Poisson-Lie interpretation of a case of the Ruijsenaars duality
Meeting Room 3, CMS
By performing a suitable symplectic reduction of the standard Heisenberg double of the group U(n), we give a geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider model. The reduced phase space is realized in terms of two global cross sections in the inverse image of the moment map value associated with the reduction. Two natural commutative families of U(n) Poisson-Lie symmetric Hamiltonian flows on the double descend upon reduction to the respective commuting flows of the mutually dual models. The reduced flows are automatically complete, and reproduce the original direct completion of the dual flows due to Ruijsenaars.
The talk is based on a forthcoming joint paper with C. Klimcik, which continues arXiv:0809.1509 and arXiv:0901.1983, and will also include a brief discussion of the quantum mechanical version of the construction outlined above.