Quantization of Poisson manifolds and noncommutative geometry

在与哥本哈根大学的R.Nest和明斯特大学的I.Peter联合进行的一项研究中，主要研究者证明了在拓扑条件下，每个闭辛流形都有严格的量子化。严格量子化是一种解析形变理论。一个代数形变理论(形变量子化的存在性)早在80年代就已为人所知。这个项目的目的是证明Poisson流形的严格量子化的存在性，它是辛流形的推广。Poisson流形的形变量子化一直是一个重要的问题，1997年M.Kontsevich终于证明了这一点。该项目分为三个步骤。第一步是重新考察辛流形严格量子化的存在性证明，以便对其存在机制有一个更深入的理解。特别地，对B.Fedosov关于形变Qu…存在的证明进行了重新考察，该证明在我们的证明中起了关键作用更多的反歧视是重要的一步。第二步是理解Poisson流形形变量子化的存在性证明，并从Fedosov的观点对其进行重写。最后一步涉及到严格量子化的实际构建。通过与Nest的多次讨论，存在的机制变得相当清楚，我们得到了我们结果的精炼版本。由于最近出现了一个比Kontsevich更简单的Poisson流形存在形变量子化的证明，我们有希望实现第二步。在上面讨论的项目中，在与纽约州立大学布法罗分校的C.L.Olsen的联合项目中，主要研究员致力于与Nest和Peter的结果不包括的案例。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，我们得到了新的“非对易2球”。这些C^*-代数是非交换Poisson流形的新例子。如上所述，不幸的是，我们不能达到项目的目标，即Poisson流形存在严格量子化。我们当然打算继续在这个项目上工作。我们希望在一年左右的时间内完成这个项目。较少

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