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

Abstract

AbstractWe construct a master dynamical system on a $${\text {U}}(n)$$ U ( n ) quasi-Poisson manifold, $${\mathcal {M}}_d$$ M d , built from the double $${\text {U}}(n) \times {\text {U}}(n)$$ U ( n ) × U ( n ) and $$d\ge 2$$ d ≥ 2 open balls in $$\mathbb {C}^n$$ C n , whose quasi-Poisson structures are obtained from $$T^* \mathbb {R}^n$$ T ∗ R n by exponentiation. A pencil of quasi-Poisson bivectors $$P_{\underline{z}}$$ P z ̲ is defined on $${\mathcal {M}}_d$$ M d that depends on $$d(d-1)/2$$ d ( d - 1 ) / 2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the $${\text {U}}(n)$$ U ( n ) -invariant functions. The master system on $${\mathcal {M}}_d$$ M d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle $$T^*\!{\text {U}}(n) \times \mathbb {C}^{n\times d}$$ T ∗ U ( n ) × C n × d . Its commuting Hamiltonians are pullbacks of the class functions on one of the $${\text {U}}(n)$$ 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 $${\mathcal {M}}_d/{\text {U}}(n)$$ M d / 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 $$P_{\underline{z}}$$ P z ̲ . The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of $${\text {U}}(n)$$ 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 $$d=1$$ d = 1 case.