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.

Outline:

  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.

References:

Books:

  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).

Articles:

  1. Lichnerowicz A. "Les varietes de Poisson et leurs algebres de Lie associees". J. Diff. Geom. 12 (1977) 253.
  2. Weinstein A. "The local structure of Poisson manifolds". J. Diff. Geom. 18 (1983) 523.
  3. Drinfel'd V.G. "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations". Soviet Math. Dokl. 27 (1983), no. 1, 68--71.
  4. Drinfel'd V.G. "Quantum groups". Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798--820, Amer. Math. Soc., Providence, RI, 1987.
  5. Dubrovin, B. A. "Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory." Russian Math. Surv. 44 (1989) 35.
  6. Dubrovin, B. A. "Differential-geometric Poisson brackets on a lattice." Funct. Anal. Appl. 23 (1989), no. 2, 131--133.
  7. Weinstein, A.; Lu, J-H. "Poisson-Lie groups, dressing transformations, and Bruhat decompositions." J. Differential Geometry 31 (1990) 501-526.