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的一个联合项目中，首席研究员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变形理论。代数形变理论（形变量子化的存在性）自80年代以来就已为人所知，本项目的目的是证明Poisson流形的严格量子化的存在性，从而推广了辛流形。Poisson流形的形变量子化的存在性一直是一个重要的问题，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，以加深对存在性机制的理解。特别是重新审查了B.Fedosov关于形变Qu存在性的证明，它在我们的证明中起了关键作用 ...更多信息 反化是重要的一步。第二步是理解Poisson流形的形变量子化的存在性证明，并从Fedosov的观点改写它。最后一步涉及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晰，我们得到了我们结果的改进版本。由于最近出现的一个更简单的证明存在的变形量子化的泊松流形比Kontsevich的，我们有一个前景，以实现第二步。而工作的项目上面讨论的，在一个联合项目与C.L.奥尔森的州立大学的纽约在布法罗，首席研究员工作的情况下，不包括与巢和彼得的结果。特别地，我们证明了具有特定泊松结构的2-球面具有严格的量子化。在构造严格量子化的过程中，我们得到了新的“非对易2-球面”。这些C^*-代数是非交换Poisson流形的新例子，正如上面所解释的，不幸的是我们不能实现项目的目标，即Poisson流形的严格量子化的存在性。我们当然打算继续进行这个项目。我们有望在一年左右的时间内完成这项工程。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