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

TL;DR

This paper shows that the rational and hyperbolic Ruijsenaars-Schneider models share the same R-operators with Calogero-Moser models, due to a quadratic r-matrix Poisson bracket structure and a common dynamical R-operator.

Contribution

It provides a geometric derivation explaining why these hierarchies are governed by the same R-operators, revealing a unifying structure in their Lax equations.

Findings

Quadratic r-matrix Poisson bracket as an exact quadratization of the linear one.

Shared dynamical R-operator governs the hierarchies.

Geometric derivation of Lax equations for arbitrary flows.

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.