We derive a Hamiltonian structure for the N-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable Lie group with another symplectic manifold that is a certain deformation of the standard canonical relations for Nℓ conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group GL_ℓ(ℂ), which explains its superintegrability. Our results are obtained in the formalism of the classical r-matrix, and they are compatible with the recent findings on the different Hamiltonian structure of the model established in the framework of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.

中文翻译：

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泊松归约法的双曲自旋Ruijsenaars-Schneider模型

通过合适的初始相空间的泊松约简，我们推导了N粒子双曲自旋Ruijsenaars-Schneider模型的哈密顿结构。该相空间被实现为可分解Lie群的Heisenberg对偶与另一个辛流形的直接乘积，该辛流形是动态变量Nℓ共轭对的标准正则关系的某种变形。我们证明该模型具有自旋群_GLℓ（ℂ）的泊松－李对称性，这说明了其超可积性。我们的研究结果是在经典的形式主义获得[R-矩阵，并且它们与关于在应用于拟泊松流形的拟哈密顿化约简框架中建立的模型的不同哈密顿结构的最新发现兼容。