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:
- Poisson manifolds:
- Symplectic and Poisson manifolds,
- symplectic stratification and generalized Darboux theorem,
- Schouten brackets and Poisson cohomology,
- Lie-Poisson brackets on the dual of a Lie algebra and symplectic leaves as coadjoint orbits.
- Poisson-Lie groups and Lie bialgebras:
- Lie algebra cohomology, non-homogeneous linear Poisson brackets,
- Poisson-Lie groups, Lie bialgebras, Manin triples,
- coboundary Lie bialgebras and Poisson-Lie groups, classical Yang-Baxter equation.
- Relation with quantum groups:
- Hopf algebras, examples of quantized function algebras and quantum enveloping algebras, semiclassical limit.
- Differential geometric Poisson brackets:
- DGPBs on the line, systems of hydrodynamic type, DGPBs on a lattice and Poisson-Lie groups.
References:
Books:
- Dubrovin, B.A. "Geometry of Hamiltonian evolutionary systems." Monographs and Textbooks in Physical Science. Lecture Notes, 22. Bibliopolis, Naples, 1991. iv+131 pp.
- Olver P. J. "Applications of Lie groups to differential equations". Springer-Verlag (1993).
- Chari V.; Pressley A. "A guide to quantum groups" Cambridge University Press (1994).
Articles:
- Lichnerowicz A. "Les varietes de Poisson et leurs algebres de Lie associees". J. Diff. Geom. 12 (1977) 253.
- Weinstein A. "The local structure of Poisson manifolds". J. Diff. Geom. 18 (1983) 523.
- 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.
- 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.
- Dubrovin, B. A. "Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory." Russian Math. Surv. 44 (1989) 35.
- Dubrovin, B. A. "Differential-geometric Poisson brackets on a lattice." Funct. Anal. Appl. 23 (1989), no. 2, 131--133.
- Weinstein, A.; Lu, J-H. "Poisson-Lie groups, dressing transformations, and Bruhat decompositions." J. Differential Geometry 31 (1990) 501-526.