Poisson algebras, Weyl algebras and Jacobi pairs

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