Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric $\mathrm{BC}_n$ Sutherland system

TL;DR

This paper constructs and analyzes dual integrable many-body systems related to the Poisson-Lie deformation of the BC_n Sutherland system, revealing new global phase space models and duality features.

Contribution

It introduces global models for dual systems derived via Hamiltonian reduction of the Heisenberg double, highlighting their non-trivial duality and deformation properties.

Findings

Global phase space models are described for dual systems.

Action variables generate standard torus actions on ^n.

Dual systems are non-degenerate with maximal conserved quantities.

Abstract

Integrable many-body systems of Ruijsenaars--Schneider--van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space $\mathbb{C}^n\simeq\mathbb{R}^{2n}$ underlies both global models, it is seen that for both systems the action variables generate the standard torus action on $\mathbb{C}^n$, and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system.