一簇拟量子群的表示及相关问题

拟Hopf代数（也称拟量子群）起源于Drinfeld对Knizhnik-Zamolodchikov方程单值性的研究，是研究有限整张量范畴的基础，与物理及数学的许多领域密切相关。本项目拟从表示论的角度，研究小拟量子群与小量子群的关系，包括小量子群的Green环、小拟量子群的表示和Green环、以及这两类Green环的联系。研究具有仿射中心的素Hopf代数的Azumaya奇点，即Taft代数和2维非Abel限制李代数的限制包络代数的联系与共性，包括后一个Hopf代数的表示和Green环，以及这两类Hopf代数Green环之间的联系。研究半单群代数的Hopf-Ore扩张的表示和Green环、2维非Abel李代数的包络代数的广义Hopf-Ore扩张的不可约表示。这些研究有助于人们理解拟Hopf代数与相应Hopf代数的内在联系，分类出更多的拟Hopf代数，可以从更高的视角整体理解Azumaya奇点。

Quasi-Hopf algebra (or quasi-quantum group) was invented by Drinfeld in relation with his treatment of the monodromy of the Knizhnik-Zamolodchikov equations. Quasi-Hopf algebras are fundamental in the study of finite integral tensor categories, and closely related with physics and many fields of mathematics. In this project, based on the representation theory point of view, we study the relation between small quasi-quantum groups and small quantum groups. We focus on the Green rings of small quantum groups, the representations and Green rings of quasi-quantum groups, and the relations between the two classes of Green rings. We study the relations and common properties of Azumaya singularities of prime Hopf algebras with affine centers. These Azumaya singularities are reduced to the two classes of Hopf algebras: Taft algebras and the restricted enveloping algebra of the 2-dimensional non-abelian restricted Lie algebra. We focus on the representations and Green ring of the later algebra, and the relations between the Green rings of the two classes of Hopf algebras. We also study the representations and Green rings of the Hopf-Ore extensions of semisimple group algebras, and the irreducible representations of the generalized Hopf-Ore extensions of the universal enveloping algebra of the 2-dimensional non-abelian Lie algebra. These studies will be helpful for understanding the relations between the Hopf algebras with the quasi-Hopf algebras constructed from them, and for classifying more quasi-Hopf algebras. These studies can help us understand the Azumaya singularities of prime Hopf algebras as a whole from a higher point of view.