Poisson-Lie Interpretation of Trigonometric Ruijsenaars Duality

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider model is presented. The phase spaces of the models in duality are realized as two different gauge slices in the same inverse image of the moment map defining a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the models in duality are shown to descend from two families of ‘free’ Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars’ proof of the crucial symplectomorphism property of the duality map.