nLab
Poisson cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Symplectic geometry

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

For (X,π)(X, \pi) a Poisson manifold, its Poisson cohomology (Lichnerowicz) is equivalently the Lie algebroid cohomology H (CE(𝔓))H^\bullet(CE(\mathfrak{P})) of the corresponding Poisson Lie algebroid 𝔓\mathfrak{P}, hence the cochain cohomology of its Chevalley-Eilenberg algebra 𝔓\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 0H^0 and H 1H^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.