Nambu structures and associated bialgebroids

  • Samik BasuEmail author
  • Somnath Basu
  • Apurba Das
  • Goutam Mukherjee


We investigate higher-order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu–Poisson manifold (of order \(n>2\)) gives rise to a special bialgebroid structure which is referred to as a weak Lie–Filippov bialgebroid (of order n). It is further demonstrated that such bialgebroids canonically induce a Nambu–Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie–Filippov bialgebroid over a point.


n-Ary operation Nambu–Poisson bracket Gerstenhaber bracket Lie bialgebroid 

2010 Mathematics Subject Classification

Primary: 17B62 17B63 Secondary: 53C15 53D17 


  1. 1.
    Alekseevsky D and Guha P, On decomposability of Nambu–Poisson tensor, Acta Math. Univ. Comenian. (N.S.) 65(1) (1996) 1–9MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bayen F and Flato M, Remarks concerning Nambu’s generalized mechanics, Phys. Rev. D (3) 11 (1975) 3049–3053MathSciNetGoogle Scholar
  3. 3.
    Ciccoli N, Nambu–Lie group actions, Acta Math. Univ. Comenian. (N.S.) 70(2) (2001) 251–263Google Scholar
  4. 4.
    Courant T J, Dirac manifolds, Trans. Amer. Math. Soc. 319(2) (1990) 631–661MathSciNetzbMATHGoogle Scholar
  5. 5.
    Das A, Gondhali S and Mukherjee G, Nambu structures on Lie algebroids and their modular classes, to appear in Proc. Ind. Acad. Sci, (Math. Sci.) Google Scholar
  6. 6.
    Das A, Reduction of Nambu–Poisson manifolds by regular distributions, Math. Phys. Anal. Geom. 21(1) (2018) Art. 5, 21 pp.Google Scholar
  7. 7.
    Das A, Singular reduction of Nambu–Poisson manifolds, Int. J. Geom. Methods Mod. Phys. 14(9) (2017) 1750128, 13 pp.Google Scholar
  8. 8.
    Dorfman I, Dirac Structures and Integrability of Nonlinear Evolution Equations (1993) (Chichester: John Wiley and Sons Ltd)Google Scholar
  9. 9.
    Dufour J-P and Zung N T, Poisson Poisson, structures and their normal forms, Progress in Mathematics 242 (2005) (Basel: Birkhäuser Verlag)zbMATHGoogle Scholar
  10. 10.
    Etingof P and Varchenko A, Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys. 192(1) (1998) 77–120MathSciNetzbMATHGoogle Scholar
  11. 11.
    Filippov V T, \(n\)-Lie algebras, Sibirsk. Mat. Zh. 26 (1985) 126–140, 191Google Scholar
  12. 12.
    Gautheron P, Some remarks concerning Nambu mechanics, Lett. Math. Phys. 37(1) (1996) 103–116MathSciNetzbMATHGoogle Scholar
  13. 13.
    Grabowski J and Marmo G, On Filippov algebroids and multiplicative Nambu–Poisson structures, Differential Geom. Appl. 12(1) (2000) 35–50MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hagiwara Y, Nambu–Dirac manifolds, J. Phys. A 35(5) (2002) 1263–1281MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hagiwara Y, Nambu–Jacobi structures and Jacobi algebroids, J. Phys. A 37(26) (2004) 6713–6725MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ibáñez R, de León M, Marrero J C and Padrón E, Leibniz algebroid associated with a Nambu–Poisson Structure, J. Phys. A 32(46) (1999) 8129–8144MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ibáñez R, de León M, López B, Marrero J C and Padrón E, Duality and modular class of a Nambu–Poisson structure, J. Phys. A 34(17) (2001) 3623–3650MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jurčo B, Schupp P and Vysoký J, Nambu–Poisson gauge theory, Phys. Lett. B 733 (2014) 221–225zbMATHGoogle Scholar
  19. 19.
