Twisted isotropic realisations of twisted Poisson structures

TL;DR

This paper explores the global structure of integrable Hamiltonian systems on almost symplectic manifolds, classifies twisted isotropic realisations of twisted Poisson structures, and identifies cohomological obstructions to their construction.

Contribution

It introduces a classification of twisted isotropic realisations and extends previous results by providing a cohomological framework for understanding their existence.

Findings

Classification of twisted isotropic realisations up to smooth isomorphism

Identification of cohomological obstructions to realisation construction

Extension of local integrability results to global twisted Poisson structures

Abstract

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in \cite{fasso_sansonetto}, which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in \cite{bursztyn_crainic_weinstein_zhu}. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising the main results of \cite{daz_delz}.