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The Lie bialgebroid of a Poisson-Nijenhuis manifold. (English) Zbl 1005.53060
The authors prove that a Poisson structure and a Nijenhuis structure constitute a Poisson-Nijenhuis structure if and only if the cotangent and the tangent bundles are a Lie bialgebroid when equipped, respectively, with the bracket of 1-forms defined by the Poisson tensor \(P\) and with the deformed bracket of vector fields defined by the Nijenhuis tensor \(N\).
The authors also give a review of the notions of Lie algebroid and Lie bialgebroid.

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
17B63 Poisson algebras
58H05 Pseudogroups and differentiable groupoids
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