Hopf代数形变下的同调性质及相关问题的研究

Calabi-Yau algebras originated from the study of the homological mirror symmetry on a Calabi-Yau manifold. In recent years, they have attracted lots of attention due to their applications in mathematical physics, representation theory etc. In this project we mainly study Calabi-Yau Hopf algebras. Our reserch is divided into two parts. First, cocycle deformations are important in the Hopf algebra theory. We plan to study when cocycle deformations preserve global dimensions or the Calabi-Yau property. Second, we plan to study universal quantum general linear groups. They are Hopf algebras coacts on algebras universally. We plan to give a description of the representations (finite dimensional comodules) and the Calabi-Yau property of the universal quantum general linear group of an N-Koszul AS regular algebra and their special Hopf quotients, to see when a Hopf algebra coacts on an algebra A will have similar homological properties as that of A. Since a cocycle deformation of a Hopf algebra only deforms its algebra structure, the research of this project will help us to analyse of the function of the coalgebra structure in the homologial properties of Hopf algebras. Moreover, by the study of the Hopf algebras coact on Calabi-Yau algebras, we will have a better understanding of Calabi-Yau algebras.

Calabi-Yau（简称为CY）代数源于对CY流形上同调镜像对称的研究。近年来，由于它在数学物理、代数表示理论等领域的应用而备受人们关注。本项目主要研究对象是CY Hopf代数，研究内容分两方面：一、cocycle形变是Hopf代数理论中重要的形变，拟刻画cocycle形变保持整体维数、CY等同调性质的条件。二、拟研究对余作用具备泛性质的泛量子一般线性群，刻画N-Koszul AS正则代数的泛量子一般线性群及其特殊Hopf商代数的表示（有限维余模）及CY等同调性质，判断余作用于代数A的Hopf代数是否会具备和A相似的同调性质。本项目的研究一方面由于cocycle形变改变Hopf代数的代数结构而保持余代数结构不变，将有助于分析余代数结构在Hopf代数同调性质中所起的作用；另一方面通过对余作用于CY代数的Hopf代数的研究，将进一步加深对CY代数的理解。

Calabi-Yau（简称为CY）代数源于对CY流形上同调镜像对称的研究。近年来，由于它在数学物理、代数表示理论等领域的应用而备受人们关注。本项目主要研究Hopf代数形变下的Calabi-Yau性质等同调性质以及相关的问题。得到的主要研究成果有：一、证明了扭CY Hopf代数的分次扭曲仍然是扭CY Hopf代数，并计算了它的Nakayama自同构；二、证明了一类余表示半环和GL(2)余表示半环同构的Hopf代数G(A,B)是扭CY代数，并具体计算了它们的Nakayama自同构；三、证明了特征为0的域上的Gelfand-Kirillov维数有限的连通分次Hopf代数都是域的迭代Hopf Ore扩张，也是它的任意分次Hopf子代数的迭代Hopf Ore扩张。本项目的研究丰富了Hopf代数形变保持扭CY性质的例子，有助于研究余代数结构在Hopf代数同调性质中所起的作用。另一方面对连通分次Hopf代数结构的研究推广改进了已有的对连通分次Hopf代数的分类，使得人们对连通分次Hopf代数的结构更加清楚。