Poisson algebras, Weyl algebras and Jacobi pairs

TL;DR

This paper investigates Jacobi pairs and their properties within Poisson and Weyl algebra structures, introducing automorphisms to analyze pairs violating the Jacobian conjecture, and extending results to Dixmier pairs in Weyl algebra.

Contribution

It introduces automorphisms of Poisson algebras to study Jacobi pairs and generalizes findings to Weyl algebras, providing new insights into pairs satisfying specific algebraic relations.

Findings

Automorphisms help analyze Jacobi pairs in Poisson algebras.

Variable changes produce pairs with specific forms satisfying additional conditions.

Results extend to Weyl algebra and properties of Dixmier pairs are obtained.

Abstract

We study Jacobi pairs in details and obtained some properties. We also study the natural Poisson algebra structure $(\PP,[...,...],...)$ on the space $\PP:=\C[y]((x^{-\frac1N}))$ for some sufficient large $N$, and introduce some automorphisms of $(\PP,[...,...],...)$ which are (possibly infinite but well-defined) products of the automorphisms of forms $e^{\ad_H}$ for $H\in x^{1-\frac1N}\C[y][[x^{-\frac1N}]]$ and $\tau_c:(x,y)\mapsto(x,y-cx^{-1})$ for some $c\in\C$. These automorphisms are used as tools to study Jacobi pairs in $\PP$. In particular, starting from a Jacobi pair $(F,G)$ in $\C[x,y]$ which violates the two-dimensional Jacobian conjecture, by applying some variable change $(x,y)\mapsto\big(x^{b},x^{1-b}(y+a_1 x^{-b_1}+...+a_kx^{-b_k})\big)$ for some $b,b_i\in\Q_+,a_i\in\C$ with $b_i<1<b$, we obtain a \QJ pair still denoted by $(F,G)$ in $\C[x^{\pm\frac1N},y]$ with the form $F=x^{\frac{m}{m+n}}(f+F_0)$, $G=x^{\frac{n}{m+n}}(g+G_0)$ for some positive integers $m,n$, and $f,g\in\C[y]$, $F_0,G_0\in x^{-\frac1N}\C[x^{-\frac1N},y]$, such that $F,G$ satisfy some additional conditions. Then we generalize the results to the Weyl algebra $\WW=\C[v]((u^{-\frac1N}))$ with relation $[u,v]=1$, and obtain some properties of pairs $(F,G)$ satisfying $[F,G]=1$, referred to as Dixmier pairs.