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Calculus on Poisson manifolds. (English) Zbl 0611.58002

A Poisson manifold is a differentiable manifold M together with a skewsymmetric contravariant 2-tensor G on M satisfying \([G,G]=0\) (where [, ] denotes the alternating Schouten product [A. Nijenhuis, Indagationes Math. 17, 390-403 (1955; Zbl 0068.150)]. One can define a cohomology operator \(\partial\) on \(\Phi\) (M), the space of skewsymmetric contravariant tensors on M by \(\partial P=[G,P]\) where \(P\in \Phi (M)\). The purpose of this article is to show that this operator \(\partial\) is part of an entire calculus of skewsymmetric contravariant tensors which is dual to the Cartan calculus of forms. The key to this calculus is the possibility of defining a Lie bracket on the space of 1-forms on a Poisson manifold. The relation between this cohomology operator \(\partial\) and the homology operator \(\delta\) defined by J. L. Brylinski (IHES Preprint March 1986) is established.

MSC:

58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis
14F40 de Rham cohomology and algebraic geometry

Citations:

Zbl 0068.150
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