预投射代数及其相关问题

The mian purpose of this project is to study homology properties and representations of (deformed) preprojective algebras and it's Hopf quotient algebras, to construct quantum group by Hopf quotient algebras of deformed preprojective algebras. We begin with the study of representations of (deformed) preprojective algebras and it's quotient algebras by representations of quivers. On the other hand, we consider the Hopf algebra structure of the quotient algebras of deformed preprojective algebras, construct quantum groups by the quotient algebras of defoemed preprojective algebras, and give some indecomposable representations of restridted quantum groupos by representations of quivers. Furthermore, by calculating the Hochschild homology, cohomology and cohomology ring of quadratic dual of two classes of preprojective algebras, study the homology properties of preprojective algebras.

本项目主要目的是研究（形变）预投射代数及其Hopf商代数的同调性质和表示，利用形变预投射代数的Hopf商代数实现量子群。首先，利用（形变）预投射代数对应的箭图的表示来构造（形变）预投射代数及其商代数的表示。其次，考虑形变预投射代数的商代数的Hopf代数结构，利用这些商代数来实现限制型量子群，进而利用箭图的表示给出限制型量子群某些不可分解表示的刻画。此外，我们通过计算两类预投射代数的二次对偶代数的Hochschild同调和上同调群，刻画上同调环的结构，来研究预投射代数的同调性质。

本项目主要研究了（形变）预投射代数的同调性质和表示范畴的结构，利用形变预投射代数的商代数来构造的Hopf代数、量子群。我们计算了两类预投射代数的二次对偶代数的Hochschild同调和上同调群，刻画了上同调环的结构。我们研究了预投射代数的模范畴中的一些特殊对象和一些特殊函子。在此基础上，对于一些特殊的限制型量子群的表示进行了初步的研究。