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**Poisson structures and their normal forms.**
*(English)*
Zbl 1082.53078

Progress in Mathematics 242. Basel: Birkhäuser (ISBN 3-7643-7334-2/hbk). xv, 321 p. (2005).

This book offers a good account on known results about the theory of normal forms of Poisson geometry. Poisson manifolds play a central role in Hamiltonian dynamics and they have been extensively studied in the literature. They form a bridge from commutative geometry to noncommutative geometry. In this work, the authors begin with a self-contained introduction to Poisson geometry and related objects including Lie groupoids and Lie algebroids. The problem of the normal form in Poisson geometry is then studied in details. Books on Poisson geometry are relatively rare. This work is in fact an excellent reference, which contains results non available in other books. The appendix gives finally a collection of discussions, which make the book more self-contained. It contains in particular a good overview of Kontsevich’s formality theorem.

Reviewer: Angela Gammella (Creil)

### MSC:

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

53D10 | Contact manifolds (general theory) |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |