Hopf代数的分类、本原上同调及相关问题

非交换性是数学及其它学科中常见和日益重要现象，Hopf代数(量子群)是典型且重要的非交换代数，Hopf代数的分类是非交换代数的一个重要课题和热点问题. 非交换代数几何与非交换代数的最新进展为Hopf代数的研究提供了新思想和方法. 随着对非交换性更多细节的进一步揭示，对无限维Hopf代数结构的进一步研究是非常必要的。本项目研究的主要内容为：Gelfand-Kirillov维数有限的无限维Hopf代数的分类和结构 及相关代数性质; Hopf代数的本原上同调和相关问题;拟三角Hopf代数和有限张量范畴的分类和结构。本项目的关键研究手段为非交换代数和同调代数方法、表示论中的组合方法和Tannaka-Krein对偶理论。目标是得到某些重要类型Hopf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。本项目的研究具有重要的理论意义和应用价值。

Noncommutativity is a very common and increasingly important phenomenon in mathematics and other disciplines，Hopf algebras (quantum groups) is a typical and important noncommutative algebra, the classification of Hopf algebras is a very important project and hot issue of noncommutative algebra. Recent developments in noncommutative algebraic geometry and noncommutative algebra furnish new ideas and methods for the study of Hopf algebras. As more details of noncommutativity are to be revealed, understanding the structure of infinite dimensional Hopf algebras is an essential part of research. The mian content of this project is to study the classification and structure of infinite dimensional Hopf algebras with finite Gelfand-Kirillov dimension and related algebraic properties, primitive cohomology of Hopf algebras and related topics, the classification and structure of quasi-triangular Hopf algebras and finite tensor categories. The main idea of our study is based on noncommutative algebras and homological algebras methods,the combinatorial methods and Tannaka-Krein duality of representation theory. The main aim of the project is to obtain some systematic means of construction, some general informations of structure and interesting properties, and some classification results of Hopf algebras and tensor categories. The study of the project will have important theoretical significance and application value.