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Analytic deformation of Poisson manifolds and noncominutative geometry

泊松流形的解析变形和非计算几何

基本信息

批准号：
13640208
负责人：
NATSUME Toshikazu
金额：
$ 2.05万
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：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

The aim of the project is to give a constructive proof of existence of analytic deformation of Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization(algebraic deformation) for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The relationship between algebraic deformation and analytic deformation is similar to the relationship between a formal power series and a smooth function that realizes the given formal power series.Symplectic manifolds are special examples of Poisson manifolds, and its structures are well known. In a joint project with R. Nest of the University of Copenhagen and I.Peter of Munster University, we investigated symplectic manifolds and showed that any closed symplectic manifold has an analytic deformation provided that the second homotopy group is trivial. This result is published as "Strict quantization of symplectic manifolds (to appear in Letters hi Mathematical Physics)".The second homotopy group of the 2-sphere is nontrivial. Thus the result above cannot be applied to the 2-sphere. In a joint project with C.L.Olsen of the State University of New York at Buffalo, we studied the 2-sphere. The 2-sphere possesses interesting Poisson structures besides the standard rotation invariant symplectic structure. We constructed an analytic deformation for a Poisson structure degenerate at the North and South poles. This result is published as "A new family of noncommutative 2-sphers".

该项目的目的是提供泊松流形解析变形存在的建设性证明，从而推广辛流形。泊松流形的变形量化（代数变形）的存在一直是一个重要的问题，最终由M.Kontsevich于1997年证明。代数变形和解析变形之间的关系类似于形式幂级数和实现给定形式幂级数的光滑函数之间的关系。辛流形是泊松流形的特殊例子，其结构是众所周知的。在与哥本哈根大学的 R. Nest 和蒙斯特大学的 I.Peter 的联合项目中，我们研究了辛流形，并表明只要第二同伦群是平凡的，任何闭辛流形都具有解析变形。该结果发表为“辛流形的严格量子化（出现在数学物理快报中）”。2-球体的第二个同伦群是非平凡的。因此上面的结果不能应用于2-球体。在与纽约州立大学布法罗分校的 C.L.Olsen 的联合项目中，我们研究了 2 球体。除了标准旋转不变辛结构之外，2-球体还具有有趣的泊松结构。我们构建了在北极和南极退化的泊松结构的解析变形。该结果作为“非交换 2-球体的新族”发表。