nLab Poisson cohomology

Idea

For $(X, \pi)$ a Poisson manifold, its Poisson cohomology (Lichnerowicz) is equivalently the Lie algebroid cohomology $H^\bullet(CE(\mathfrak{P}))$ of the corresponding Poisson Lie algebroid $\mathfrak{P}$, hence the cochain cohomology of its Chevalley-Eilenberg algebra $\mathfrak{P}$ (Huebschmann).

Similarly if one generalizes from the the Poisson algebra of functions on a smooth manifold is generalized to any Poisson algebra, the corresponding Poisson cohomology is that of the corresponding Poisson Lie-Rinehart pair.

References

The notion of Poisson cohomology was introduced in

• André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), 253-300 MR0501133 euclid

A textbook reference on Poisson cohomology is section 3.6 (for $H^0$ and $H^1$) and sec. 18.4 (in terms of a cohomology for Lie algebroids) in

• A. Cannas da Silva, Alan Weinstein, Geometric models for noncommutative algebras, Amer. Math. Soc. 1999 (Berkeley Mathematics Lecture Notes) pdf errata

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

The observation that Poisson cohomology is just the Lie algebroid cohomology of the corresponding Poisson Lie algebroid is originally due to 1990 article

• Johannes Huebschmann, Poisson cohomology and quantization J. Reine Angew. Math. 408 (1990), 57-113, MR1058984 doi, reprinted at arXiv:1303.3903

A different flavour is variational Poisson cohomology

• A. De Sole, Victor Kac, Essential variational Poisson cohomology, Comm. Math. Phys. 313 (2012), no. 3, 837-864 arxiv/1106.5882; The variational Poisson cohomology, Jpn. J. Math. 8, (2013), 1-145 arxiv/1106.0082