Remarks on the Lichnerowicz-Poisson cohomology

Izu Vaisman

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 4, page 951-963
  • ISSN: 0373-0956

Abstract

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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-Hochshild 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.

How to cite

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Vaisman, Izu. "Remarks on the Lichnerowicz-Poisson cohomology." Annales de l'institut Fourier 40.4 (1990): 951-963. <http://eudml.org/doc/74907>.

@article{Vaisman1990,
abstract = {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-Hochshild 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.},
author = {Vaisman, Izu},
journal = {Annales de l'institut Fourier},
keywords = {Lichnerowicz-Poisson cohomology; Serre-Hochschild spectral sequence; Poisson manifold},
language = {eng},
number = {4},
pages = {951-963},
publisher = {Association des Annales de l'Institut Fourier},
title = {Remarks on the Lichnerowicz-Poisson cohomology},
url = {http://eudml.org/doc/74907},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Vaisman, Izu
TI - Remarks on the Lichnerowicz-Poisson cohomology
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 4
SP - 951
EP - 963
AB - 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-Hochshild 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.
LA - eng
KW - Lichnerowicz-Poisson cohomology; Serre-Hochschild spectral sequence; Poisson manifold
UR - http://eudml.org/doc/74907
ER -

References

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  10. [LT] J. A. A. LÓPEZ and Ph. TONDEUR, Hodge decomposition along the leaves of a Riemannian foliation, Preprint, Urbana-Illinois, 1989. 
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  14. [X] P. XU, Poisson cohomology of regular Poisson manifolds, Preprint, Berkeley, 1990. 

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