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arXiv · 1503.07339

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

Francesco Bonechi·Jian Qiu·Marco Tarlini
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Abstract

We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.

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Francesco Bonechi, Jian Qiu, Marco Tarlini. 2015-03-25. Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces. https://arxiv.org/abs/1503.07339

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