Duality between the trigonometric BC(n) Sutherland system and a completed rational Ruijsenaars-Schneider-van Diejen system

TL;DR

This paper uncovers a new duality between the BC(n) trigonometric Sutherland system and a rational Ruijsenaars-Schneider-van Diejen system, using Hamiltonian reduction and group-theoretic methods.

Contribution

It introduces a novel dual pair linking BC(n) Sutherland and a rational integrable system via Hamiltonian reduction on U(2n).

Findings

Established a duality between BC(n) Sutherland and a rational system.

Generalized previous results on A(n) and hyperbolic BC(n) systems.

Used Hamiltonian reduction on U(2n) to demonstrate the duality.

Abstract

We present a new case of duality between integrable many-body systems, where two systems live on the action-angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide group-theoretic interpretation for several dualities discovered originally by Ruijsenaars. In the group-theoretic framework one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space. Taking the cotangent bundle of U(2n) as the upstairs space, we demonstrate how this mechanism leads to a new dual pair involving the BC(n) trigonometric Sutherland system. Thereby we generalize earlier results pertaining to the A(n) trigonometric Sutherland system as well as a recent work by Pusztai on the hyperbolic BC(n) Sutherland system.