Skip to main content

Noncommutative integrable systems on b-symplectic manifolds

Regular and Chaotic Dynamics volume 21, pages 643–659 (2016)Cite this article

Abstract

In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Atiyah, M. F., Convexity and Commuting Hamiltonians, Bull. London Math. Soc., 1982, vol. 14, no. 1, pp. 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  2. Besse, A. L., Manifolds All of Whose Geodesics Are Closed, Ergeb. Math. Grenzgeb. (2), vol. 93, Berlin: Springer, 2012.

    MATH  Google Scholar 

  3. Bolsinov, A.V. and Jovanović, B., Non-Commutative Integrability, Moment Map and Geodesic Flows, Ann. Glob. Anal. Geom., 2003, vol. 23, no. 4, pp. 305–322.

    Article  MATH  Google Scholar 

  4. Delshams, A., Kiesenhofer, A. and Miranda, E., Examples of Integrable and Non-Integrable Systems on Singular Symplectic Manifolds, J. Geom. Phys., 2016 (in press).

    Google Scholar 

  5. Fernandes, R. L., Laurent-Gengoux, C., and Vanhaecke, P., Global Action–Angle Variables for Non- Commutative Integrable Systems, arXiv:1503.00084 (2015).

    Google Scholar 

  6. Gualtieri, M., Li, S., Pelayo, Á., and Ratiu, T., The Tropical Momentum Map: A Classification of Toric Log Symplectic Manifolds, Math. Ann., 2016, 42 pp.

    Google Scholar 

  7. Guillemin, V., Miranda, E., and Pires, A.R., Codimension One Symplectic Foliations and Regular Poisson Structures, Bull. Braz. Math. Soc. (N. S.), 2011, vol. 42, no. 4, pp. 607–623.

    Article  MathSciNet  MATH  Google Scholar 

  8. Guillemin, V., Miranda, E., and Pires, A.R., Symplectic and Poisson Geometry on b-Manifolds, Adv. Math., 2014, vol. 264, pp. 864–896.

    Article  MathSciNet  MATH  Google Scholar 

  9. Guillemin, V., Miranda, E., Pires, A. R., and Scott, G., Toric Actions on b-Symplectic Manifolds, Int. Math. Res. Not. IMRN, 2015, no. 14, pp. 5818–5848.

    Article  MathSciNet  MATH  Google Scholar 

  10. Guillemin, V., Miranda, E., Pires, A. R., and Scott, G., Convexity for Hamiltonian Torus Actions on b-Symplectic Manifolds, Math. Res. Lett., 2016 (to appear).

    Google Scholar 

  11. Guillemin, V. and Sternberg, Sh., Convexity Properties of the Moment Mapping, Invent. Math., 1982, vol. 67, no. 3, pp. 491–513.

    Article  MathSciNet  MATH  Google Scholar 

  12. Guillemin, V. and Sternberg, Sh., Symplectic Techniques in Physics, 2nd ed., Cambridge: Cambridge Univ. Press, 1990.

    MATH  Google Scholar 

  13. Kiesenhofer, A. and Miranda, E., Cotangent Models for Integrable Systems, Commun. Math. Phys., 2016, 23 pp.

    Google Scholar 

  14. Kiesenhofer, A., Miranda, E., and Scott, G., Action–Angle Variables and a KAM Theorem for b-Poisson Manifolds, J. Math. Pures Appl. (9), 2016, vol. 105, no. 1, pp. 66–85.

    Article  MathSciNet  MATH  Google Scholar 

  15. Kirwan, F., Convexity Properties of the Moment Mapping: 3, Invent. Math., 1984, vol. 77, no. 3, pp. 547–552.

    Article  MathSciNet  MATH  Google Scholar 

  16. Laurent-Gengoux, C., Miranda, E., and Vanhaecke, P., Action–Angle Coordinates for Integrable Systems on Poisson Manifolds, Int. Math. Res. Not. IMRN, 2011, no. 8, pp. 1839–1869.

    MathSciNet  MATH  Google Scholar 

  17. Miranda, E. and Solha, R., On a Poincaré Lemma for Foliations, in Foliations 2012, P. Walczak, J. Álvarez López, S. Hurder, R. Langevin, and T. Tsuboi (Eds.), Hackensack,N.J.: World Sci., 2013, pp. 115–137.

    Chapter  Google Scholar 

  18. Martínez Torres, D. and Miranda, E., Weakly Hamiltonian Actions, J. Geom. Phys., 2016 (in press).

    Google Scholar 

  19. Lie, S., Theorie der Transformationsgruppen: Vol. 1, Leipzig: Teubner, 1888.

    Google Scholar 

  20. Nekhoroshev, N. N., Action–Angle Variables and Their Generalization, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180–198; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181–198.

    MATH  Google Scholar 

  21. Sjamaar, R., Convexity Properties of the Moment Mapping Re-Examined, Adv. Math., 1998, vol. 138, no. 1, pp. 46–91.

    Article  MathSciNet  MATH  Google Scholar 

  22. Souriau, J.-M., Structure des systèmes dynamiques, Maîtrises de mathématiques, Paris: Dunod, 1970.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Universitat Politècnica de Catalunya, EPSEB, Avinguda del Doctor Marañón 44–50, Barcelona, Spain

    Anna Kiesenhofer & Eva Miranda

  2. Barcelona Graduate School of Mathematics, Campus de Bellaterra, Edifici C, 08193, Bellaterra, Barcelona, Spain

    Eva Miranda

Authors
  1. Anna Kiesenhofer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eva Miranda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Kiesenhofer.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kiesenhofer, A., Miranda, E. Noncommutative integrable systems on b-symplectic manifolds. Regul. Chaot. Dyn. 21, 643–659 (2016). https://doi.org/10.1134/S1560354716060058

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354716060058

Keywords

  • Poisson manifolds
  • b-symplectic manifolds
  • noncommutative integrable systems
  • action-angle coordinates

MSC2010 numbers

  • 53D05
  • 53D17
  • 70H06