Twisted isotropic realisations of twisted Poisson structures

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

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,, Italy
2.	CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

Received July 2012 Revised March 2013 Published July 2013

Abstract
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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].

Keywords: almost symplectic manifolds., Twisted Poisson manifolds, isotropic realisations, Hamiltonian integrable systems.

Mathematics Subject Classification: 37J35, 53D15, 53D1.

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

References:

[1]	M. Adler, P. van Moerbeke and P. Vanhaecke, "Algebraic integrability, Painlevé Geometry and Lie Algebras,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 47 (2004).
[2]	P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits,, Nonlinearity, 10 (1997), 595. doi: 10.1088/0951-7715/10/3/002.
[3]	P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267. doi: 10.1007/s00205-012-0512-9.
[4]	L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?,, in, 44 (1999), 29. doi: 10.1016/S0034-4877(99)80142-6.
[5]	G. Blankenstein and T. S. Ratiu, Singular reduction of implicit Hamiltonian systems,, Rep. Math. Phys., 53 (2004), 211. doi: 10.1016/S0034-4877(04)90013-4.
[6]	A. M. Bloch and D. V. Zenkov, Dynamics of the n-Dimensional Suslov problem,, J. Geom. Phys., 34 (2000), 121. doi: 10.1016/S0393-0440(99)00058-3.
[7]	O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19. doi: 10.1007/s002200050412.
[8]	H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Math. J., 123 (2004), 549. doi: 10.1215/S0012-7094-04-12335-8.
[9]	A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures,, J. Geom. Phys., 49 (2004), 187. doi: 10.1016/S0393-0440(03)00086-X.
[10]	M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math. (2), 157 (2003), 575. doi: 10.4007/annals.2003.157.575.
[11]	M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, J. Diff. Geom., 66 (2004), 71.
[12]	R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eq., 172 (2001), 42. doi: 10.1006/jdeq.2000.3852.
[13]	P. Dazord and P. Delzant, Le problème général des variables actions-angles,, J. Diff. Geom., 26 (1987), 223.
[14]	J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy,, Ann. Physics, 324 (2009), 1953. doi: 10.1016/j.aop.2009.03.008.
[15]	J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687. doi: 10.1002/cpa.3160330602.
[16]	A. El Kacimi-Alaoui, Sur la cohomologie feuilletée,, Compositio Math., 49 (1983), 195.
[17]	F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2783937.
[18]	F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93. doi: 10.1007/s10440-005-1139-8.
[19]	, F. Fassò, A. Giacobbe, L. Garcia-Naranjo and N. Sansonetto,, in preparation., ().
[20]	F. Fassò, F. and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system,, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007). doi: 10.3842/SIGMA.2007.051.
[21]	M. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185. doi: 10.1090/S0002-9947-1980-0561832-4.
[22]	W. M. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations,, Proc. Amer. Math. Soc., 86 (1982), 511. doi: 10.1090/S0002-9939-1982-0671227-8.
[23]	J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493. doi: 10.1088/0951-7715/8/4/003.
[24]	C. Klimčík and T. Strobl, WZW-Poisson manifolds,, J. Geom. Phys., 43 (2002), 341. doi: 10.1016/S0393-0440(02)00027-X.
[25]	Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory,, in, 232 (2005), 363. doi: 10.1007/0-8176-4419-9_12.
[26]	C. Laurent-Gengoux, E. Miranda and P. Vanhaecke, Action-angle coordinates for integrable systems on Poisson manifolds,, Int. Math. Res. Not., 8 (2011), 1839. doi: 10.1093/imrn/rnq130.
[27]	K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'', London Mathematical Society Lecture Notes Series, 213 (2005).
[28]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'', Texts in Applied Mathematics, 17 (1994). doi: 10.1007/978-1-4612-2682-6.
[29]	A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries,, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5.
[30]	I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'', Cambridge Studies in Advanced Mathematics, 91 (2003). doi: 10.1017/CBO9780511615450.
[31]	N. N. Nehorošev, Action-angle variables and their generalizations,, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180.
[32]	J.-S. Park, Topological open p-branes,, in, (2001), 311. doi: 10.1142/9789812799821_0010.
[33]	A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems,, preprint, ().
[34]	C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology,, preprint, ().
[35]	, N. Sansonetto and D. Sepe,, in preparation., ().
[36]	D. Sepe, Almost Lagrangian obstruction,, Diff. Geom. Appl., 29 (2011), 787. doi: 10.1016/j.difgeo.2011.08.007.
[37]	P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, in, 144 (2001), 145. doi: 10.1143/PTPS.144.145.
[38]	I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'', Progress in Mathematics, 118 (1994). doi: 10.1007/978-3-0348-8495-2.
[39]	A. Weinstein, Symplectic groupoids and Poisson manifolds,, Bull. Amer. Math. Soc., 16 (1988), 101. doi: 10.1090/S0273-0979-1987-15473-5.
[40]	D. V. Zenkov, The geometry of the Routh problem,, Jour. Nonlin. Science, 5 (1995), 503. doi: 10.1007/BF01209025.

