We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable ‘free’ system carried by a spin extension of the Heisenberg double of the \({\mathrm{U}}(n)\) Poisson–Lie group. The Poisson structure on the unreduced real phase space \({\mathrm{GL}}(n,{\mathbb {C}})\times {\mathbb {C}}^{nd}\) is the direct product of that of the Heisenberg double and \(d\ge 2\) copies of a \({\mathrm{U}}(n)\) covariant Poisson structure on \({\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}\) found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars–Schneider system and we prove its degenerate integrability.

我们研究Krichever和Zabrodin在1995年在运动方程一级引入的自旋Ruijsenaars-Schneider系统的三角实形式。这项开创性的工作和对系统的哈密顿解释的所有较早研究都是在复杂的全纯环境中进行的。 ; 了解真实形式是一个不平凡的问题。我们解释的三角真正形式从哈密顿减少由海森堡两倍的旋转扩展携带的明显积“免费”系统的出现\（{\ mathrm【U}}（N）\）泊松李群。未经归约的实相空间\（{\ mathrm {GL}}（n，{\ mathbb {C}}）\乘以{\ mathbb {C}} ^ {nd} \）上的泊松结构是的海森堡双和\（d \ ge 2 \）Zakrzewski找到的\（{\ mathbb {C}} ^ n \ simeq {\ mathbb {R}} ^ {2n} \）上\（{\ mathrm {U}}（n）\）协变Poisson结构的副本，同样是在1995年。我们通过将群值矩图固定到身份的倍数来进行约简，并详细分析所得的约简系统。特别是，我们在减少的相空间上推导了三角自旋Ruijsenaars-Schneider系统的哈密顿结构，并证明了它的简并可积性。