Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization).
The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.

Autorentext

C. Laurent-Gengoux research focus lies on Poisson geometry, Lie-groups and integrable systems. He is the author of 14 research articles. Furthermore, he is committed to teaching and set up several mathematics projects with local high schools. In 2002 he earned his doctorate in mathematics with a dissertation on " Quelques problèmes analytiques et géométriques sur les algèbres et superalgèbres de champs et superchamps de vecteurs.

A. Pichereau earned her doctorate in mathematics with a dissertation on Poisson (co)homology and isolated singularities in low dimensions, with an application in the theory of deformations under the supervision of P. Vanheacke in 2006. She has since published four journal articles on Poisson structures and contributed to the Proceedings of "Algebraic and Geometric Deformation Spaces.

P. Vanheacke's research focus lies on integrable systems, Abelian varieties, Poisson algebra/geometry and deformation theory. In 1991 he earned his doctorate in mathematics with a dissertation on Explicit techniques for studying two-dimensional integrable systems and has published numerous research articles since.

Inhalt

Part I Theoretical Background:1.Poisson Structures: Basic Definitions.- 2.Poisson Structures: Basic Constructions.- 3.Multi-Derivations and Kähler Forms.- 4.Poisson (Co)Homology.- 5.Reduction.- Part II Examples:6.Constant Poisson Structures, Regular and Symplectic Manifolds.- 7.Linear Poisson Structures and Lie Algebras.- 8.Higher Degree Poisson Structures.- 9.Poisson Structures in Dimensions Two and Three.- 10.R-Brackets and r-Brackets.- 11.PoissonLie Groups.- Part III Applications:12.Liouville Integrable Systems.- 13.Deformation Quantization.- A Multilinear Algebra.- B Real and Complex Differential Geometry.- References.- Index.- List of Notations.