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Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?

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arxiv hep-th/9602160 v2 pith:PPPH627N submitted 1996-02-28 hep-th nlin.SIsolv-int

Yuri B. Suris (Bremen , Germany) This is my paper · ORCID
classification hep-thnlin.SIsolv-int
keywords bracketgovernedhierarchieshyperbolicmatrixmodelspoissonrational
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abstract

We demonstrate that in a certain gauge the Lax matrices of the rational and hyperbolic Ruijsenaars--Schneider models have a quadratic $r$-matrix Poisson bracket which is an exact quadratization of the linear $r$--matrix Poisson bracket of the Calogero--Moser models. This phenomenon is explained by a geometric derivation of Lax equations for arbitrary flows of both hierarchies, which turn out to be governed by the same dynamical $R$--operator.

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  1. Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models

    math-ph 2019-08 conditional novelty 7.0 of 10

    The trigonometric spin Ruijsenaars-Sutherland hierarchy is obtained by Poisson reduction of a bi-Hamiltonian free system on T*U(n), yielding explicit compatible reduced Poisson brackets.

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