On a Poisson–Lie deformation of the BC n Sutherland system

Abstract

A deformation of the classical trigonometric BC n Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson-Lie analogue of the Kazhdan-Kostant-Sternberg type reduction of the free particle on SU(2n) that leads to the BC n Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation parameter, can be recovered (at least on a dense submanifold) as a singular limit of the standard 5-coupling deformation due to van Diejen. Our findings complement and further develop those obtained recently by Marshall on the hyperbolic case by reduction of the Heisenberg double of SU(n, n).

The action–angle dual of an integrable Hamiltonian system of Ruijsenaars–Schneider–van Diejen type

“…Since tr L is gauge invariant, we obtain Φ red 1 as a function of λ, θ if we substitute (3.21) and (3.31). In agreement with [19], let us replace α = e −µ , x = e −v , y = e −u , (3.58) where u, v, µ are real parameters, µ > 0. We shall prove the following…”

“…By now it has become widely known [16,17] that several dual pairs of models arise by applying Hamiltonian reduction to suitable pairs of "free systems" on a higher dimensional master phase space, and, whenever available, this interpretation provides a powerful tool for the analysis of the dual pairs. The term free system is a loose one: a free Hamiltonian induces a complete flow, which often can be written down explicitly, and participates in a large Abelian Poisson algebra invariant under a group of symmetries.The goal of this paper is to exhibit action-angle duality for an integrable Ruijsenaars-Schneider-van Diejen (RSvD) type model derived recently [18,19] by Hamiltonian reduction of the Heisenberg double [20] of the Poisson Lie group SU(2n). The model in question has three free parameters and is a deformation of the trigonometric BC n Sutherland model.…”

“…Its derivation [19] closely followed the analogous reduction [18] of the Heisenberg double of SU(n, n). The papers [18,19] (see also [21,22]) applied Poisson-Lie analogues of the reduction of the cotangent bundle of SU(n, n) that yields the hyperbolic BC n Sutherland model with three arbitrary coupling constants [23]. Other relevant predecessors of the present work are the paper of Pusztai [24], where the action-angle dual of the hyperbolic BC n Sutherland model was constructed by reduction of T * SU(n, n), and its adaptation [25] to the trigonometric case.A key ingredient of every Hamiltonian reduction is the choice of symmetry group, which in the above examples is the group K + × K + with K + = SU(n, n) ∩ SU(2n).…”

“…The pertinent Heisenberg doubles carry two natural (K + × K + )-symmetric free systems, and the previous works investigated reductions of those systems corresponding to geodesic motion. In the present article, we analyse the same reduction of the Heisenberg double of SU(2n) as in [19], but develop a new model of the reduced phase space, wherein it is the other free system whose reduction admits a many-body interpretation. In combination with the earlier results, this allows us to establish action-angle duality between the model treated in [19] and the many-body model that we obtain here.…”

“…In the present article, we analyse the same reduction of the Heisenberg double of SU(2n) as in [19], but develop a new model of the reduced phase space, wherein it is the other free system whose reduction admits a many-body interpretation. In combination with the earlier results, this allows us to establish action-angle duality between the model treated in [19] and the many-body model that we obtain here. The Hamiltonians of this pair of RSvD type models are given in equations (3.59) and (4.6) below, and their duality with one another is discussed in Section 4.As in [18], we adopt the modest aim of finding a model for a dense open subset of the reduced phase space.…”