3 years ago

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

L. Feher, I. Marshall

We continue the recent studies of Ruijsenaars--Schneider--van Diejen type integrable many-body systems in action-angle duality derived by Hamiltonian reduction of the Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. In particular, we complete the local description developed in our previous paper into a global model of the reduced phase space. In combination with its dual global model found in an earlier work, this reveals non-trivial features of the two systems in duality with one another. For example, we prove that for both systems the action variables generate the standard torus action on the symplectic vector space $\mathbb{C}^n\simeq \mathbb{R}^{2n}$, which is demonstrated to underlie the global models. The fixed point given by the origin corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are shown 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. We also clarify the relation of the dual of the deformed Sutherland system to the systems of van~Diejen.

Publisher URL: http://arxiv.org/abs/1710.08760

DOI: arXiv:1710.08760v1

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