不具有对偶Chevalley性质的有限维Hopf代数的分类问题

One of the central questions in Hopf algebra theory is to classify finite-dimensional ones. The research in this direction is very rich. Most of the classification results consist of Hopf algebras that are basic or have the dual Chevalley property (that is, its coradical is a subalgebra) . But there are very few results on finite-dimensional Hopf algebras without the dual Chevalley property in the literature, which is a very difficult problem in the classification. We propose, by means of the generalized lifting method, to study the construction and classification of finite-dimensional Hopf algebras without the dual Chevalley property. It will involve the representation of non-semisimple Hopf algebras, the classification of (non-diagonal) Nichols algebras, and the deformation of their bosonization.

有限维Hopf代数的分类是Hopf代数结构理论研究中的一个中心课题，研究成果非常丰富。已有的分类结果几乎都是由具有对偶Chevalley性质（即余根基是子代数）的或基本的Hopf代数构成的。而不具有对偶Chevalley性质的有限维Hopf代数的分类是一个困难的问题，相关的研究还很少。本项目拟应用广义提升范式的思想，研究不具有对偶Chevalley性质的有限维Hopf代数的构造和分类问题。 研究过程将涉及到非半单Hopf代数的表示、（非对角型）Nichols代数的分类以及它们的玻色化的形变等问题。