Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$

Abstract

We construct a master dynamical system on a $U(n)$ quasi-Poisson manifold, ${\mathcal{M}}_d$, built from the double $U(n)\times U(n)$ and $d\ge 2$ open balls in ${\mathbb{C}}^n$, whose quasi-Poisson structures are obtained from $T^{\ast }{\mathbb{R}}^n$ 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 $U(n)$-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 $T^{\ast }U(n)\times {\mathbb{C}}^{n\times d}$. Its commuting Hamiltonians are pullbacks of the class functions on one of the $U(n)$ 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 ${\mathcal{M}}_d/U(n)$ 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 $U(n)$ 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.

arXiv comment56 pages, 2 figures, v2: a typo is removed from equation (4.25)

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