Infinitesimal Takesaki duality of Hamiltonian vector fields on a symplectic manifold

Abstract

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