Krichever和Zabrodin自旋RS模型的三角实型
Trigonometric real form of the spin RS model of Krichever and Zabrodin
摘要：
我们研究了Krichever和Zabrodin在1995年在运动方程水平上引入的自旋Ruijsenaars-Schneider系统的三角实形式。这一开创性的工作和所有早期的研究的哈密顿解释的系统进行了复杂的全纯设置;了解实形式是一个不平凡的问题。我们解释说，三角实形式出现从一个明显的可积的“自由”系统进行自旋扩展的海森堡双的${\rm U}(n)$泊松李群的哈密顿约化。未约化实相空间${\rm GL}(n,\mathbb{C}) \times \mathbb{C}^{nd}$上的泊松结构是Zakrzewski在1995年发现的$\mathbb{C}^n \simeq \mathbb{R}^{2n}$上的${\rm U}(n)$协变泊松结构的海森堡二重和$d\geq 2$副本的直接乘积。我们通过固定一个群值矩映射到单位元的倍数来约简系统，并详细分析了所得到的约简系统。特别地，我们在约化相空间上导出了三角自旋Ruijsenaars-Schneider系统的哈密顿结构，并证明了它的退化可积性。
英文摘要：
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 ${\rm U}(n)$ Poisson-Lie group. The Poisson structure on the unreduced real phase space ${\rm GL}(n,\mathbb{C}) \times \mathbb{C}^{nd}$ is the direct product of that of the Heisenberg double and $d\geq 2$ copies of a ${\rm U}(n)$ 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.
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