We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group $\mathrm{U}(n)$ in such a way that the invariant functions of the $\mathfrak{u}(n)^*$-valued momenta generate a bi-Hamiltonian hierarchy. One of the Poisson structures is the canonical one and the other one arises from embedding the Heisenberg double of the Poisson-Lie group $\mathrm{U}(n)$ into $T^*\mathrm{U}(n)$, and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ using the conjugation action of $\mathrm{U}(n)$, for which the ring of invariant functions is closed under both Poisson brackets. We demonstrate that the reduced hierarchy belongs to the overlap of well-known trigonometric spin Sutherland and spin Ruijsenaars--Schneider type integrable many-body models, which receive a bi-Hamiltonian interpretation via our treatment.

中文翻译：

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减少 $$T^*\mathrm{U}(n)$$T∗U(n) 上的双汉密尔顿层次结构以旋转 Ruijsenaars-Sutherland 模型

我们首先在酉群 $\mathrm{U}(n)$ 的余切丛上展示两个兼容的泊松结构，使得 $\mathfrak{u}(n)^*$ 值动量的不变函数生成双汉密尔顿层次结构。泊松结构之一是规范结构，另一个来自将泊松-李群 $\mathrm{U}(n)$ 的海森堡双标嵌入 $T^*\mathrm{U}(n)$ 中，随后将嵌入的泊松结构扩展到完整的余切丛。然后，我们使用 $\mathrm{U}(n)$ 的共轭作用将泊松归约应用于 $T^*\mathrm{U}(n)$ 上的双汉密尔顿层次结构，其中不变函数环是闭合的在两个泊松括号下。