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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

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, 3. Folge, A Series of Modern Surveys in Mathematics, 47, Springer-Verlag, Berlin, 2004.

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. 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), 267-310. doi: 10.1007/s00205-012-0512-9.

[4]

L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?, in "Proceedings of the XXX Symposium on Mathematical Physics'' (Toruń, 1998), Rep. Math. Phys., 44 (1999), 29-35. 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), 211-260. 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), 121-136. doi: 10.1016/S0393-0440(99)00058-3.

[7]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. doi: 10.1007/s002200050412.

[8]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607. 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), 187-196. 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), 575-620. doi: 10.4007/annals.2003.157.575.

[11]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Diff. Geom., 66 (2004), 71-137.

[12]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy, J. Diff. Eq., 172 (2001), 42-58. doi: 10.1006/jdeq.2000.3852.

[13]

P. Dazord and P. Delzant, Le problème général des variables actions-angles, J. Diff. Geom., 26 (1987), 223-251.

[14]

J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy, Ann. Physics, 324 (2009), 1953-1982. doi: 10.1016/j.aop.2009.03.008.

[15]

J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.

[16]

A. El Kacimi-Alaoui, Sur la cohomologie feuilletée, Compositio Math., 49 (1983), 195-215.

[17]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13 pp. doi: 10.1063/1.2783937.

[18]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. 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), 051, 12 pp. doi: 10.3842/SIGMA.2007.051.

[21]

M. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205. 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), 511-518. doi: 10.1090/S0002-9939-1982-0671227-8.

[23]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. doi: 10.1088/0951-7715/8/4/003.

[24]

C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys., 43 (2002), 341-344. 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 "The Breadth of Symplectic and Poisson Geometry," Progress in Mathematics, 232, Birkhäuser Boston, Boston, MA, (2005), 363-389. 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), 1839-1869. 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, Cambridge University Press, Cambridge, 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, Springer-Verlag, New York, 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), 5-54.

[30]

I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'' Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615450.

[31]

N. N. Nehorošev, Action-angle variables and their generalizations, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180-198.

[32]

J.-S. Park, Topological open p-branes, in "Symplectic Geometry and Mirror Symmetry'' (Seoul, 2000), World Sci. Publ., River Edge, NJ, (2001), 311-384. doi: 10.1142/9789812799821_0010.

[33]

A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems, preprint, arXiv:1108.0328.

[34]

C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology, preprint, arXiv:math/0404449.

[35]

, N. Sansonetto and D. Sepe, in preparation.

[36]

D. Sepe, Almost Lagrangian obstruction, Diff. Geom. Appl., 29 (2011), 787-800. doi: 10.1016/j.difgeo.2011.08.007.

[37]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, in "Noncommutative Geometry and String Theory'' (Yokohama, 2001), Progr. Theoret. Phys. Suppl., 144 (2001), 145-154. doi: 10.1143/PTPS.144.145.

[38]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'' Progress in Mathematics, 118, Birkäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.

[39]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1988), 101-104. doi: 10.1090/S0273-0979-1987-15473-5.

[40]

D. V. Zenkov, The geometry of the Routh problem, Jour. Nonlin. Science, 5 (1995), 503-519. 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, 3. Folge, A Series of Modern Surveys in Mathematics, 47, Springer-Verlag, Berlin, 2004.

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. 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), 267-310. doi: 10.1007/s00205-012-0512-9.

[4]

L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?, in "Proceedings of the XXX Symposium on Mathematical Physics'' (Toruń, 1998), Rep. Math. Phys., 44 (1999), 29-35. 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), 211-260. 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), 121-136. doi: 10.1016/S0393-0440(99)00058-3.

[7]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. doi: 10.1007/s002200050412.

[8]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607. 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), 187-196. 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), 575-620. doi: 10.4007/annals.2003.157.575.

[11]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Diff. Geom., 66 (2004), 71-137.

[12]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy, J. Diff. Eq., 172 (2001), 42-58. doi: 10.1006/jdeq.2000.3852.

[13]

P. Dazord and P. Delzant, Le problème général des variables actions-angles, J. Diff. Geom., 26 (1987), 223-251.

[14]

J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy, Ann. Physics, 324 (2009), 1953-1982. doi: 10.1016/j.aop.2009.03.008.

[15]

J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.

[16]

A. El Kacimi-Alaoui, Sur la cohomologie feuilletée, Compositio Math., 49 (1983), 195-215.

[17]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13 pp. doi: 10.1063/1.2783937.

[18]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. 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), 051, 12 pp. doi: 10.3842/SIGMA.2007.051.

[21]

M. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205. 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), 511-518. doi: 10.1090/S0002-9939-1982-0671227-8.

[23]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. doi: 10.1088/0951-7715/8/4/003.

[24]

C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys., 43 (2002), 341-344. 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 "The Breadth of Symplectic and Poisson Geometry," Progress in Mathematics, 232, Birkhäuser Boston, Boston, MA, (2005), 363-389. 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), 1839-1869. 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, Cambridge University Press, Cambridge, 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, Springer-Verlag, New York, 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), 5-54.

[30]

I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'' Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615450.

[31]

N. N. Nehorošev, Action-angle variables and their generalizations, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180-198.

[32]

J.-S. Park, Topological open p-branes, in "Symplectic Geometry and Mirror Symmetry'' (Seoul, 2000), World Sci. Publ., River Edge, NJ, (2001), 311-384. doi: 10.1142/9789812799821_0010.

[33]

A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems, preprint, arXiv:1108.0328.

[34]

C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology, preprint, arXiv:math/0404449.

[35]

, N. Sansonetto and D. Sepe, in preparation.

[36]

D. Sepe, Almost Lagrangian obstruction, Diff. Geom. Appl., 29 (2011), 787-800. doi: 10.1016/j.difgeo.2011.08.007.

[37]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, in "Noncommutative Geometry and String Theory'' (Yokohama, 2001), Progr. Theoret. Phys. Suppl., 144 (2001), 145-154. doi: 10.1143/PTPS.144.145.

[38]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'' Progress in Mathematics, 118, Birkäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.

[39]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1988), 101-104. doi: 10.1090/S0273-0979-1987-15473-5.

[40]

D. V. Zenkov, The geometry of the Routh problem, Jour. Nonlin. Science, 5 (1995), 503-519. doi: 10.1007/BF01209025.

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