Poisson Reductions of Master Integrable Systems on Doubles of Compact Lie Groups

Abstract

We consider three ‘classical doubles’ of any semisimple, connected and simply connected compact Lie group G: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of ‘master integrable systems’ and investigate their Poisson reductions. In the simplest cotangent bundle case, the reduction is defined by taking quotient by the cotangent lift of the conjugation action of G on itself, and this naturally generalizes to the other two doubles. In each case, we derive explicit formulas for the reduced Poisson structure and equations of motion and find that they are associated with well known classical dynamical r-matrices. Our principal result is that we provide a unified treatment of a large family of reduced systems, which contains new models as well as examples of spin Sutherland and Ruijsenaars–Schneider models that were studied previously. We argue that on generic symplectic leaves of the Poisson quotients the reduced systems are integrable in the degenerate sense, although further work is required to prove this rigorously.

Poisson–Lie analogues of spin Sutherland models revisited

“…Section 5.1 contains the derivation of the dynamical r-matrix form of the reduced Poisson brackets. The result is given by theorem 5.2, which can be considered as an improvement of a previous result found in [21]. In section 5.2, we describe the reduced Hamiltonian vector fields and present a quadrature leading to their integral curves.…”

“…The principal goal of the present paper is to complement and enhance our previous results [19,21] on the structure of Poisson-Lie analogues of those spin Sutherland models that result by reductions of cotangent bundles of semisimple Lie groups via the conjugation action. Here, we consider these models in association with every (connected and simply connected) compact Lie group G having a simple Lie algebra.…”

“…This means that the phase space was initially extended by a dressing orbit of G in B, and then the reduction was defined by setting the relevant B-valued moment map to the identity value. The resulting systems were further investigated in [21] using Poisson reduction, i.e. by directly taking the quotient of the phase space by the action of the symmetry group G. Via restriction to symplectic leaves after reduction, the two methods give the same models.…”

“…In [19,21] we collected heuristic arguments in favour of the degenerate integrability of the reduced systems that descend from the master systems of free motion supported by the Heisenberg doubles, but have not obtained a full proof. The main achievement of this paper is that we will establish in a mathematically exact manner the degenerate integrability of the reduced systems after restriction to a dense open subset of the Poisson quotient.…”

“…Besides the main achievement, the analyses of [19,21] will be further developed in several other respects as well. For example, we shall present two useful, alternative descriptions of the reduced Poisson brackets.…”