Reduction of Nambu-Poisson Manifolds by Regular Distributions

  • Apurba Das


The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ibáñez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.


Reduction Nambu-Poisson manifolds Gauge transformations 

Mathematics Subject Classification (2010)

17B63 53C15 53D17 



The author wish to thank Prof. Goutam Mukherjee for his carefully reading the manuscript. The author would also like to thank the referee for his comments and suggestions on the earlier version of the paper that have improved the exposition.


  1. 1.
    Bi, Y.H., Sheng, Y.: On higher analogues of Courant algebroids. Sci. China Math. 54, 437–447 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chatterjee, R.: Dynamical symmetries and Nambu mechanics. Lett. Math. Phys. 36, 117–126 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chatterjee, R., Takhtajan, L.: Aspects of classical and quantum Nambu mechanics. Lett. Math. Phys. 37(4), 475–482 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Curtright, T., Zachos, C.: Classical and quantum Nambu mechanics. Phys. Rev. D 68, 085001 (2003)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Das, A.: Singular reduction of Nambu-Poisson manifolds. Int. J. Geom. Methods Mod. Phys. 14, 1750128 (2017). [13 pages]MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dito, G., Flato, M., Sternheimer, D., Takhtajan, L.: Deformation quantization and Nambu mechanics. Commun. Math. Phys. 183(1), 1–22 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dufour, J.-P., Zung, N.T.: Poisson Structures and Their Normal Forms. Birkhäuser Verlag, Basel-Boston-Berlin (2005)zbMATHGoogle Scholar
  8. 8.
    Falceto, F., Zambon, M.: An extension of the Marsden-Ratiu reduction for Poisson manifolds. Lett. Math. Phys. 85, 203–219 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gautheron, P.: Some remarks concerning Nambu mechanics. Lett. Math. Phys. 37, 103–116 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ibáñez, R., de León, M., Marrero, J.C., Diego, D.M.: Reduction of generalized Poisson and Nambu-Poisson manifolds. Rep. Math. Phys. 42, 71–90 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ibáñez, R., de León, M., Marrero, J.C., Padrón, E.: Leibniz algebroid associated with a Nambu-Poisson structure. J. Phys. A: Math. Gen. 32, 8129–8144 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jurco, B., Schupp, P.: Nambu-sigma model and effective membrane actions. Phys. Lett. B 713(3), 313–316 (2012)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Jurco, B., Schupp, P.: Nambu-sigma model and branes. In: Proceedings of the 11th Hellenic School and Workshops on Elementary Particle Physics and Gravity, pp. 45–53. Corfu Summer Institute (Corfu2011) (2011)Google Scholar
  14. 14.
    Marmo, G., Ibort, A.: A generalized reduction procedure for dynamical systems. In: Proceeding of the IV Workshop on Differential Geometry and Its Applications. Santiago de Compostela 95 RSEF Monografias, vol. 3 (1995)Google Scholar
  15. 15.
    Marsden, J.E., Ratiu, T.: Reduction of Poisson manifolds. Lett. Math. Phys. 11, 161–169 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nambu, Y.: Generalized Hamiltonian Dynamics. Phys. Rev. D 7(8), 2405–2412 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ortega, J.-P., Ratiu, T.: Momentum Maps and Hamiltonian reduction, Progress in Mathematics (Boston Mass.), vol. 222. Birkhäuser, Boston (2004)Google Scholar
  18. 18.
    Ševera, P., Weinstein, A.: Poisson geometry with a 3-form background. Prog. Theor. Phys. Suppl. 144, 145–154 (2001)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Takhtajan, L.: On foundation of the Generalized Nambu Mechanics. Commun. Math. Phys. 160, 295–315 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wade, A.: Nambu-Dirac structures for Lie algebroids. Lett. Math. Phys. 61, 85–99 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical InstituteKolkataIndia

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