Poisson-Lie groups and Hamiltonian structures for integrable systems

Guido Carlet

The aim of this course is to give an introduction to Poisson manifolds, Poisson-Lie groups and R-matrix theory.


  1. Poisson manifolds:
    1. Symplectic and Poisson manifolds,
    2. symplectic stratification and generalized Darboux theorem,
    3. Schouten brackets and Poisson cohomology,
    4. Lie-Poisson brackets on the dual of a Lie algebra and symplectic leaves as coadjoint orbits.
  2. Poisson-Lie groups and Lie bialgebras:
    1. Lie algebra cohomology, non-homogeneous linear Poisson brackets,
    2. Poisson-Lie groups, Lie bialgebras, Manin triples,
    3. coboundary Lie bialgebras and Poisson-Lie groups, classical Yang-Baxter equation.
  3. Relation with quantum groups:
    1. Hopf algebras, examples of quantized function algebras and quantum enveloping algebras, semiclassical limit.
  4. Differential geometric Poisson brackets:
    1. DGPBs on the line, systems of hydrodynamic type, DGPBs on a lattice and Poisson-Lie groups.



  1. Dubrovin, B.A. "Geometry of Hamiltonian evolutionary systems." Monographs and Textbooks in Physical Science. Lecture Notes, 22. Bibliopolis, Naples, 1991. iv+131 pp.
  2. Olver P. J. "Applications of Lie groups to differential equations". Springer-Verlag (1993).
  3. Chari V.; Pressley A. "A guide to quantum groups" Cambridge University Press (1994).


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