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arxiv: 1503.07339 · v1 · pith:RBWFMFLWnew · submitted 2015-03-25 · 🧮 math.SG · hep-th· math-ph· math.DG· math.MP· math.QA

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

classification 🧮 math.SG hep-thmath-phmath.DGmath.MPmath.QA
keywords definedbruhat-poissoncompactcompleteeigenvalueshermitianintegrabilitynijenhuis
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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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