Global Description of Action-Angle Duality for a Poisson\u2013Lie Deformation of the Trigonometric varvec{mathrm {BC}_n} Sutherland System

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}^nsimeq 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.