Calabi-Yau代数的同调和表示与Poisson代数的同调

The purpose of this project is to research the homological properties and representation theory of noncommutative algebras, by using deformation quantization and Poisson order theory. A commutative Poisson algebra can be deformed to a noncommutative associative algebra by deformation quantization, while the homological properties of the two algebras are relevant to each other. The representation theory of the noncommutative algebras equipped with the structure of Poisson orders, are closely related to their Poisson geometric constructions. In this project, we plan to study the Hochschild (co)homology, De Rham cohomology and cyclic homology of some noncommutative algebras, such as graded Calabi-Yau algebras, by deformation quantization theory. We will also enrich the technique in calculation of Poisson (co)homology, then develop the duality theory between Poisson homology and cohomology, and try to generalize the Batalin-Vilkovisky algebra structures over the cohomology rings. Furthermore, we will study the irreducible representations, Azumaya locus and noncommutative discriminant of the Calabi-Yau algebras with structures of Poisson orders, by analyzing their symplectic cores. The subject of this project is a forward subject being widely concerned. This project will bring important impacts on the research of related questions in mathematics and mathematical physics.

形变量子化理论使得Poisson代数与非交换代数的同调性质紧密相连；而另一类具有Poisson order结构的非交换代数的表示理论与其中心上的Poisson几何结构密切相关。本项目计划以形变量子化理论为工具研究一些非交换代数，如分次Calabi-Yau代数的Hochschild（上）同调、De Rham上同调和循环同调；同时丰富Poisson（上）同调的相关结论，发展Poisson代数的同调对偶理论以及Poisson上同调环的Batalin-Vilkovisky代数结构；还将运用Poisson order相关理论，进一步研究Calabi-Yau代数的不可约表示、Azumaya locus以及非交换判别式等问题。这是数学中广受关注的前沿课题，对非交换代数几何、数学物理中相关问题的研究具有重要意义。

本项目以形变量子化理论为工具研究一些非交换代数, 如分次Calabi-Yau代数的同调性质, 证明了形变过程中Brylinski谱序列保持上链/上同调环上的Gerstenhaber代数结构以及Batalin-Vilkovisky结构。同时发展了Poisson代数的同调理论，得到光滑Poisson代数的Poisson上同调环具有Batalin-Vilkovisky代数结构当且仅当这个Poisson结构是pseudo unimodular, 并给出其Batalin-Vilkovisky算子的具体形式。本项目还研究了非交换代数和非交换代数几何领域中关于Poisson代数不可约表示的Dixmier-Moeglin等价性定理。