Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces

• 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.

Subject headings

NATURVETENSKAP  -- Matematik -- Geometri (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Geometry (hsv//eng)

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art (subject category)