Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of {text {U}}(n)

Abstract

We construct a master dynamical system on a {text {U}}(n) quasi-Poisson manifold, {mathcal {M}}_d, built from the double {text {U}}(n) times {text {U}}(n) and dge 2 open balls in mathbb {C}^n, whose quasi-Poisson structures are obtained from T^* mathbb {R}^n by exponentiation. A pencil of quasi-Poisson bivectors P_{underline{z}} is defined on {mathcal {M}}_d that depends on d(d-1)/2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the {text {U}}(n)-invariant functions. The master system on {mathcal {M}}_d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T^*!{text {U}}(n) times mathbb {C}^{ntimes d}. Its commuting Hamiltonians are pullbacks of the class functions on one of the {text {U}}(n) factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space {mathcal {M}}_d/{text {U}}(n) associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors P_{underline{z}}. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of {text {U}}(n) provide a new real form of the complex, trigonometric spin Ruijsenaars–Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the d=1 case.