Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction

TL;DR

This paper extends the Kazhdan-Kostant-Sternberg reduction to a Poisson-Lie setting, deriving the Ruijsenaars-Schneider model via symplectic reduction of free motion on the Heisenberg double of U(n).

Contribution

It introduces a Poisson-Lie generalization of the reduction process, providing a new geometric framework for integrable models like Ruijsenaars-Schneider.

Findings

Derived the Ruijsenaars-Schneider model from Poisson-Lie symmetric free motion.

Established a simple Lax matrix that reduces to the known Ruijsenaars-Schneider Lax matrix.

Demonstrated the effectiveness of symplectic reduction in Poisson-Lie contexts.

Abstract

The trigonometric Ruijsenaars-Schneider model is derived by symplectic reduction of Poisson-Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars-Schneider Lax matrix upon reduction.