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Reduction of a bi-Hamiltonian hierarchy on \(T^*\mathrm{U}(n)\) to spin Ruijsenaars–Sutherland models

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Abstract

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.

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Notes

  1. This is so despite the fact that the holomorphic hyperbolic/trigonometric RS model is well-understood in more than one reduction approaches [1, 7, 18, 30]. Treating real forms of holomorphic integrable systems is a highly non-trivial task in general.

  2. If not specified otherwise, our spaces of \(C^\infty \)-functions always denote spaces of real functions.

  3. Our calculations do not use the Jacobi identity of the \(m_1\)-related brackets. Hence, the Jacobi identity of the bracket \(\{\ ,\ \}_+^1\) (2.18) follows from that of \(\{\ ,\ \}_+\) (2.15).

  4. Incidentally, one can also work out a direct proof of the closure of \(C^\infty ({\mathfrak {M}})^{\mathrm{U}(n)}\) under \(\{\ ,\ \}_2\) (3.5).

  5. We here use some well-known results about free proper actions; see, e.g. paragraph 6.5 in [36].

  6. A complicated explicit formula for \(b_+(Q,\lambda )\) can be obtained along the lines of Section 5.2 in [13].

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Acknowledgements

I wish to thank I. Marshall for generous help with some calculations. I am also grateful to J. Balog, T.F. Görbe and M. Fairon for useful comments on the manuscript. This research was performed in the framework of the Project GINOP-2.3.2-15-2016-00036 co-financed by the European Regional Development Fund and the budget of Hungary.

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  1. Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, Szeged, 6720, Hungary

    L. Fehér

  2. Department of Theoretical Physics, WIGNER RCP, RMKI, P.O.B. 49, Budapest, 1525, Hungary

    L. Fehér

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Fehér, L. Reduction of a bi-Hamiltonian hierarchy on \(T^*\mathrm{U}(n)\) to spin Ruijsenaars–Sutherland models. Lett Math Phys 110, 1057–1079 (2020). https://doi.org/10.1007/s11005-019-01252-1

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