Universal enveloping algebras and universal derivations of Poisson algebras

TL;DR

This paper explores the structure of universal enveloping algebras of Poisson algebras, establishing isomorphisms with Weyl algebras, constructing bases, and linking automorphisms to the Jacobian Conjecture.

Contribution

It demonstrates isomorphisms between universal enveloping algebras of Poisson and Weyl algebras, constructs bases for free Poisson algebras, and connects automorphism tameness to the Jacobian Conjecture.

Findings

Universal enveloping algebras of Poisson symplectic and Weyl algebras are isomorphic.

A basis for the universal enveloping algebra of a free Poisson algebra is constructed.

Automorphisms of free Poisson algebras are shown to be tame, linked to the Jacobian Conjecture.

Abstract

Let $k$ be an arbitrary field of characteristic $0$. It is shown that for any $n\geq 1$ the universal enveloping algebras of the Poisson symplectic algebra $P_n(k)$ and the Weyl algebra $A_n(k)$ are isomorphic and the canonical isomorphism between them easily leads to the Moyal product. A basis of the universal enveloping algebra $P^e$ of a free Poisson algebra $P=k\{x_1,...,x_n\}$ is constructed and proved that the left dependency of a finite number of elements of $P^e$ over $P^e$ is algorithmically recognizable. We prove that if two elements of a free Poisson algebra do not generate a free two generated subalgebra then they commute. 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.