Poisson Structures: Basic Constructions

Abstract

In this chapter we give a few basic, general constructions which allow one to build new Poisson structures from given ones. These constructions are fundamental and will be used throughout the book. First we consider the tensor product of Poisson algebras, which geometrically corresponds to the construction of a Poisson structure on the product of two Poisson manifolds. Then we investigate the notion of a Poisson ideal, whose geometrical counterpart is that of a Poisson submanifold. We also present some other constructions, such as the relations between real and complex Poisson structures, localization and germification of Poisson structures. Throughout the chapter we reformulate all our algebraic constructions in geometrical terms, or, conversely, present the geometrical construction in general algebraic terms.