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.
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Acknowledgements
We wish to thank Alexei Rosly and Simon Ruijsenaars for useful discussions. We are also grateful to Tamás Görbe and Gábor Pusztai for comments on the manuscript. L.F. is indebted to Youjin Zhang for hospitality at Tsinghua University during the final stage of the work. This work was supported in part by the Hungarian Scientific Research Fund (OTKA) under the grant K-111697.
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Communicated by Jean-Michel Maillet.
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Fehér, L., Marshall, I. Global Description of Action-Angle Duality for a Poisson–Lie Deformation of the Trigonometric \(\varvec{\mathrm {BC}_n}\) Sutherland System. Ann. Henri Poincaré 20, 1217–1262 (2019). https://doi.org/10.1007/s00023-019-00782-7
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DOI: https://doi.org/10.1007/s00023-019-00782-7