On a Poisson-Lie deformation of the BC(n) Sutherland system

TL;DR

This paper introduces a new Poisson-Lie deformation of the BC(n) Sutherland system, proving its integrability and connecting it to known deformations via a limiting process, thus advancing the understanding of integrable many-body systems.

Contribution

It constructs a novel Poisson-Lie deformation of the BC(n) Sutherland system through Hamiltonian reduction, establishing its integrability and relation to existing models.

Findings

The deformed system is Liouville integrable.

A globally valid phase space model is constructed.

The system can be obtained as a limit of van Diejen's deformation.

Abstract

A deformation of the classical trigonometric BC(n) Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson-Lie analogue of the Kazhdan-Kostant-Sternberg type reduction of the free particle on SU(2n) that leads to the BC(n) Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation parameter, can be recovered (at least on a dense submanifold) as a singular limit of the standard 5-coupling deformation due to van Diejen. Our findings complement and further develop those obtained recently by Marshall on the hyperbolic case by reduction of the Heisenberg double of SU(n,n).