分次Calabi-Yau代数的有限群作用及其不变子代数

Calabi-Yau(简记为CY)代数源于对CY流形上同调镜像对称的研究，而后在非交换代数几何以及代数表示理论中得到了广泛应用，目前是非交换代数领域中重要研究对象之一。本项目主要研究连通分次CY代数的有限群作用及其不变子代数，依据不变子代数的正则性与否，分三个方面开展研究：Gorenstein不变子代数的Cohen-Macaulay模范畴以及非交换Auslander定理；正则不变子代数以及斜群代数的PBW形变代数的本原理想的结构；整体维数较低的CY代数及其不变子代数的PBW形变代数的自同构群。主要目标是在CY代数上给出非交换Auslander定理成立的等价条件、进而寻求适当的代数来刻画Cohen-Macaulay模范畴，并计算低维CY代数的形变代数的自同构群及探求斜群代数的形变代数的本原理想的结构，为研究非交换轨道空间的Gorenstein奇点及其非交换形变的结构与表示提供理论依据。

Calabi-Yau algebra firstly appeared at the study of the homological mirror symmetry on a Calabi-Yau manifold. It seeks its extensive applications in noncommutative algebraic geometry and algebra representation theory. Nowadays, it is one of the major research objects in noncommutative algebra. We mainly study the finite group actions on connected graded Calabi-Yau algebras and their invariant subalgebras. According to the regularity of the invariant subalgebras, the project is divided into three topics: the category of Cohen-Macaulay modules over Gorenstein invariant subalgebras and noncommutative Auslander Theorem, the structure of the primitive ideals of the PBW deformations of regular invariant subalgebras and skew group algebras, the automorphism groups of the PBW deformations of Calabi-Yau algebras with lower global dimensions and their invariant subalgebras. The aim of the project is to give equivalent conditions for the Auslander Theorem, then try to find certain algebras to represent the category of Cohen-Macaulay modules, and compute the automorphism groups and study the structures of the primitive ideals of the PBW deformations of CY algebras with lower dimensions. The research of the project will provide theoretical supports on the study of the Gorenstein singularities over noncommutative orbit spaces and the structures and representations of their noncommutative deformations.