Let  $(M,\omega)$  be  a  symplectic  manifold  with the volum 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 $\omega$?In the  other words, if two Lie  algebras $L(M,\omega)$ and $L(M,\omega')$ are isomorphic Lie  algebra, does this imply that 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  $M,\omega$?


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 $Z(M,\omega)=\{X\in \chi^{\infty}(M)\mid L_X \omega=0\}$  which is  an ideal in $E(M,\omega)$.