Infinitesimal Takesaki duality of Hamiltonian vector fields on a symplectic manifold

For an infinitesimal symplectic action of a Lie algebra $\text{goth}g$ on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and Lie algebra $\text{goth}g$. We obtain its second crossed product in case $\text{goth}g=R$ and show an infinitesimal version for a theorem type of Takesaki duality.