Abstract
We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of ‘free motion’ on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the ‘reduced main Hamiltonians’ reproduce the spin Sutherland model by keeping only their leading terms. The solutions of the equations of motion emerge from geodesics on the compact Lie group via the standard projection method and possess many first integrals. Similar hyperbolic spin Ruijsenaars–Schneider type models were obtained previously by L.-C. Li using a different method, based on coboundary dynamical Poisson groupoids, but their relation with spin Sutherland models was not discussed.
Highlights
Integrable systems of particles moving in one dimension have been studied intensively for nearly 50 years, beginning with the pioneering papers of Calogero [6], Sutherland [51] and Moser [35]. Thanks to their fascinating mathematics and diverse applications [11,37,38,39,44,52], the interest in these models shows no sign of diminishing
New connections to mathematics and new applications are still coming to light in the current literature, see e.g. [4,7,8,22,24,46,53]
It is natural to expect that the analogous reduction of the Heisenberg double of any compact simple Lie group, along an arbitrary dressing orbit, will lead to a generalization of the spin Sutherland model (1.1)
Summary
Integrable systems of particles moving in one dimension have been studied intensively for nearly 50 years, beginning with the pioneering papers of Calogero [6], Sutherland [51] and Moser [35]. The models can involve a collective spin variable that typically belongs to a coadjoint orbit, and is not assigned separately to the particles An example of this second type is the trigonometric spin Sutherland model defined classically by a Hamiltonian of the following form: HSuth(eiq , p, ξ ). It is natural to expect that the analogous reduction of the Heisenberg double of any compact simple Lie group, along an arbitrary dressing orbit, will lead to a generalization of the spin Sutherland model (1.1). In the SU(n) case, using analytic continuation from trigonometric to hyperbolic functions, our models reproduce the spin RS type equations of motion derived by Braden and Hone [5] from the soliton solutions of An−1 affine Toda theory with imaginary coupling. Further results are the simple derivation of the reduced equations of motion and their solutions in Section 6.1, and the arguments put forward in Section 6.2 that indicate their integrability
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