量子齐次空间上同调的非交换Hodge分解及形变意义

项目名称： 量子齐次空间上同调的非交换Hodge分解及形变意义

项目编号： No.11501492

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 刘立宇

作者单位： 扬州大学

项目金额： 18万元

中文摘要： 本项目拟研究一类非交换代数——量子齐次空间的Hochschild上同调及其非交换Hodge分解，并研究其二阶上同调群的各个Hodge分支的形变意义。人们已经证明了交换代数的Hochschild上同调具有所谓的Hodge分解，由此导出了几类光滑概型的上同调的HKR分解。最近有不少学者致力于非交换Hodge理论。本项目研究量子齐次空间上同调的非交换Hodge分解。量子齐次空间是Hopf代数的右余理想子代数，通常是非交换的。申请人希望通过同调积分、形变等工具计算它们的Hochschild上同调，并引入适当的滤，使得导出的谱序列收敛到相应的上同调群。这样的谱序列即可视为非交换Hodge分解。其中，我们特别关心二阶上同调群的分解，因为二阶上同调类一一对应了形变等价类。申请人希望通过研究，能够类比交换情形，详细解释二阶上同调群各个非交换Hodge分支对应的形变行为，并阐述它们的形变意义。

中文关键词： Hochschild上同调；非交换Hodge分解；Hopf代数；量子齐次空间；形变

英文摘要： The project is devoted to the Hochschild cohomology of a class of non-commutative algebras---quantum homogeneous spaces, as well as the non-commutative Hodge decomposition of their cohomology, also to the deformational significance of each Hodge component in the second cohomology group. It turns out that the Hochschild cohomology of a commutative algebra admits the so-called Hodge decomposition. By virtue of it, people deduced the HKR decomposition for the cohomology of several classes of smooth schemes. Recently, some scholars work on the non-commutative theory. This project is to study non-commutative Hodge decomposition for quantum homogeneous space cohomology. Quantum homogeneous spaces are right coideal subalgebras of Hopf algebras, usually being non-commutative. The applicant hopes to compute their Hochschild cohomology by some tools, such as homological integrals, deformations. He also hopes to introduce an appropriate filtration for every quantum homogeneous space so that the induced spectral sequence converges to the corresponding cohomology groups. Such a spectral sequence can be regarded as non-commutative Hodge decomposition. We are especially concerned about the decomposition of the second Hochschild cohomology group, since there is a one-to-one correspondence between second Hochschild cohomology classes and equivalence classes of deformations. The applicant would like to explain the deformational behavior for every non-commutative component in the second cohomology group in detail, and set forth their deformational significance.

英文关键词： Hochschild cohomology;non-commutative Hodge decomposition;Hopf algebra;quantum homogeneous space;deformation