Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of U(n) - Archive ouverte HAL
Article Dans Une Revue Annales Henri Poincaré Année : 2023

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

Résumé

We construct a master dynamical system on a U(n) quasi-Poisson manifold, Md, built from the double U(n) x U(n) and d >= 2 open balls in Cn, whose quasi-Poisson structures are obtained from T *R-n by exponentiation. A pencil of quasi-Poisson bivectors P-z is defined on 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 M-d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T * U(n) x C-nxd. 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 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-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.

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hal-04411998 , version 1 (16-09-2024)

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Maxime Fairon, L. Fehér. Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of U(n). Annales Henri Poincaré, 2023, 24 (10), pp.3461-3529. ⟨10.1007/s00023-023-01344-8⟩. ⟨hal-04411998⟩
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