| 1. |
- Bonechi, F., et al.
(author)
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A bi-Hamiltonian system on the Grassmannian
- 2016
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In: Theoretical and mathematical physics. - 0040-5779 .- 1573-9333. ; 189:1, s. 1401-1410
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Journal article (peer-reviewed)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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| 2. |
- Bonechi, F., et al.
(author)
-
Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces
- 2018
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In: The Journal of Symplectic Geometry. - 1527-5256 .- 1540-2347. ; 16:5, s. 1167-1208
-
Journal article (peer-reviewed)abstract
- Poisson-Nijenhuis (PN) structures have been proven to be relevant for the quantization of Poisson manifolds, through the notion of multiplicative integrable model on the symplectic groupoid. We study in this paper a class of PN structures defined by the compatible Bruhat-Poisson structure and KKS symplectic form on compact hermitian symmetric spaces. We determine the spectrum of the Nijenhuis tensor and prove complete integrability. In the case of Grassmannians, this leads to a bihamiltonian approach to Gelfand-Tsetlin variables. Our results provide a tool for the quantization of the Bruhat-Poisson structure on compact hermitian symmetric spaces.
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| 3. |
- Bonechi, F., et al.
(author)
-
Nijenhuis tensor and invariant polynomials
- 2022
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In: Journal of Geometry and Physics. - : Elsevier. - 0393-0440 .- 1879-1662. ; 180
-
Journal article (peer-reviewed)abstract
- We discuss the diagonalization problem of the Nijenhuis tensor in a class of Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We study its action on the ring of invariant polynomials of a Thimm chain of subalgebras. The existence of phi- minimal representations defines a suitable basis of invariant polynomials that completely solves the diagonalization problem. We prove that such representations exist in the classical cases AIII, BDI, DIII and CI, and do not exist in the exceptional cases EIII and EVII. We discuss a second general construction that in these two cases computes partially the spectrum and hints at a different behavior with respect to the classical cases.(c) 2022 Elsevier B.V. All rights reserved.
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