Artin-Schelter正则代数及相关课题的研究

Noncommutativity is a very common and increasingly important phenomenon in mathematics and other disciplines. The subject of noncommutative algebras is the mathematical foundation for characterizing and understanding noncommutativity. Recent developments in noncommutative algebraic geometry furnish new ideas and methods for the study of noncommutative algebras. The study of Artin-Schelter regular algebras is the important realization of this new ideas. The proposal concerns mainly the following two topics in the area of noncommutative algebra: (1) Artin-Schelter regular algebras, one of goals is to classify the case, in which 4-dim is a continuous work of former project (grant no. 10971188); another goal we interested in is new phenomenon in the case of 5-dim and the method of gaining new Artin-Schelter regular algebras from the Hilbert series. (2) To give a homotopic invariant for connected graded algebras by using A-infinity algebra structures on the EXT-algebra, to reveal some deeper properties of noncommutative algebras such as Koszul-type algebras and Calabi-Yau algebras, and describe their homological theory. More understanding of noncommutativity will benefit to the study of mathematics (especially to noncommutative algebraic geometry) and some field of physics.

非交换性是数学及其它领域中普遍存在且愈显重要的现象，而非交换代数是理解非交换性的数学基础；非交换代数几何中的新发展为非交换代数的研究提供了新的思想和方法；Artin-Schelter正则代数是几何思想运用于非交换代数研究的重要实现。本项目研究内容主要有两个方面：一是研究Artin-Schelter正则代数，其中整体维数为4的Artin-Schelter正则代数的分类问题是前一课题（批准号：10971188）的延续，将进而讨论它们的同调性质，另一关注的目标是5维Artin-Schelter正则代数所出现的新特性，以及从Hilbert序列获取正则代数的方法；二是利用EXT-代数上自然的A无穷结构给出同伦不变量，揭示一些非交换代数（如Koszul型代数，Calabi-Yau代数等）深层次信息，并描述它们的同调理论。对非交换代数理论更深入的理解将有助于数学（尤其是非交换代数几何）和物理的一些领域。

项目主要研究了Artin-Schelter正则代数的分类问题和同调性质，讨论了Artin-Schelter正则代数、Koszul型代数，Hopf代数，以及Calabi-Yau代数之间的联系等。项目总体上按计划进行。我们完成了两个生成元时, 双分次意义下的整体维数5的Artin-Schelter 正则代数的完全分类：一方面我们给出了二十类5维Artin-Schelter正则代数，它们是双分次意义下的完全分类，另一方面也在双分次范围内完全回答了挪威学者Floystad和Vatne的问题；完成了整体维数4的Artin-Schelter正则代数Jordan型代数的完全分类；给出了Nakayama自同构的刻画；获得了齐次PBW形变理论和Artin-Schelter正则性的判别，和非双分次代数的正则性判别定理；发现了分段Koszul代数的A无穷代数对偶定理；给出了有限维Hopf代数块结构方法，为分类问题提供了一条新的途径；Koszul型代数和Calabi-Yau代数的研究方面也获得一些新的成果。项目在同伦不变量方面未获得理想的进展。