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

Authors: Francesco Bonechi, Jian Qiu, Marco Tarlini

Preprint number: UUITP-08/15

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-Cetlin 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.