Poisson代数及其在形变量子化中的应用

Poisson algebra appears from Poisson geometry.With the development of noncommutative geometry, noncommutative Poisson algebra was widely investigated by many mathematicians. Poisson algebras have enough context for the study, such as representation theory,deformation theory and cohomology theory, and they can be applied to study the structures of associative algebras and Poisson geometry. This project is devoted to the study of Poisson algebras, strongly homotopy noncommutative Poisson structures and their application on noncommutative geometry. More precisely, we will study the classification of Poisson structures on some associative algebras, and characterize some conditions of associative algebra with outer Poisson structures; we will discuss the algebraic structures of quasi-Poisson cohomology and FGV-Poisson cohomology; we will investigate the strongly homotopy structures of noncommutative Poisson algebras and their application on cohomology theory and formal deformation theory; we will study the application of FGV-Poisson cohomology in deformation quantization and characterize the homological condition of deformation quantization. The study we carry out in this project will complete the system of noncommutative Poisson algebras, and help us to understand the application of noncommutative Poisson algebra on the geometry.

Poisson代数源于Poisson几何的研究。 随着非交换几何的发展，非交换Poisson代数得到了广泛的关注. Poisson代数不仅本身有着丰富的研究内容，如表示理论、形变理论与上同调理论，同时它在结合代数结构理论和Poisson几何领域也有着重要的应用。本项目拟研究Poisson代数的结构理论，强同伦结构及其在非交换几何中的应用。具体来说，项目拟研究一些结合代数上Poisson结构的分类，刻画具有外Poisson结构的结合代数的条件；研究拟Poisson上同调与FGV-Poisson上同调的代数结构；研究非交换Poisson代数的强同伦结构及其在上同调理论和形变理论中的应用；研究FGV-Poisson上同调在形变量子化的应用，给出形变量子化同调性质的刻画。这些问题的解决将进一步完善Poisson代数的研究体系，有助于将Poisson代数应用到几何问题中.

Poisson代数源于Poisson几何的研究。传统的Poisson代数的结合乘法是交换的. 随着非交换几何的发展，结合乘法不必要交换的Poisson代数得到了广泛的关注. 本项目主要研究非交换Poisson代数上拟Poisson上同调与FGV-Poisson上同调的代数结构，强同伦非交换Poisson结构，以及FGV-Poisson上同调在形变量子化中的应用. 经过本项目的研究，我们得到了一些具体Poisson代数的拟Poisson上同调群的微分分次代数结构；一般Poisson代数上FGV-Poisson复形的微分分次Lie代数结构，并以此定义了强同伦非交换Poisson代数；建立了高阶FGV-Poisson上同调群与Poisson代数高阶形式形变之间的联系；建立了Poisson代数非交换形式形变与形变量子化之间的联系，并利用FGV-Poisson上同调群给出了形变量子化存在的若干条件；对于交换Poisson代数，比较了FGV-Poisson与Lichnerowicz-Poisson上同调，并对于有限生成光滑代数建立二者之间的联系；给出了联系FGV-Poisson上同调群，拟Poisson上同调群与Lie代数上同调群的长正合列定理，并用其计算了几类具体Poisson代数的FGV-Poisson上同调群。这些成果进一步完善了Poisson代数的研究体系，并有助于将Poisson代数应用到非交换几何问题中。