Universal enveloping algebras and universal derivations of Poisson algebras

Let k be an arbitrary field of characteristic 0. It is shown that for any n⩾1 the universal enveloping algebras of the Poisson symplectic algebra Pn(k) and the Weyl algebra An(k) are isomorphic and the canonical isomorphism between them easily leads to the Moyal product. A basis of the universal enveloping algebra Pe of a free Poisson algebra P=k{x1,…,xn} is constructed and it is proved that the left dependence of a finite number of elements of Pe over Pe is algorithmically recognizable. We describe the Poisson dependence of any two elements of a free Poisson algebra in characteristic 0 in the language of universal derivatives. The Fox derivatives on free Poisson algebras are defined and it is proved that an analogue of the Jacobian Conjecture for two generated free Poisson algebras is equivalent to the two-dimensional classical Jacobian Conjecture. A new proof of the tameness of automorphisms of two generated free Poisson algebras is also given.