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

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.

LaTeX, 31 pages

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Cited by in corpus (10)

• Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit
• The action-angle dual of an integrable Hamiltonian system of Ruijsenaars--Schneider--van Diejen type
• Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters
• Superintegrability of Calogero-Moser systems associated with the cyclic quiver
• Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${\mathrm{BC}}_n$ Sutherland system
• On a Poisson-Lie deformation of the BC(n) Sutherland system
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system
• Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions
• Stable equilibria for the roots of the symmetric continuous Hahn and Wilson polynomials