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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.03551
 Bibcode:
 2020arXiv200603551S
 Keywords:

 Mathematics  Rings and Algebras;
 17B63;
 16R10;
 17B30;
 17B50;
 17B01