Analytic deformation of Poisson manifolds and noncominutative geometry

点击翻译按钮获取中文摘要

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".