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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中的几个问题以及量化问题。特别是，我们的项目专注于从几何角度处理量化问题。非交换微分几何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取得了一些成果。鸊鷉自然地从二次形式的星指数函数的构造中出现。也就是说，我们考虑复韦尔代数中的二次形式集合，它形成与 sp(n, C) 同构的李代数。当我们考虑这些对象的响应函数时，我们可能会期望元波李群的复杂版本。由于这是简单的连接，我们无法处理它。我们通过显式计算对该物体进行了投资，并将其归入具有乘法的鷊鷉类别中。然而，可以根据连接来描述更多的几何形状。第二个问题是研究不变变形量化问题并构造ax+b组情况的收敛星积。我们得到了通用明星产品配方。第三个成果是研究封闭式明星产品。我们描述了如何通过 Stokes 公式获得 Hochschild 余循环的一般设置。该结果仍在研究中，我们预计它应该与无限维情况下的变形量化问题有关。为了获得这些成果，我们邀请国内外研究人员与研究合作伙伴一起举办了很多研讨会。结论是，我们取得了丰硕的研究成果，在国际上得到了高度评价，并通过此次资助建立了该领域的国际研究网络。该项目仍在进行中，并将继续进行下一个项目。