Laurent-Gengoux C., Pichereau A., Vanhaecke P. Poisson Structures

Springer-Verlag, Berlin, Heidelberg, 2013. — 470 p. — (Comprehensive Studies in Mathematics 347). — ISBN: 3642310893.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.Poisson Structures: Basic Definitions
Poisson Structures: Basic Constructions
Multi-Derivations and Kähler Forms
Poisson (Co)Homology
Reduction
Constant Poisson Structures, Regular and Symplectic Manifolds
Linear Poisson Structures and Lie Algebras
Higher Degree Poisson Structures
Poisson Structures in Dimensions Two and Three
R-Brackets and r-Brackets
Poisson–Lie Groups
Liouville Integrable Systems
Deformation Quantization