A bi-Hamiltonian system on the Grassmannian

F. Bonechi^a, J. Qiu^b, M. Tarlini<sup>a

a</sup> National Institute of Nuclear Physics, Sezione di Firenze, Firenze, Italy
^b Uppsala University, Department of Mathematics, Uppsala, Sweden

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.

Keywords: symplectic geometry, integrable system, Poisson–Nijenhuis geometry, Poisson manifold quantization, symplectic groupoid.

PACS: 02.40.-k

MSC: 53Dxx

DOI: 10.4213/tmf9087

 English version: Theoretical and Mathematical Physics, 2016, 189:1, 1401–1410

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