Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson $P$-module is equivalent to a left $P^e$-module.
My question is whether $P^e$ is (left) noetherian or not. (For the background, see http://arxiv.org/pdf/1102.0366v1.pdf)
For next question, let $I$ be a Poisson ideal of $P$ as above. Then $I$ is a $P^e$-module. Then is $I$ a finitely generated $P^e$-module?