Solvability of Poisson algebras

Abstract

Let $P$ be a Poisson algebra with a Lie bracket $\{,\}$ over a field $\F$ of characteristic $p\ge 0$. In this paper, the Lie structure of $P$ is investigated. In particular, if $P$ is solvable with respect to its Lie bracket, then we prove that the Poisson ideal $\mathcal{J}$ of $P$ generated by all elements $\{\{\{x_1,x_2\},\{x_3,x_4\}\},x_5\}$ with $x_1,\dots ,x_5\in P$ is associative nilpotent of index bounded by a function of the derived length of $P$. We use this result to further prove that if $P$ is solvable and $p\ne 2$, then the Poisson ideal $\{P,P\}P$ is nil.