Remarks on the Lichnerowicz-Poisson cohomology. (English) Zbl 0708.58010

The paper begins with some general remarks which include the Mayer- Vietoris exact sequence, a covariant version of the Lichnerowicz-Poisson cohomology, and the definition of an associated Serre-Hochschild spectral sequence. Then we consider the regular case, and we discuss the Poisson cohomology by using a natural bigrading of the Lichnerowicz cochain complex. Furthermore, if the symplectic foliation of the Poisson manifold is either transversally Riemannian or transversally symplectic, the spectral sequence mentioned above is determined by the leafwise cohomologies of the foliation, and if, moreover, the Poisson structure is transversally constant, the spectral sequence defines the Lichnerowicz- Poisson cohomology in a straightforward manner.
Reviewer: I.Vaisman


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58A12 de Rham theory in global analysis
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