论文标题
Krichever和Zabrodin的旋转RS模型的三角实体形式
Trigonometric real form of the spin RS model of Krichever and Zabrodin
论文作者
论文摘要
我们研究了1995年Krichever和Zabrodin在运动方程级别引入的自旋Ruijsenaars-Schneider系统的三角实体形式。这项开创性的工作以及对汉密尔顿对系统解释的所有早期研究均在复杂的Holomorphic Andings中进行；了解真实形式是一个非平凡的问题。我们解释说，三角实体形式来自哈密顿的减少，这是由于$ {\ rm u}（n）$ poisson $ poisson-lie group的Heisenberg double的旋转扩展所带来的明显集成的“自由”系统。未还原的真实相空间上的泊松结构$ {\ rm gl}（n，\ mathbb {c}）\ times \ times \ mathbb {c}^{nd} $是Heisenberg double和$ d \ geq 2 $ cop of a $ {\ rm um u} $ covar的直接产物$ \ mathbb {c}^n \ simeq \ mathbb {r}^{2n} $由Zakrzewski找到，也是1995年。我们通过将值的算置映射固定到该身份的多个身份，并详细分析了所得减少的系统。特别是，我们在减少的相空间上得出了三角旋转ruijsenaars-Schneider系统的哈密顿结构，并证明了其退化的集成性。
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.
