Hideki Omori

A CONSTRUCTION OF A STAR PRODUCT ON A POISSON MANIFOLD

For an arbitrary Poisson manifold a new formula of deformation quantization is obtained. [x i , x j ] = ω ij (x). (1) Let˜A = A[[t]]] be the space of formal power series in a variable t with coefficients in A = C ∞ (X). A star product is a bilinear associative product oñ A, defined for f, g ∈ A by f ⋆ g = gf + t 2 [f, g] + ∞ k=2 t k c k (f, g), (2) where c k (f, g) are bidifferential operators. The star product is a basic object of the deformation quantization [1]. A scheme for construction of the star product on an arbitrary Poisson manifold was found by Kontsevich [2]. For symplectic Poisson manifolds one can also use Fedosovs construction of deformation quantization [3]. The star product for linear Poisson structures can be derived from the Campbell-Baker-Hausdorff (CBH) formula [4]. It is different from Kontse-vichs product being a part of the last one.