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Poisson-Nijenhuis structures. (English) Zbl 0707.58048
A Poisson-Nijenhuis manifold is a manifold equipped with both a Poisson structure, defined by a bivector \({\mathcal P}\) whose Schouten bracket vanishes, and a (1,1)-tensor \({\mathcal N}\) whose Nijenhuis torsion vanishes which satisfy a compatibility condition. One studies the deformation and the dualization of the derivations of the algebra of forms and of the Schouten bracket of multivectors, obtaining generalizations of the differential geometric results together with new proofs. The article comprises the study of the Nijenhuis operators on the twilled Lie algebras, an “N-matrix version” of the Konstant-Symes theorem, and an application to Hamiltonian systems of Toda type on semisimple Lie algebras.
Reviewer: M.Rahula

MSC:
58H15 Deformations of general structures on manifolds
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
51H25 Geometries with differentiable structure
17B99 Lie algebras and Lie superalgebras
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