Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 005, 30 pages      arXiv:0710.3098      http://dx.doi.org/10.3842/SIGMA.2008.005
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Yvette Kosmann-Schwarzbach
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France

Received August 31, 2007, in final form January 02, 2008; Published online January 16, 2008

Abstract
After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.

Key words: Poisson geometry; Poisson cohomology; modular classes; twisted Poisson structures; Lie algebroids; Gerstenhaber algebras; Lie algebroid cohomology; triangular r-matrices; quasi-Frobenius algebras; pure spinors.

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