Poisson Geometry, Contact Geometry and Quantization Problems

泊松几何、接触几何和量化问题

基本信息

批准号：
15204005
负责人：
MAEDA Yoshiaki
金额：
$ 10.73万
依托单位：
KEIO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

This research project aims several problems in Poisson geometry and contact geometry, and quantization problems. Specially, our project focus on the treatments for the quantization problems geometrical point of view. The noncommutative differential geometry is one of our target of our research project. In this project, we had several results on convergent deformation quantization problem. Grebes appears naturally from the construction of the star exponential functions of quadratic forms. Namely, we consider the set of quadratic forms in the complex Weil algebra which forms a Lie algebra isomorphic to sp(n, C). When we consider the esponential functions for these objects, we might expect the complex version of the metaplectic Lie group. Since this is simply connected, we could not handle it. We invested this object by using the explicit computations and have it is in the category of grebes with multiplications. However, it can be described more geometry in terms of the connections. The second problem is to study the invariant deformation quantization problems and construct a convergent star product for ax+b group case. We obtain the universal star product formula. The third result is to study the closed star product. We describe a general settings for obtain how to get the Hochschild cocycle via Stokes formula. This results is still working on and we expect it should be related to the deformation quantization problems for infinite dimensional case. To obtain these results, we have lot of workshops by inviting overseas and domestic researchers together with the research partners. As conclusions, we have fruitful research results which have high evaluation internationally, and also establish the international research network for this area by this grant. This project is still working and will continue for the next project.

该研究项目针对泊松几何和接触几何形状以及量化问题的几个问题。特别是，我们的项目专注于量化问题的处理几何学观点。非交换性差异几何形状是我们研究项目的目标之一。在这个项目中，我们在收敛变形量化问题上有几个结果。 Grebes自然而然地来自二次形式的星指数函数的构建。也就是说，我们考虑了复杂的Weil代数中的一组二次形式，该代数形成了与SP（N，C）同构的谎言代数同构。当我们考虑这些对象的特别函数时，我们可能会期望Metapclect Lie组的复杂版本。由于这是仅连接的，因此我们无法处理。我们通过使用明确的计算来投资此对象，并将其放在具有乘法的Grebes类别中。但是，可以描述更多的几何形状。第二个问题是研究不变的变形量化问题，并为AX+B组病例构建收敛星产品。我们获得通用星级产品公式。第三个结果是研究封闭的恒星产品。我们描述了通过Stokes公式获得如何获得Hochschild Cocycle的一般设置。该结果仍在进行，我们预计它应该与无限尺寸案例的变形量化问题有关。为了获得这些结果，我们通过与研究合作伙伴一起邀请海外和国内研究人员邀请国内研究人员来参加很多研讨会。作为结论，我们的研究成果在国际上具有很高的评估，还通过该赠款建立了该领域的国际研究网络。该项目仍在工作，将继续进行下一个项目。