Solvability of Poisson algebras

Abstract: Let $P$ be a Poisson algebra with a Lie bracket ${, }$ over a field $\F$ of characteristic $p\geq 0$. 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 $x_1,\ldots ,x_5 \in P$ 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 $p\neq 2$, then the Poisson ideal ${P,P}P$ is nil.