# American Institute of Mathematical Sciences

June  2013, 5(2): 233-256. doi: 10.3934/jgm.2013.5.233

## Twisted isotropic realisations of twisted Poisson structures

 1 Dipartimento di Informatica, Università degli Studi di Verona, Ca' Vignal 2, Strada Le Grazie 15, 37134 Verona,, Italy 2 CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

Received  July 2012 Revised  March 2013 Published  July 2013

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in [3], 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 [17], 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 [8]. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising some of the main results of [13].
Citation: Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233
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