量子齐次空间上同调的非交换Hodge分解及形变意义

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.

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。

Hochschild上同调理论和形变理论是非交换几何的重要成分，二者之间又有极为密切的联系。本项目以量子齐次空间为主要研究对象，考察其上同调的非交换Hodge分解以及相应的形变意义。具体地，我们首先利用Gerstenhaber-Schack复形等工具定义了代数扭预层的扭形变，证明了Gerstenhaber-Schack复形的二阶上同调类一一对应于扭形变等价类，特别地，二阶上同调的三个Hodge分支分别对应了扭形变的三个组成部分，即局部乘法、限制映射、扭元素。接下来定义了代数扭预层上的拟凝聚模范畴，证明了在几何条件下该范畴是一个Grothendieck阿贝尔范畴；着眼于代数几何，我们发现很多扭预层具有所谓的中心扭条件，因此，当代数扭预层满足中心扭条件时，又定义了扭拟凝聚预层范畴，进一步证明了这个范畴与拟凝聚模范畴是阿贝尔等价的。我们还计算了射影超曲面的Hochschild上同调，根据超曲面的次数和射影空间的维数的不同大小关系，给出了任意阶上同调群的表达式；刻画了超曲面的二阶上同调群的典范分解，描述了该分解与经典的HKR型分解的关联；对于局部完备交证明了HKR定理的逆命题。最后，我们研究了二元多项式环上广义Weyl代数的同调光滑性，通过它的定义多项式和它的两个偏导数给出了广义Weyl代数具有同调光滑性的充分必要条件，进一步利用定义自同构的雅可比行列式给出了广义Weyl代数是Calabi-Yau代数的充要条件；将这一结论应用于几个具体的量子群上，证明了相应的量子齐次空间都具有同调光滑性和斜Calabi-Yau性质。

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