Poisson (Co)Homology

  • Camille Laurent-Gengoux
  • Anne Pichereau
  • Pol Vanhaecke
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 347)


A Poisson bracket on a commutative associative algebra Open image in new window , or a Poisson structure on a manifold M, leads in a natural way to cohomology spaces, derived from the multi-derivations of Open image in new window (multivector fields on M), and to homology spaces, derived from the Kähler differentials of Open image in new window (differential forms on M). These spaces give information on the derivations, normal forms, deformations and several invariants of the Poisson structure. In some specific, but important, cases they are related to classically known cohomology spaces, like de Rham cohomology or Lie algebra cohomology, as will be shown in subsequent chapters. In general, Poisson cohomology is finer, but is also more difficult to compute. We construct in this chapter the various complexes which lead to these homologies and cohomologies, we describe a few natural operations in Poisson cohomology and homology and we show that the Poisson cohomology and homology spaces of a Poisson manifold are, under certain conditions, isomorphic to each other. We describe in particular the modular class, which is a cohomology class associated to the Poisson structure; its vanishing is shown to imply the existence of an isomorphism between Poisson homology and cohomology.


Poisson Bracket Volume Form Poisson Structure Poisson Algebra Poisson Manifold 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Camille Laurent-Gengoux
    • 1
  • Anne Pichereau
    • 2
  • Pol Vanhaecke
    • 3
  1. 1.CNRS UMR 7122, Laboratoire de MathématiquesUniversité de LorraineMetzFrance
  2. 2.CNRS UMR 5208, Institut Camille JordanUniversité Jean MonnetSaint-EtienneFrance
  3. 3.CNRS UMR 7348, Lab. Mathématiques et ApplicationsUniversité de PoitiersFuturoscope ChasseneuilFrance

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