We suggest a Hamiltonian formulation for the spin Ruijsenaars-Schneider system in the trigonometric case. Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars-Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly.

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关于三角自旋 Ruijsenaars-Schneider 系统的哈密顿公式

我们建议在三角函数情况下为自旋 Ruijsenaars-Schneider 系统使用哈密顿公式。在这种解释中，相空间是通过在（余切丛到）框架约旦箭袋的表示空间上执行的准哈密顿约简获得的。对于任意箭袋，Crawley-Boevey 和 Shaw 引入了类似的变体，Van den Bergh 将它们解释为准汉密尔顿商。使用范登伯格的形式主义，我们在相空间上构造交换哈密顿函数，并使用自旋 Ruijsenaars-Schneider 系统识别流之一。然后我们计算局部坐标之间的所有泊松括号，从而回答了 Arutyunov 和 Frolov 的一个老问题。我们还构建了一套完整的通勤哈密顿量并显式地整合了所有流。