The local structure of Poisson manifolds

Table of Contents (*) {/,*}= In the early nineteenth century, Poisson noticed that the vanishing of {/, g} and {/, h] imply that of {/, {g, A}}; almost thirty years later Jacobi discovered the identity {/, {g, h}} = {{/, g}, h) + {g, {/, h}} which "explains" Poisson's theorem. In his study of general composition laws satisfying the Jacobi identity, Lie [29] defined in local coordinate form what is now known as a Poisson structure. On R r such a structure is given by functions w^x^-,jc r ) satisfying the identities w .. + W = o,