Simple Poisson Ore extensions

Abstract

Let A be a Poisson algebra over a field $\mathbf{k}$ with characteristic zero, let $\gamma$, $\alpha$ be Poisson derivations on A such that $\gamma \alpha =\alpha \gamma$ and $0\ne \rho \in \mathbf{k}$. Here the notion of a $\gamma$-Poisson normal element is introduced, it is proved that the polynomial algebra A[y, x] has a Poisson structure defined by $\{y,a\}=\alpha (a)y, \{x,a\}=\beta (a)x, \{x,y\}=\beta (y)x+\delta (y)$ for $a\in A$, where $\beta$ is a Poisson derivation on A[y] defined by $\beta |_A=\gamma -\alpha$, $\beta (y)=\rho y$ and $\delta$ is a derivation on A[y] such that $\delta |_A=0$, and its Poisson simplicity criterion is established and endorsed by examples.