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.
Highlights
Integrable Hamiltonian systems have important applications in diverse fields of physics and are in the focus of intense investigation by a great variety of mathematical methods
We are interested in the family of classical many-body systems introduced in their simplest form by Calogero [2], Sutherland [38] and Ruijsenaars and Schneider [35]. The relevance of these systems to numerous areas of mathematics and physics is apparent from the reviews devoted to them [4,21,22,23,31,34,39,42]
Before outlining the content of the paper, let us recall from [8,13] the local description of our many-body systems in duality, which arises by restricting attention to dense open submanifolds of the reduced phase space
Summary
Integrable Hamiltonian systems have important applications in diverse fields of physics and are in the focus of intense investigation by a great variety of mathematical methods. The goal of this paper is to present a thorough analysis of a dual pair of integrable many-body systems recently derived in [8,13] by reduction of the Heisenberg double of the standard Poisson–Lie group SU(2n). Before outlining the content of the paper, let us recall from [8,13] the local description of our many-body systems in duality, which arises by restricting attention to dense open submanifolds of the reduced phase space. These systems have 3 real parameters, μ > 0 and u and v, whose range will be specified below. The first one is purely technical, while in the second we clarify the connection between the Hamiltonian H (1.11) and van Diejen’s five parametric integrable trigonometric Hamiltonians
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