Solvability of Poisson algebras

Abstract

Let P be a Poisson algebra with a Lie bracket {,} over a field F of characteristic p≥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 J of P generated by all elements {{{x1,x2},{x3,x4}},x5} with x1,…,x5∈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≠2, then the Poisson ideal {P,P}P is nil.