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A bi-Hamiltonian system on the Grassmannian

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Abstract

Considering the recent result that the Poisson–Nijenhuis geometry corresponds to the quantization of the symplectic groupoid integrating a Poisson manifold, we discuss the Poisson–Nijenhuis structure on the Grassmannian defined by the compatible Kirillov–Kostant–Souriau and Bruhat–Poisson structures. The eigenvalues of the Nijenhuis tensor are Gelfand–Tsetlin variables, which, as was proved, are also in involution with respect to the Bruhat–Poisson structure. Moreover, we show that the Stiefel bundle on the Grassmannian admits a bi-Hamiltonian structure.

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Correspondence to F. Bonechi.

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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 1, pp. 3–14, October, 2016.

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Bonechi, F., Qiu, J. & Tarlini, M. A bi-Hamiltonian system on the Grassmannian. Theor Math Phys 189, 1401–1410 (2016). https://doi.org/10.1134/S0040577916100019

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