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 $${\text {U}}(n)$$ 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.

Poisson–Lie analogues of spin Sutherland models revisited

“…Moreover, there appeared interesting applications of the reduction method [23-25, 29, 41, 48] in which the nature of the underlying symmetry had been promoted from Hamiltonian group actions to their generalizations based on Poisson-Lie groups [10,11,62,63] and on quasi-Poisson/quasi-Hamiltonian geometry L Fehér [3,4]. The reviews [6,13,47,52] and the recent papers [8,9,16,17,35,55,56] show that this subject possesses close connections to important areas of physics and mathematics, and enjoys considerable current activity.…”

“…Our proof of degenerate integrability was motivated by ideas that we learned from papers by Reshetikhin [53,54]. It also relies on techniques introduced in our joint work with Fairon [16] and on the recent note [22] dealing with reduced integrability on T * G/G.…”