show all references

References:

[1]	M. Adler, P. van Moerbeke and P. Vanhaecke, "Algebraic integrability, Painlevé Geometry and Lie Algebras,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 47 (2004).
[2]	P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits,, Nonlinearity, 10 (1997), 595. doi: 10.1088/0951-7715/10/3/002.
[3]	P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267. doi: 10.1007/s00205-012-0512-9.
[4]	L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?,, in, 44 (1999), 29. doi: 10.1016/S0034-4877(99)80142-6.
[5]	G. Blankenstein and T. S. Ratiu, Singular reduction of implicit Hamiltonian systems,, Rep. Math. Phys., 53 (2004), 211. doi: 10.1016/S0034-4877(04)90013-4.
[6]	A. M. Bloch and D. V. Zenkov, Dynamics of the n-Dimensional Suslov problem,, J. Geom. Phys., 34 (2000), 121. doi: 10.1016/S0393-0440(99)00058-3.
[7]	O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19. doi: 10.1007/s002200050412.
[8]	H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Math. J., 123 (2004), 549. doi: 10.1215/S0012-7094-04-12335-8.
[9]	A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures,, J. Geom. Phys., 49 (2004), 187. doi: 10.1016/S0393-0440(03)00086-X.
[10]	M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math. (2), 157 (2003), 575. doi: 10.4007/annals.2003.157.575.
[11]	M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, J. Diff. Geom., 66 (2004), 71.
[12]	R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eq., 172 (2001), 42. doi: 10.1006/jdeq.2000.3852.
[13]	P. Dazord and P. Delzant, Le problème général des variables actions-angles,, J. Diff. Geom., 26 (1987), 223.
[14]	J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy,, Ann. Physics, 324 (2009), 1953. doi: 10.1016/j.aop.2009.03.008.
[15]	J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687. doi: 10.1002/cpa.3160330602.
[16]	A. El Kacimi-Alaoui, Sur la cohomologie feuilletée,, Compositio Math., 49 (1983), 195.
[17]	F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2783937.
[18]	F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93. doi: 10.1007/s10440-005-1139-8.
[19]	, F. Fassò, A. Giacobbe, L. Garcia-Naranjo and N. Sansonetto,, in preparation., ().
[20]	F. Fassò, F. and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system,, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007). doi: 10.3842/SIGMA.2007.051.
[21]	M. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185. doi: 10.1090/S0002-9947-1980-0561832-4.
[22]	W. M. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations,, Proc. Amer. Math. Soc., 86 (1982), 511. doi: 10.1090/S0002-9939-1982-0671227-8.
[23]	J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493. doi: 10.1088/0951-7715/8/4/003.
[24]	C. Klimčík and T. Strobl, WZW-Poisson manifolds,, J. Geom. Phys., 43 (2002), 341. doi: 10.1016/S0393-0440(02)00027-X.
[25]	Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory,, in, 232 (2005), 363. doi: 10.1007/0-8176-4419-9_12.
[26]	C. Laurent-Gengoux, E. Miranda and P. Vanhaecke, Action-angle coordinates for integrable systems on Poisson manifolds,, Int. Math. Res. Not., 8 (2011), 1839. doi: 10.1093/imrn/rnq130.
[27]	K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'', London Mathematical Society Lecture Notes Series, 213 (2005).
[28]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'', Texts in Applied Mathematics, 17 (1994). doi: 10.1007/978-1-4612-2682-6.
[29]	A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries,, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5.
[30]	I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'', Cambridge Studies in Advanced Mathematics, 91 (2003). doi: 10.1017/CBO9780511615450.
[31]	N. N. Nehorošev, Action-angle variables and their generalizations,, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180.
[32]	J.-S. Park, Topological open p-branes,, in, (2001), 311. doi: 10.1142/9789812799821_0010.
[33]	A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems,, preprint, ().
[34]	C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology,, preprint, ().
[35]	, N. Sansonetto and D. Sepe,, in preparation., ().
[36]	D. Sepe, Almost Lagrangian obstruction,, Diff. Geom. Appl., 29 (2011), 787. doi: 10.1016/j.difgeo.2011.08.007.
[37]	P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, in, 144 (2001), 145. doi: 10.1143/PTPS.144.145.
[38]	I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'', Progress in Mathematics, 118 (1994). doi: 10.1007/978-3-0348-8495-2.
[39]	A. Weinstein, Symplectic groupoids and Poisson manifolds,, Bull. Amer. Math. Soc., 16 (1988), 101. doi: 10.1090/S0273-0979-1987-15473-5.
[40]	D. V. Zenkov, The geometry of the Routh problem,, Jour. Nonlin. Science, 5 (1995), 503. doi: 10.1007/BF01209025.

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