Poisson-Lie analogues of spin Sutherland models

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 Integrable systems of particles moving in one dimension have been studied intensively for nearly 50 years, beginning with the pioneering papers of Calogero [5], Sutherland [51] and Moser [35]. Thanks to their fascinating mathematics and diverse applications [10,37,38,44,52], 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. [6,7,22,24,46,53].The richness of these models is also due to their many generalizations and deformations. These are associated with different interaction potentials (from rational to elliptic), root systems and extensions with internal degrees of freedom. We call 'Sutherland models' the systems defined by trigonometric or hyperbolic potentials. For all these systems, classical and quantum mechanical versions are studied separately, and one needs to pay attention to the distinct features of the systems with real particle positions and their complexifications. The investigations of Ruijsenaars-Schneider (RS) type deformations [44,45] is motivated, for example, by relations to solitons, spin chains, special functions and double affine Hecke algebras.The internal degrees of freedom are colloquially called 'spin', and can be of two rather different kinds. First, the point particles can carry spins varying in a vector space, as is the case for the Gibbons-Hermsen models [20] and their RS type generalizations introduced by Krichever and Zabrodin [28]. Second, 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:

Reduction of a bi-Hamiltonian hierarchy on $$T^*\mathrm{U}(n)$$ to spin Ruijsenaars–Sutherland models

“…We now briefly discuss another change of variables, which is suited for the second reduced whereby every function f (Q, L) is represented by a function F (Q, p, λ). It can be shown (both by direct calculation or by applying Theorem 4.3 of [13]) that the reduced second Poisson bracket acquires the following decoupled form in terms of the new variables:…”

“…The proper analogues of Proposition 4.6 and Proposition 4.7 hold for the reduction of a suitable free system on the Heisenberg double of any compact simple Lie group as well [13], but (at least at present) we do not have a bi-Hamiltonian structure in such general case.…”

“…See, for example, the reviews [29,36,46] and references therein. The spin extensions of these models [19,47,24] are also important, and are currently subject to intense studies [2,8,13,14,21,33,34,38].…”

“…This explains how the spinless RS model appears on a symplectic leaf of the Ruijsenaars-Sutherland hierarchy with respect to its second Poisson structure. The reader may consult [13], too, where, we studied symplectic reductions of Heisenberg doubles at arbitrary moment map values; but without dealing with any bi-Hamiltonian aspect.…”

“…A complicated explicit formula for b + (Q, λ) can be obtained along the lines of Section 5.2 in[13].…”