Poisson-Lie interpretation of trigonometric Ruijsenaars duality

Understand

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented.

• The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U(n).
• The collections of commuting Hamiltonians of the systems 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.