Poisson Structures: Basic Definitions

Abstract

In this chapter we give the basic definitions of a Poisson algebra, of a Poisson variety, of a Poisson manifold and of a Poisson morphism. A Poisson algebra is a (typically infinite-dimensional) vector space equipped with a commutative, associative product and a Lie bracket; these two structures are demanded to be compatible. This definition is easily transported to affine varieties, considering as vector space its algebra of regular functions: thus, an affine Poisson variety consists of an affine variety, with a compatible Lie bracket on its algebra of functions. For real or complex manifolds, it is more natural to start out from a bivector field on the manifold and demand that it induces on local functions a Lie algebra structure; the bivector character is tantamount to the compatiblity between the two algebra structures on local functions. We treat the case of affine Poisson varieties and of Poisson manifolds separately; as we will show, Poisson varieties and Poisson manifolds can be treated uniformly up to some point, but quickly the techniques and results diverge, past this point. We prove Weinstein’s splitting theorem, which yields both the local and global structure of a (real or complex) Poisson manifold. At the end of the chapter, we specialize some of the results to the case of Poisson brackets on the algebra of polynomial, smooth or holomorphic functions on a finite-dimensional vector space.