Artin-Schelter 正则代数(或AS-正则代数)是由 Artin和 Schelter在1987年提出的,并因其在非交换代数几何以及数学物理领域的广泛应用而备受关注. 近来,很多学者推广了AS-正则代数的定义：Etingof 和Ginzburg提出了twisted Calabi-Yau代数的概念, 而Minamoto和Mori则提出了广义AS-正则代数的概念. Poisson 代数是一类重要的交换代数, 它被用来描述Poisson流形上光滑函数全体的代数结构, 是量子力学的一个主要研究对象. 本项目一方面研究广义AS-正则代数的同调理论,推广AS-正则代数的相关结果；另一方面将AS-正则代数与Poisson代数联系起来, 主要考察Poisson代数的同调和上同调之间的对偶,及其形变量子化代数的twisted Calabi-Yau性质. 在一定条件下, 建立了这两者之间的联系.

Artin-Schelter regular algebras(AS-regular for short) were introduced by Artin and Schelter in 1987. Since then, it attracts much attention for its wide range of applications in noncommutative algebraic geometry and mathematical physics. Recently, many researchers generalized this notion. Etingof and Ginzburg introduced the concept of twisted Calabi-Yau algebras, Minamoto and Mori defined generalized AS-regular algebras. Poisson algebras are an important class of commutative algebras. It is used to describe the algebraic structure of smooth functions on Poisson manifolds. It is also a main object in the study of quantum mechanics. In this project，we explore the homology theory for generalized AS-regular algebras. We also relate the twisted Poincare duality between Poisson homologies and cohomologies of some Poisson algebras to the twisted Calabi-Yau property of their deformation quantizations. Under some condition, we explicitly establish a link between these two properties.