We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of \(\mathrm{GL}(n,\mathbb {C})\), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard \(\mathrm{GL}(n,\mathbb {C})\) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

中文翻译：

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Spin Sutherland 模型的 Bi-Hamiltonian 结构：全纯情况

我们基于集体自旋变量为全纯自旋萨瑟兰层次结构构建了一个双汉密尔顿结构。该构造依赖于\(\mathrm{GL}(n,\mathbb {C})\)的全纯余切丛上双哈密尔顿结构的泊松约简，它本身来自于正则辛结构和泊松结构标准\(\mathrm{GL}(n,\mathbb {C})\) Poisson–Lie 群的海森堡双倍。在全纯自旋萨瑟兰模型的实切片上恢复先前获得的双曲和三角实型的双哈密顿结构。