Solvability of Poisson algebras

TL;DR

This paper investigates the Lie algebra structure within Poisson algebras, proving that certain ideals are nilpotent or nil under solvability conditions, especially when the characteristic is not 2.

Contribution

It establishes bounds on the nilpotency of specific Poisson ideals in solvable Poisson algebras, extending understanding of their algebraic structure.

Findings

Poisson ideal generated by certain brackets is associative nilpotent with bounded index.

If the algebra is solvable and characteristic is not 2, then the ideal generated by brackets is nil.

Provides conditions under which Poisson ideals exhibit nilpotency or nil properties.

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.