非交换代数和几何整体技巧

With the aid of integral techniques of commutative projective algebraic geometry, the study of noncommutative algebras has made a series of profound progress. Among which, Artin-Schelter regular algebras and Calabi-Yau algebras are two important research objects. The former arose from the study of noncommutative projective spaces, which can be viewed as a quantization of the polynomial algebras; the latter came from the study of homological mirror symmetry on a Calabi-Yau manifold. It is also natural to these two objects from a physical point of view. The main research objects of this project will be these two algebras, and based on the former results and the questions, the study will be done with emphasis on using geometry integral techniques. The project is divided into two aspects: one is to classify 4 and 5 dimensional Artin-Schelter regular algebras by using the method mentioned above, to study the moduli spaces of their point modules and line modules, and find out some geometry invariants; the other one is by the properties of the fixed subalgebras that a finite group acting on a Calabi-Yau algebra, to find out concern conditions and study the structures of the category of singularities and the primitive spectrum. There are close relations between these two objects due to Artin-Schelter regular algebras are twisted Calabi-Yau algebras.

借助于交换射影代数几何的整体技巧，非交换代数的研究取得了一系列深刻的进展。Artin-Schelter正则代数和Calabi-Yau代数是其中两类重要的研究对象。前者源于非交换射影空间的研究，可看作多项式代数的量子化；后者源于对Calabi-Yau流形上同调镜像对称的研究。从物理的角度看这两类非交换代数也是十分自然的。本项目主要研究对象是这两类代数，根据之前工作的进展和发现的问题，将侧重运用几何整体技巧来研究。项目主要分两个方面：一是运用几何整体技巧分类整体维数为4和5的Artin-Schelter正则代数，研究相应的点模和线模的模空间，寻求其中一些几何不变量；二是依据Calabi-Yau代数在有限群作用下的不变子代数的性质，寻求相关的条件，并研究相应的奇点范畴和本原谱的结构。由于Artin-Schelter正则代数是扭曲的Calabi-Yau代数，这两类对象之间存在密切的联系。

项目主要研究了Artin-Schelter正则代数和Calabi-Yau代数两类对象的结构、同调性质和不变量问题，以及非交换代数的相关问题. 项目总体上按计划进行, 获得的主要成果如下：通过运用组合的语言和方法建立了连通分次Hopf代数的结构定理，从理论和结果两方面看都是Hopf代数研究领域的一个创新和重要进展；给出了消去问题的Morita等价和导出等价意义下的理论，将经典的判别式不变量发展成为非交换代数判别式理论,对深入认识非交换代数具有重要的意义；建立了非交换Auslander定理，证明了关于Auslander条件在不同分次下的关系，并由此获得一些重要的同调不变量；证明了斜Calabi-Yau性质在正规扩张下的保持性，得到了描述斜Calabi-Yau分次代数的Nakayama自同构的同调恒等式；通过对Hopf稠密Galois扩张的观察，研究了非交换分次的孤立奇异性；讨论了Poisson-Ore扩张问题等.在Artin-Schelter正则代数的不变量Nakayama自同构的刻画，以及斜Calabi-Yau性质在正规扩张下的保持性问题方面都实现了项目预期的目标.