Poisson-Lie interpretation of trigonometric Ruijsenaars duality

TL;DR

This paper provides a geometric and Poisson-Lie theoretical interpretation of the duality in the trigonometric Ruijsenaars-Schneider system, simplifying the understanding of its symplectomorphism properties.

Contribution

It introduces a Poisson-Lie framework for the duality, viewing phase spaces as models of a single reduced space from symplectic reduction of the Heisenberg double.

Findings

Dual Hamiltonians descend from dual free Hamiltonian families

Phase spaces are models of the same reduced space

Simplifies Ruijsenaars' proof of symplectomorphism

Abstract

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.