Hopf代数的上同调、形变和量子齐次空间

本项目以上同调为主线，基于形变观点，研究Hopf代数及其量子齐次空间的同调性质。在一定条件下，量子包络代数和有限维Hopf代数的Hochschild上同调具有BV代数结构，美中不足的是，这些BV代数结构的显式表达大多数是未知的。本项目拟通过构造合适的投射分解来计算这两类代数以及前者的量子齐次空间的Hochschild上同调，进而给出BV代数结构。另外，Hopf代数的2-上闭链形变是研究余模范畴等价和构造Calabi-Yau代数的重要手段，本项目将计算与此相关的lazy上同调，并研究其形变特性以及与Calabi-Yau代数的联系。Nakayama自同构是代数的一种很精妙的不变量，正日益受到代数学家关注，但是它的计算具有很大难度。本项目将计算量子坐标代数及其量子齐次空间的Nakayama自同构，从而更好地理解Nakayama自同构在形变中所扮演的角色。

Based on the deformation point of view, with cohomology as the main line, this project is devoted to the homological properties of Hopf algebras and their quantum homogeneous spaces. Under certain conditions, the Hochschild cohomology of a quantized enveloping algebra or a finite dimensional Hopf algebra has a BV algebra structure. However, the explicit expressions of these BV algebra structures are mostly unknown. This project intends to compute the Hochschild cohomology of the two kinds of algebras and of the quantum homogeneous spaces of the former by constructing suitable projective resolutions, and then to give the BV algebra structures. In addition, 2-cocycle deformations of Hopf algebras are important approaches to study the equivalence of categories of comodules and to construct Calabi-Yau algebras. In this project we will calculate lazy cohomology related to 2-cocycles, and study its deformation properties as well as its relation to Calabi-Yau algebras. Nakayama automorphisms are subtle invariants of algebras, which are gaining more and more attention from algebraists, but their calculation is very difficult. We will calculate Nakayama automorphisms of quantized coordinate algebras and their quantum homogeneous spaces to better understand the role of Nakayama automorphisms in deformation theory.