Infinitesimal Takesaki duality of Hamiltonian vector fields on a symplectic manifold

TL;DR

This paper develops an infinitesimal version of Takesaki duality for Hamiltonian vector fields on symplectic manifolds, involving crossed products with Lie algebras, extending classical duality concepts to the infinitesimal setting.

Contribution

It introduces an infinitesimal crossed product construction for Hamiltonian vector fields and Lie algebras, and establishes an infinitesimal Takesaki duality theorem.

Findings

Constructed an infinitesimal crossed product of Hamiltonian vector fields and Lie algebras.

Derived a second crossed product when the Lie algebra is R.

Proved an infinitesimal version of Takesaki duality theorem.

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.