自量子群理论创立以来，Hopf代数和量子群理论已成为当今国际数学与理论物理研究的热点问题之一， Hopf代数的结构、表示以及与之相关的许多代数结构的表示和张量范畴的分类问题是代数界普遍关注的课题. 本课题主要研究非半单pointed Hopf代数的Green环的结构、表示和分类. 课题的重点是研究几类有限维非半单pointed Hopf代数的Green环特别是对所有有限表示型pointed Hopf 代数的Green环的结构、表示及分类进行明确刻画. 拟研究Weyl群和一些有限单群上的Pointed Hopf代数的表示理论及其Green环.拟研究有限表示型Hopf代数的Green环的的几何性质、幂等元的刻画和Green环的自同构群.拟探讨有限张量范畴在Green环、Picard群及Brauer群等不变量框架下的分类问题以及其它一些相关问题.

The study of quantum groups dates back to 1980's. Since then, it has been developing into an important branch of of Mathematics and Physics. The study of structures, representations of quantum groups and (more general) Hopf algebras becomes one of the most active research areas in Algbera. The proposed research mainly focus on the structure, the representation theory and the classification of the Green rings of finite dimensional non-semisimple pointed Hopf algebras. In particular, we shall investigate the Green rings of finite dimensional non-semisimple Hopf algebras of finite representation type. Among the aforementioned Hopf algebras, we shall pay more attension to those with the coradicals being group algebras of Abel groups, of Weyl groups or of finite simple groups。 Moreover, the geometric properties， the idempotent elements and the automorphisms of the Green rings of those Hopf algebras of finite representation type will be investigated. Finally, we shall explore the classification of finite pointed tensor categories using the invariants such as the Green ring, the Picard group, the Brauer group and so on.