Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ with the volume form $\omega^n.$

In this question we associate a Lie algebra $L(M,\omega)$ to $(M,\omega)$. Then we are interested to know:

1) Does the Lie structure of $L(M,\omega)$ depend on symplectic structure $\omega$? At the other extreme can one prove that if two Lie algebras $L(M,\omega)$ and $L(M,\omega')$ are isomorphic Lie algebras, then there is a symplectomorphism $f:(M,\omega) \to (M,\omega')$?

2)In the literature, are there some precise computation of $L(M,\omega)$ for some symplectic manifolds $(M,\omega)$? What can be said about dimension of $L(M,\omega)$?

Here is the Lie algebra we are considering:

$$L(M,\omega)=E(M,\omega)/Z'(M,\omega)$$

where $$E(M,\omega)=\left\{ X\in \chi^{\infty}(M)\mid L_X \omega=(1/n)Div(X)\omega\right \}=\{X\in \chi^{\infty}(M)\mid L_X \omega=f\omega,\;\;\text{ for some }f\in C^{\infty}(M)\}$$

and $Z'(M,\omega)$ is the normalizer of $Z(M,\omega)=\{X\in \chi^{\infty}(M)\mid L_X \omega=0\}$ in $E(M,\omega)$.

4more comments