Poisson结构、Hopf作用及其不变量理论

Poisson structures originated from classical mechanics, and are a common interesting topic in the field of algebra and geometry. Some generalized Poisson structures have been widely applied in the study of noncommutative geometry and mathematical physics. In this project, we will apply the FGV-Poisson cohomology and operad theory to investigate the existence of deformation quantization; investigate the representation theory and cohomology theory of restricted Poisson algebras, and apply the restricted Poisson algebra to give some new invariants of Poisson structures over a field of characteristic zero by reduction mod-p and apply it to study the structure and representation of Poisson algebra over a field of characteristic zero; introduce the invariants of fixed subalgebras of Hopf actions on Poisson algebra, and study the algebraic structures of the smash product of a Poisson algebra and a Hopf algebra and Poisson modules of fixed algebras; apply the induced Poisson structure to study the category of MCM modules, the singularity category and the related triangulated categories of the fixed subalgebra of Hopf action on a filtered algebra. These investigations have the direct significance to construct the relation between Poisson structures and noncommutative geometry, and promote the interactive development of the related fields.

Poisson代数源自经典力学的研究，一直都是代数学和几何学共同关心的课题，由此发展出来的各种Poisson结构在非交换几何、数学物理等领域得到广泛应用。本项目主要研究内容包括：运用FGV-Poisson上同调与Operad理论研究形变量子化的存在性；研究限制Poisson代数的表示与上同调理论，用模p约化法给出零特征域上新的Poisson不变量，并应用于零特征域上Poisson代数的结构与表示理论的研究；研究Poisson代数上Hopf作用的不变量以及Poisson代数与Hopf代数smash积的结构，并用于对应不变子代数上Poisson模的研究；利用诱导Poisson结构研究非交换代数在Hopf作用下的不变量，以及不变子代数上极大CM模范畴、奇点范畴及其它相关三角范畴。这些研究对于建立Poisson结构与非交换几何之间的联系有着直接意义，从而推动相关领域的交互发展。

Poisson代数源自经典力学的研究，是代数学和几何学共同关心的课题，由此发展的各种Poisson结构在非交换几何、数学物理等领域有着广泛应用。.本项目主要围绕Poisson代数及其在非交换几何领域中的应用开展研究。利用Operad理论从运算角度给出Poisson结构的形变量子化理论的一种新理解，具体研究2-酉Operad的截面理想，证明了结合代数Operad关于截面理想链的滤子Operad同构于Poisson Operad；计算了若干具体Poisson代数的FGV-Poisson上同调群，从FGV-Poisson上同调和非交换Poisson代数的形式形变理论给出了形变量子化存在的若干条件；引入了限制Poisson代数上限制Poisson模的定义，构造了其对应的限制Poisson包络代数；研究了有限群的群代数Tate-Hochschild上同调环的代数结构；研究了带奇点的非交换Noether代数上非交换拟消解。