Universal Star Products

One defines the notion of universal deformation quantization: given any manifold M, any Poisson structure Λ on M and any torsionfree linear connection ? on M, a universal deformation quantization associates to this data a star product on (M, Λ) given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor Λ, the curvature tensor R and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poissoncohomology.