A Lecture on Poisson—Nijenhuis Structures

  • Izu Vaisman
Part of the Progress in Mathematics book series (PM, volume 145)


This is an expository paper. In it, the Poisson—Nijenhuis structures are motivated and defined in the general algebraic framework of Gel’fand and Dorfman. Then, in the particular case of Lie algebroids and differentiable manifolds, the Poisson—Nijenhuis structures are related to the notion of a complementary 2-form, that has been introduced and studied by the author in [20], and several examples of complementary forms and Poisson—Nijenhuis manifolds are given.


Symplectic Manifold Poisson Structure Hamiltonian Structure Poisson Manifold Poisson Bivector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. V. Beltrân and J. Monterde, Poisson—Nijenhuis Structures and the Vinogradov Bracket, Ann. Global Analysis and Geom. 12 (1994), 65–78.zbMATHCrossRefGoogle Scholar
  2. [2]
    R. Brouzet, Systèmes bihamiltoniens et complète intégrabilité en dimension 4,C. R. Acad. Sc. Paris 311–1 (1990) 895–898; thesis, Montpellier, 1991.MathSciNetzbMATHGoogle Scholar
  3. [3]
    R. Brouzet, Feuilletage de Libermann et variétés bihamiltoniennes en dimension 4, Sém. G. Darboux, Montpellier, 1988–89, 97–114.Google Scholar
  4. [4]
    R. Brouzet, P. Molino, F. J. Turiel, Géométrie des systèmes bihamiltoniens, Indag. Math. 4 (3) (1993), 269–296.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, J. Wiley & Sons, New York, 1993.Google Scholar
  6. [6]
    I. M. Gel’fand and I. Ya. Dorfman, The Schouten bracket and Hamiltonian operators, Funct. Anal. Appl. 14 (1980), 223–226.MathSciNetCrossRefGoogle Scholar
  7. [7]
    I. M. Gel’fand and L. A. Dikii, The structure of a Lie algebra in the formal calculus of variations,Funct. Anal. Appl. 10 (1976), 16–22.zbMATHCrossRefGoogle Scholar
  8. [8]
    I. M. Gel’fand and I. Zakharevich, On the local geometry of a bihamiltonian structure, in: The G elf and Mathematical Seminars 1990–1992, (L. Corwin, I. Gelfand, J. Lepowsky, eds.), pp. 51–112, Birkhäuser, Boston-Basel, 1993.CrossRefGoogle Scholar
  9. [9]
    P. Casati, F. Magri and M. Pedroni, Bihamiltonian Manifolds and Sato’s Equations, in: Integrable Systems, The Verdier Memorial Conference. Actes du Colloque International de Luminy, Progress in Math. 115 (O. Babelon, P. Cartier and Y. Kosmann-Schwarzbach, eds.), pp. 251–272, Birkhäuser, Boston, 1993.Google Scholar
  10. [10]
    Y. Kosmann-Schwarzbach, The Lie bialgebroid of a Poisson—Nijenhuis manifold, preprint, Ecole Polytechnique Palaiseau, France, 1996.Google Scholar
  11. [11]
    Y. Kosmann-Schwarzbach and F. Magri, Poisson—Nijenhuis structures, Ann. Inst. H. Poincaré, série A (Physique théorique) 53 (1990), 35–81.MathSciNetzbMATHGoogle Scholar
  12. [12]
    J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in: E. Cartan et les Mathématiques d’Aujourd’hui, Soc. Math, de France, Astérisque, hors série, 1985, 257–271.Google Scholar
  13. [13]
    K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Notes Series 124, Cambridge Univ. Press, Cambridge, 1987.Google Scholar
  14. [14]
    F. Magri and C. Morosi, A geometrical characterization of integrable Hamiltonian Systems through the theory of Poisson—Nijenhuis manifolds, Quaderno S. 19, Univ. of Milan, 1984.Google Scholar
  15. [15]
    J. E. Marsden and T. Ratiu, Reduction of Poisson manifolds, Lett. Math. Physics 11 (1986), 161–169.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    P. J. Olver, Canonical forms for bihamiltonian systems, in: Integrable Systems, The Verdier Memorial Conference, Actes du Colloque de Luminy, Progress in Math. 115 (O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, eds.), pp. 239–249, Birkhâuser, Boston, 1993.Google Scholar
  17. [17]F. J. Turiel, Classification locale d’un couple de formes symplectiques Poisson compatibles, C. R. Acad. Sc. Paris 308–1 (1989), 573–578 et Sém. G. Darboux, Montpellier (1988–89), 49–76.Google Scholar
  18. [18]
    I. Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Math. 118, Birkhâuser, Boston, 1994.Google Scholar
  19. [19]
    I. Vaisman, The Poisson—Nijenhuis manifolds revisited, Rendiconti Sem. Mat. Torino 52 (1994), 377–394.MathSciNetzbMATHGoogle Scholar
  20. [20]
    I. Vaisman, Complementary 2-forms of Poisson structures, Compositio Mathematica 101 (1996), 55–75.MathSciNetzbMATHGoogle Scholar
  21. [21]
    I. Vaisman, On complementary 2-forms of Poisson-Lie groups, Anal. Sci. Univ. Jassy (Mat.) 40 (1994), 417–425.MathSciNetzbMATHGoogle Scholar
  22. [22]
    I. Vaisman, Reduction of Poisson—Nijenhuis manifolds, J. of Geometry and Physics 19 (1996), 90–98.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Izu Vaisman
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of HaifaIsrael

Personalised recommendations