Analytic deformation of Poisson manifolds and noncominutative geometry

本项目的目的是构造性地证明Poisson流形的解析形变的存在性，推广了辛流形。Poisson流形的形变量子化(代数形变)的存在性一直是一个重要的问题，1997年M.Kontsevich终于证明了这一点。代数变形与解析变形之间的关系类似于形式级数与实现给定形式级数的光滑函数之间的关系。辛流形是Poisson流形的特例，其结构是众所周知的。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，我们研究了辛流形，证明了只要第二同伦群是平凡的，任何闭辛流形都有解析形变。这一结果发表为《辛流形的严格量子化(出现在数学物理通讯中)》。2-球面的第二同伦群是非平凡的。因此，上述结果不能应用于2球体。在与纽约州立大学布法罗分校的C.L.Olsen联合开展的一个项目中，我们研究了二维球。2-球面除了具有标准的旋转不变辛结构外，还具有有趣的泊松结构。我们构造了在北极和南极简并的泊松结构的解析形变。这一结果发表在《一族新的非对易2-球体》上。

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".