Infinitesimal deformations of a formal symplectic groupoid

الأصل · EN

Given a formal symplectic groupoid G over a Poisson manifold (M, π₀), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation π₀ + επ₁ of the Poisson bivector field π₀. The source and target mappings of a deformation of G are deformations of the source and target mappings of G. To any pair of natural star products (,) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair (,). We give explicit formulas for the source and target mappings of the deformation groupoid of a pair of star products with separation of variables on a Kaehler- Poisson manifold. Finally, we give an algorithm for calculating the principal symbols of the components of the logarithm of a formal Berezin transform of a star product with separation of variables. This algorithm is based upon some deformation groupoid.