Poisson structures / Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke

This book is an excellent presentation of Poisson geometry, its applications and related topics.

The first part, theoretical background, contains the basic definitions and constructions of Poisson structures, and the basic tools such as Schouten brackets, Poisson homology and cohomology, reduction, etc. Algebraic techniques play the main role in this exposition. The detailed and well organized introduction to reduction theory is really comprehensive.

The second part collects the most commonly seen situations of Poisson structures: special Poisson structures such as symplectic, linear, homogeneous and Nambu Poisson structures, etc. Among them, the chapter on Poisson-Lie groups is the most difficult one, but also the richest one by the various applications and digressions into some earlier subjects: R-matrices, r-brackets and Lie bialgebras etc.

The third part covers the applications, namely that of integrable systems and the hard problem of deformations quantization. These are simply outlines, not the details of everything. The reader may refer to other texts for full expositions.
This book is suitable for those who have a solid foundation of differential geometry and Lie algebras. The reader will understand why Poisson geometry is such an interesting and important subject. (Zentralblatt)