Free Poisson fields and their automorphisms

TL;DR

This paper establishes that the automorphism group of a free Poisson field in two variables over a field of characteristic zero is isomorphic to the Cremona group, and explores properties of its universal enveloping algebra.

Contribution

It proves the isomorphism between the automorphism group of a free Poisson field and the Cremona group, and characterizes Poisson dependence via universal derivatives.

Findings

Automorphism group of free Poisson field in two variables is isomorphic to Cremona group.

Universal enveloping algebra of a free Poisson field is a free ideal ring.

Poisson dependence characterized by universal derivatives.

Abstract

Let $k$ be an arbitrary field of characteristic 0. We prove that the group of automorphisms of a free Poisson field $P(x,y)$ in two variables $x,y$ over $k$ is isomorphic to the Cremona group $\mathrm{Cr}_2(k)$. We also prove that the universal enveloping algebra $P(x_1,...,x_n)^e$ of a free Poisson field $P(x_1,...,x_n)$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P(x_1,...,x_n)$ via universal derivatives.