    Kosmann-Schwarzbach Y, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41(1–3) (1995) 153–165MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kosmann-Schwarzbach Y, The Lie bialgebroid of a Poisson-Nijenhuis manifold, Lett. Math. Phys. 38(4) (1996) 421–428MathSciNetzbMATHGoogle Scholar
  21. 21.
    Landsman N P, Lie groupoids and Lie algebroids in physics and noncommutative geometry, J. Geom. Phys. 56(1) (2006) 24–54MathSciNetzbMATHGoogle Scholar
  22. 22.
    de León M and Sardon C, Geometric Hamiltonian–Jacobi theory on Nambu–Poisson manifolds, J. Math. Phys. 58(3) (2017) 033508, 15 pp.Google Scholar
  23. 23.
    Liu Z-J, Weinstein A and Xu P, Manin triples for Lie bialgebroids, J. Differential Geom. 45(3) (1997) 547–574MathSciNetzbMATHGoogle Scholar
  24. 24.
    Liu Z-J and Xu P, The local structure of Lie bialgebroids, Lett. Math. Phys. 61(1) (2002) 15–28MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lu J-H, Multiplicative and affine Poisson structures on Lie groups, Ph.D. thesis (1990) (UC Berkeley)Google Scholar
  26. 26.
    Lu J-H and Weinstein A, Poisson Lie Groups, Dressing transformations, and Bruhat decompositions, J. Differential Geom. 31(2) (1990) 501–526MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mackenzie K C H, General Theory of Lie Groupoids and Lie Algebroids (2005) (Cambridge: Cambridge University Press)Google Scholar
  28. 28.
    Mackenzie K C H and Xu P, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73(2) (1994) 415–452MathSciNetzbMATHGoogle Scholar
  29. 29.
    Magri F and Morosi C, A Geometrical characterization of integrable Hamiltonian systems through the theory of Poisson–Nijenhuis manifolds, Quaderno S19 (1984) (University of Milan)Google Scholar
  30. 30.
    Marmo G, Vilasi G and Vinogradov A M, The local structure of \(n\)-Poisson and \(n\)-Jacobi manifolds, J. Geom. Phys. 25(1–2) (1998) 141–182MathSciNetzbMATHGoogle Scholar
  31. 31.
    Mukunda N and Sudarshan E C G, Structure of Dirac bracket in Classical mechanics, J. Math. Phys. 9(3) (1968) 411–417zbMATHGoogle Scholar
  32. 32.
    Nakanishi N, On Nambu–Poisson manifolds, Rev. Math. Phys. 10(4) (1998) 499–510MathSciNetzbMATHGoogle Scholar
  33. 33.
    Nambu Y, Generalized Hamiltonian Dynamics, Phys. Rev. D (3) 7 (1973) 2405–2412MathSciNetzbMATHGoogle Scholar
  34. 34.
    Schupp P and Jurčo B, Nambu-sigma model and branes, in: Proc. of the Corfu Summer Institute 2011, School and Workshops on Elementary Particle Physics and Gravity, September 4–18 (2011), Corfu, Greece, pp. 45–53Google Scholar
  35. 35.
    Takhtajan L, On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160(2) (1994) 295–315MathSciNetzbMATHGoogle Scholar
  36. 36.
    Vaisman I, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118 (1994) (Basel: Birkhäuser Verlag)Google Scholar
  37. 37.
    Vaisman I, Nambu–Lie groups, J. Lie Theory 10(1) (2000) 181–194MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wang S-H, Calculation of Nambu mechanics, J. Comput. Math. 24(3) (2006) 444–450MathSciNetzbMATHGoogle Scholar
  39. 39.
    Wade A, Nambu–Dirac Structures for Lie Algebroids, Lett. Math. Phys. 61(2) (2002) 85–99MathSciNetzbMATHGoogle Scholar
  40. 40.
    Xu P, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200(3) (1999) 545–560MathSciNetzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Stat-Math Unit, Indian Statistical InstituteKolkataIndia
  2. 2.Department of Mathematics and StatisticsIndian Institute of Science Education and ResearchMohanpurIndia

Personalised recommendations