本项目主要探讨了有限整体维数代数的Bridgeland-Hall代数结构。构造了特殊生成元并给出乘法公式，刻画了Hall代数为Bridgeland-Hall代数的子代数的等价条件。我们探讨了仿射箭图表示簇的典范分解，确定了典范分解下表示数量的渐进关系。我们还研究了循环箭图的double Hall代数的新表现，建立和Bridgeland-Hall代数的联系；并构造了Robinson-Schensted-Knuth超对应，证明了对应的超对称性。我们研究了仿射Kac-Moody代数的Bridgeland-Hall代数实现，确定了仿射Kac-Moody代数虚根对应的Bridgeland-Hall代数对象的乘法公式。