路余代数上的双代数结构和Artin代数的有限维数

在Hopf代数理论中，箭图Hopf代数的推广问题一直是一个热点问题。当一个箭图的顶点集有群结构时，对应的路余代数上的Hopf代数结构，或等价的，群上的Hopf双模结构已经研究清楚。作为推广，路余代数上的双代数结构与幺半群及幺半群上的双代数双模有一个一一对应。但关于幺半群上的双代数双模，相关结论却知之甚少。在本项目中，我们利用代数表示论的手段和某些特殊幺半群的结构，将群上的Hopf双模推广到幺半群上的双代数双模中去，从而对路余代数上的双代数结构进行分类。. 在代数表示论中，有一个著名的猜想，即有限维数猜想。有限维数猜想有一个等价形式：设B为Artin代数A的子代数，radB为A的左理想，若A的有限维数有限，则B 的有限维数有限。因此，进一步研究Artin代数的子代数的模范畴及其同调性质，有利于探索和理解有限维数猜想。

In Hopf algebra, the question of extending the Hopf quiver theory to the setting of generalized Hopf structures has always been a hot topic. When the set of vertices of a quiver has a group structure, the Hopf algebra structure on the path coalgebra, or equivalently the Hopf bimodule structures over groups is clear. As generalization, the set of graded bialgebra structure on the path coalgebra is in one-to -one correspondence with the set of monoids and bialgebra bimodules on monoids. To the best of our knowledge, not much is known about the bialgebra bimodules on monoids. In this project, we intend to generalize the classification of Hopf bimodules over groups to that of bialgebra bimodules over monoids.. In the representation theory of algebras, there is a famous finitistic dimension conjecture which says that the finitistic dimension of an arbitrary artin algebra is finite. The finitistic dimension conjecture is equivalent to the following statement: if B is a subalgebra of an Artin algebra A such that radB is a left ideal in A, then B has finite finitistic dimension whenever A has finite finitistic dimension. So a further study on the module categories and homological properties of subalgebrass is useful to investigate and understand the finitistic dimension conjecture.