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Quantization of Poisson manifolds and noncommutative geometry

泊松流形的量化和非交换几何

基本信息

批准号：
11640198
负责人：
NATSUME Toshikazu
金额：
$ 2.3万
依托单位：
Nagoya Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640198/
关键词：
Poisson manifold deformation quantization C^*-algebra strict deformation quantization Strict quantization

项目摘要

In a joint project with R.Nest of the University of Copenhagen and I.Peter of the University of Munster the pricipal investigator showed that under a topological condition every closed symplectic manifold has a strict quantization. Strict quantization is an analytic deformation theory. An algebraic deformation theory (existence of deformation quantization) has been known since 80's.The aim of the project is to show existence of strict quantizations for Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The project is divided into three steps. The first step is to re-examine the existence proof of strict quantization for symplectic manifolds, in order to have a deep understanding of mechanism of existence. In particular, re-examination of the proof by B.Fedosov, which played a crucial role in our proof, of existence of deformation qu … More antization is an important step. The second step is to understand the existence proof of deformation quantization for Poisson manifolds and to rewrite it from the viewpoint of Fedosov. The last step involves actual construction of strict quantization.Through quite a few discussions with Nest, the mechanism of existence became fairly clear, and we obtained a refined version of our result. Thanks to a recent appearance of a simpler proof of existence of deformation quantization for Poisson manifolds than Kontsevich's, we have a prospect to achieve the second step.While working on the project discussed above, in a joint project with C.L.Olsen of the State University of New York at Buffalo, the principal investigator worked on the cases that are not covered by the results with Nest and Peter. In particular, we showed that the 2-sphere with a specific Poisson structure has a strict quantization. In the process to construct strict quantization we obtained new "noncommutative 2-spheres". These C^*-algebras are new examples of noncommutative Poisson manifolds.As explained above, unfortunately we could not achieve the goal of the project, i.e. the existence of strict quantizations for poisson manifolds. We certainly intend to continue working on the project. We will hopefully complete the project within a year or so. Less

在与哥本哈根大学的r.s nest和明斯特大学的i.p oter的联合项目中，首席研究员证明了在拓扑条件下，每个封闭辛流形都有严格的量子化。严格量子化是一种解析变形理论。一种代数变形理论（变形量子化的存在性）在80年代被提出。本课题的目的是证明推广辛流形的泊松流形的严格量化存在性。泊松流形变形量化的存在性是一个长期以来的重要问题，最终由M.Kontsevich在1997年证明。该项目分为三个步骤。第一步是重新审视辛流形严格量子化的存在性证明，以便对存在性机制有更深的理解。特别是对B.Fedosov关于变形曲存在性的证明的重新检验，这一证明在我们的证明中起到了至关重要的作用。第二步是理解泊松流形变形量子化的存在性证明，并从费多索夫的观点对其进行改写。最后一步涉及严格量化的实际构造。通过与Nest的多次讨论，存在的机制变得相当清晰，我们得到了一个精炼版的结果。由于最近出现了一个比Kontsevich的更简单的泊松流形变形量化存在性的证明，我们有希望实现第二步。在与纽约州立大学布法罗分校的C.L.Olsen合作进行上述项目时，首席研究员与Nest和Peter一起研究了研究结果中未涵盖的案例。特别地，我们证明了具有特定泊松结构的2球具有严格的量子化。在构造严格量子化的过程中，得到了新的“非交换2-球”。这些C^*代数是非交换泊松流形的新例子。如上所述，不幸的是，我们无法实现项目的目标，即泊松流形的严格量化的存在。我们当然打算继续这个项目。我们有望在一年左右的时间内完成这项工程。少