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 wˣ^-, jc r) satisfying the identities w. . + W = o,