Twisted isotropic realisations of twisted Poisson structures

Nicola Sansonetto; Daniele Sepe

doi:10.3934/jgm.2013.5.233

Article Contents

2013, Volume 5, Issue 2: 233-256. Doi: 10.3934/jgm.2013.5.233

This issue Previous Article Semi-global symplectic invariants of the Euler top Next Article

Twisted isotropic realisations of twisted Poisson structures

Nicola Sansonetto^1, and
Daniele Sepe^2,

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

Received: July 2012

Revised: March 2013

Published: June 2013

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Abstract

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].

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Mathematics Subject Classification: 37J35, 53D15, 53D17.

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