量子代数和张量范畴

Quantum algebras arise from the quantization of algebraic groups, Lie groups and Lie algebras, while tensor categories are categories equipped with associative tensor product functors. The two classes of algebraic objects are closely related via the Tannaka-Krein formality. The present project is devoted to the theory of quantum algebras and tensor categories. We are mainly concerned about the theories of: Gr-categories; Nichols algebras and arithmetic root systems in twisted Yetter-Drinfeld categories; finite quasi-quantum groups; finite pointed tensor categories. The main idea of our study is based on the systematic theory of quantum algebras, the combinatorial methods (PBW bases, root systems, Weyl groups, Dynkin Diagrams, quivers and representations) and the Tannaka-Krein duality of representation theory. The main aim of the project is to obtain some systematic means of construction, some informations of general structures and interesting properties, and some classification results of quantum algebras and tensor categories. The proposed project lies in the intersection of several important research themes as quantum algebras and tensor categories, representation theory of groups and algebras, quantum groups and noncommutative algebraic geometry, etc., and its outcome will provide a quantum type advance for the theory of finite groups and cohomology, algebraic groups and Lie algebras and their representations.

量子代数来源于代数群、李群和李代数的量子化，而张量范畴是具有结合张量积函子的范畴。通过Tannaka-Krein 对偶原理，这两类重要的代数对象密切相关。本项目致力于研究量子代数和张量范畴理论，其主要的研究内容为：Gr-范畴；twisted Yetter-Drinfeld范畴中的Nichols代数和算术根系；有限拟量子群；有限点张量范畴。本项目的关键研究手段为量子代数的系统理论、表示论中的组合方法（PBW基、根系、Weyl群、Dynkin图、箭图及其表示等）以及Tannaka-Krein 对偶理论，目标是得到一些重要类型量子代数和张量范畴的系统构造方法，一些普遍性的结构信息和重要性质，并在部分类型上得到较完整的分类结果。本项目属于有限群及上同调、代数群及其李代数、表示论的量子型发展，是量子代数及张量范畴、群及代数表示论、量子群及非交换代数几何等重要研究领域的交叉课题。

本项目主要研究量子代数和张量范畴理论，其主要的研究内容为：Gr－范畴；twisted  Yetter－Drinfeld范畴中的Nichols代数和算术根系；有限拟量子群；有限点张量范畴。本项目的关键研究手段为量子代数的系统理论、表示论中的组合方法（根系、Weyl群、Dynkin图、 箭图及其表示等）以及Tannaka-Krein 对偶理论，得到了一些重要类型量子代数和张量范畴的系统构造方法，一些普遍性的结构信息和重要性质，并在部分类型上得到较完整的分类结果。特别地，我们发现Gr－范畴中的一些单代数在经典的二次型复合的Hurwtiz问题上有意想不到的应用，证实了一个30多年的猜想，并为该问题提供了崭新的研究方法。