We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group U(n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{U}(n)$$\end{document} in such a way that the invariant functions of the u(n)∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {u}}(n)^*$$\end{document}-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 U(n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{U}(n)$$\end{document} into T∗U(n)\documentclass[12pt]{minimal}
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\begin{document}$$T^*\mathrm{U}(n)$$\end{document}, and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on T∗U(n)\documentclass[12pt]{minimal}
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\begin{document}$$T^* \mathrm{U}(n)$$\end{document} using the conjugation action of U(n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{U}(n)$$\end{document}, 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.
