# nLab Poisson cohomology

Contents

cohomology

### Theorems

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

## Classical mechanics and quantization

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## 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

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

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

Last revised on March 20, 2013 at 23:02:04. See the history of this page for a list of all contributions to it.