Calabi-Yau代数的同调理论及相关课题

Calabi-Yau algebras were originated in the research of Calabi-Yau manifolds in algebraic geometry and mathematical physics. Calabi-Yau algebras and Calabi-Yau categories attract lots of attention for their profound background and wide application in noncommutative algebraic geometry, algebra representation theory, conformed field theory, superstring theory. In this project, we mainly investigate the homology theory of Calabi-Yau algebras and some related topics. More explicitly, study the A_∞-algebra structure on the Yoneda Ext algebra of the Calabi-Yau algebra, and the Calabi-Yau property of its stable module category; and calculate the Hochschild homologies and cohomologies of Calabi-Yau algebras, then to characterize the Calabi-Yau property of its PBW deformation; also, we will explore the structure and property of the invariants of the Calabi-Yau algebra under the action of a finite group. The above research, is a new exploration on the homological theory of Calabi-Yau algebras, is also a new attempt on the noncommutative invariant theory and will enrich and develop the theory of noncommutative algebras and geometry.

Calabi-Yau代数源于代数几何和数学物理中对Calabi-Yau流形的研究。Calabi-Yau代数和Calabi-Yau范畴因其在非交换代数及几何、代数表示论、共形场论、超弦理论中的深刻背景和广泛应用备受关注。该项目主要研究Calabi-Yau代数中的同调问题及相关课题。具体而言，研究Calabi-Yau代数的Yoneda Ext代数上的A_∞代数结构，及其分次稳定范畴的Calabi-Yau性质；计算分次Calabi-Yau代数的Hochschild同调与上同调,刻画其PBW形变的Calabi-Yau性质；研究Calabi-Yau代数在有限群作用下的不变子代数的结构与性质。以上研究，是Calabi-Yau代数同调理论的新探索，也是对非交换不变量理论的新尝试，将丰富和发展非交换代数及几何的理论。

Calabi-Yau代数源于代数几何和数学物理中对Calabi-Yau流形的研究。Calabi-Yau代数和Calabi-Yau范畴因其在非交换代数及几何、代数表示论、共形场论、超弦理论中的深刻背景和广泛应用备受关注。该项目主要研究了与Calabi-Yau代数有关的几个问题。具体而言，首先研究了斜Calabi-Yau代数的斜多项式扩张与Koszul对偶。证明了斜Calabi-Yau代数的斜多项式扩张的Yoneda Ext代数同构于原来代数的Yoneda Ext代数的一个斜平凡扩张，进一步刻画了斜Calabi-Yau代数的斜多项式扩张是Calabi-Yau代数的充要条件。这一结果后来被同行推广到非Koszul情形及非分次情形。其次，我们试图站在Poisson代数的形变量子化的角度来理解Calabi-Yau代数或更广一点斜Calabi-Yau代数及其同调。研究了仿射Poisson代数的Poisson(上)同调，得到了在其Poisson上同调和Poisson同调之间存在扭Poincare对偶，并由此得到了Poisson上同调环和其形变代数(对应李代数的包络代数)的Hochschild上同调环之间的同构。这一结果后来被其他学者推广到更一般的Poisson代数上，并进一步讨论Poisson上同调环的代数结构有着应用。我们还研究和讨论了Frobenius Poisson代数的Poisson(上)同调，证明了其存在和光滑情形类似的结果，并进一步证明了在unimodular Frobenius Poisson代数的Poisson上同调环上存在BV代数结构。这个结果也被本人及合作者推广。也将为日后研究和讨论形变量子化提供基础。我们还研究了Artin-Schelter正则代数的doubel Ore 扩张的Nakayama自同构，Poisson张量代数的Poisson(上)同调，Poisson包络代数的Calabi-Yau性质，这些结果尚未接受发表。以上研究，是Calabi-Yau代数同调理论的新尝试，丰富和发展非交换代数及几何的理论。