A two-cocycle on the group of symplectic diffeomorphisms

TL;DR

This paper studies a specific two-cocycle on symplectic diffeomorphism groups, exploring its properties, conditions for vanishing, and applications to symplectic bundles and group actions.

Contribution

It provides new insights into the properties of a two-cocycle on symplectic diffeomorphisms, including vanishing and non-vanishing results and applications.

Findings

Identifies conditions under which the cocycle vanishes or not.

Applies cocycle properties to foliated symplectic bundles.

Analyzes Hamiltonian actions of finitely generated groups.

Abstract

We investigate a two-cocycle on the group of symplectic diffeomorphisms of an exact symplectic manifolds defined by Ismagilov, Losik, and Michor and investigate its properties. We provide both vanishing and non-vanishing results and applications to foliated symplectic bundles and to Hamiltonian actions of finitely generated groups.