Poisson结构诱导的Artin-Schelter正则代数的相关研究

Noncommutative algebra comes from noncommutative algebraic geometry, which is one of important branches in the theory of algebra, and it is the mathematical foundation for characterizing and understanding different noncommutativities. Recent developments in noncommutative algebraic geometry furnish new ideas, methods and impetus for the study of noncommutative algebras. The study of Artin-Schelter regular algebras is the important practice of these new geometry ideas. The proposal concerns mainly the following two topics in the area of noncommutative algebra: (1) Beginning with the Poisson structure on polynomial algebras, we will construct new Artin-Schelter regular algebras. Moreover, we will pay special attention to the ring theory, homological properties and new phenomenon of these regular algebras, and will reveal some deeper properties of these regular algebras such as Koszul-type property. (2) Starting from the Nakayama automorphisms of algebras, we will study the noncommutative invariant theory, such as group actions and Hopf algebra (co)actions on Artin-Schelter regular algebras. Further, we will discuss the Zariski Cancellation Problem for regular algebras with the help of Maker-Limanov invariance theory and Nakayama automorphisms. The results that we expect in this proposal will contribute to the comprehensive and profound understanding of the theory of noncommutative algebra, and will benefit to the study of the other fields in mathematica such as noncommutative algebraic geometry, and some fields of physics.

非交换代数起源于非交换代数几何，是代数学的重要研究分支之一，是理解各种非交换性的数学基础；而非交换代数几何的新进展又为非交换代数的研究提供了新的思想、方法和动力。AS正则代数就是几何思想运用于非交换代数研究的重要实践。本项目的研究内容主要有两个方面：一是从多项式代数上的Poisson结构出发，构造新的AS正则代数，并关注由这种方式产生的正则代数的环论、同调性质及出现的新特性，揭示这些正则代数的Koszul-型性质等一些深层次信息；二是从代数的Nakayama自同构出发，研究正则代数上的群作用和Hopf代数（余）作用等非交换不变量理论，借助Maker-Limanov不变量理论，讨论正则代数的Zariski消去问题。相关成果的取得将有助于更加全面深刻地理解非交换代数理论，进而有助于数学的其它领域（如非交换代数几何等）和物理的一些领域的发展。

本项目主要围绕Poisson代数、由Poisson代数诱导的一些代数结构、代数的Nakayama自同构、代数成为Calabi-Yau代数的条件等方面展开。具体地，详细计算了由Poisson结构诱导的Artin-Schelter正则代数的Nakayama自同构，并给出了一些应用；给出了Poisson代数的泛包络代数成为Calabi-Yau代数的条件；研究了Poisson代数的泛包络代数的双重Poisson-Ore扩张；引入了微分分次Poisson（Hopf）代数，研究了这类代数和其泛包络代数的结构及相关性质。在本项目的资助下，项目组成员已在《Trans. AMS》、《Lett. Math. Phys.》、《Proc. AMS》、《Israel J. Math.》、《中国科学》等国内外杂志上发表学术论文10余篇。