Poisson Structures

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.

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.

The algebraic structure of the space of differential forms on a manifold is well-known; the algebraic structure on the space of multi-vector fields may be less well-known. We present both here, together with the algebraic counterpart of these spaces, which are respectively the spaces of Kähler forms and of skew-symmetric multi-derivations of an algebra. It leads to the Gerstenhaber structure on the space of multi-vector fields of a given manifold (or multi-derivations of a given algebra), a space in which the Poisson structure lives and which acts on the space of differential forms on the manifold (Kähler forms of the algebra), via the generalized Lie derivative.

A Poisson bracket on a commutative associative algebra , or a Poisson structure on a manifold M, leads in a natural way to cohomology spaces, derived from the multi-derivations of (multivector fields on M), and to homology spaces, derived from the Kähler differentials of (differential forms on M). These spaces give information on the derivations, normal forms, deformations and several invariants of the Poisson structure. In some specific, but important, cases they are related to classically known cohomology spaces, like de Rham cohomology or Lie algebra cohomology, as will be shown in subsequent chapters. In general, Poisson cohomology is finer, but is also more difficult to compute. We construct in this chapter the various complexes which lead to these homologies and cohomologies, we describe a few natural operations in Poisson cohomology and homology and we show that the Poisson cohomology and homology spaces of a Poisson manifold are, under certain conditions, isomorphic to each other. We describe in particular the modular class, which is a cohomology class associated to the Poisson structure; its vanishing is shown to imply the existence of an isomorphism between Poisson homology and cohomology.

We explain in this chapter a few more advanced constructions, which are related to the general concept of reduction. Roughly speaking, reduction means that the object under study (here a Poisson structure), is replaced by an object of the same type, but on a manifold of smaller dimension, in classical terms “with less degrees of freedom”. Poisson reduction deals with Poisson structures on quotients of Poisson manifolds or of coisotropic submanifolds of Poisson manifolds. Poisson–Dirac reduction is concerned with Poisson structures on submanifolds, which are not necessarily Poisson submanifolds. As examples and applications, we consider the transverse Poisson structure to an arbitrary symplectic leaf of a Poisson manifold, fixed point sets and quotient spaces of groups acting on Poisson varieties/manifolds and the reduced space of the momentum map, associated to the action. We also describe each of the constructions in the corresponding algebraic context.

Symplectic manifolds appear as phase spaces of many mechanical systems and are as such, historically, the first examples of Poisson manifolds. They can be characterized as Poisson manifolds whose rank is constant and equal to the dimension of the manifold. Since the rank of their Poisson structure is constant, symplectic manifolds are regular Poisson manifolds. Locally their Poisson structure looks like the standard Poisson structure, i.e., locally they are constant Poisson structures. Important examples of symplectic manifolds which are discussed include Kähler manifolds, cotangent bundles and quotients of symplectic vector spaces.

Together with symplectic manifolds, considered in the previous chapter, Lie algebras provide the first examples of Poisson manifolds. The dual \(\mathfrak{g}^{*}\) of a finite-dimensional Lie algebra \(\mathfrak{g}\) admits a natural Poisson structure, called its Lie–Poisson structure. It is a linear Poisson structure and every linear Poisson structure (on a finite-dimensional vector space) is a Lie–Poisson structure. We show that the leaves of the symplectic foliation are the coadjoint orbits of the adjoint group of \(\mathfrak{g}\) and we shortly discuss the linearization of Poisson structures (in the neighborhood of a point where the rank is zero). Using a non-degenerate Ad-invariant symmetric bilinear form, we get the Lie-Poisson structure on \(\mathfrak{g}\), which has several virtues, amongst which the fact that the Hamiltonian vector fields on \(\mathfrak{g}\) take a natural form, a so-called Lax form. Affine Poisson structures and their Lie theoretical interpretation are discussed at the end of the chapter.

In this chapter we consider weight homogeneous Poisson structures, the simplest of which (homogeneous Poisson structures of degree 0 or 1) have been considered in the previous chapters. Other distinguished classes of weight homogeneous Poisson structures, considered in this chapter, are quadratic Poisson structures (for which a partial classification is given, with the help of the modular vector field), rank two Poisson structures arizing from weight homogeneous Nambu–Poisson structures and the transverse Poisson structures to adjoint orbits in a semi-simple Lie algebra.

In the two-dimensional case the Jacobi identity is trivially satisfied, so it is easy to describe all Poisson structures on, for example, the affine plane. Yet, the local classification of Poisson structures in two dimensions is non-trivial, and has up to now only been accomplished under quite strong regularity assumptions on the singular locus of the Poisson structure, which can be identified with the zero locus of a local function on the manifold. This classification will be treated in detail in the first part of this chapter.

In the three-dimensional case, the Jacobi identity can be stated as the integrability condition of a distribution, which eventually leads to the symplectic foliation, or as the integrability condition of a differential one-form, dual to the Poisson structure with respect to a volume form (assuming that the manifold is orientable). Many results about three-dimensional Poisson manifolds are essentially true because the Poisson structure is of rank two (or zero). We therefore also present some general results about rank two Poisson structures. Combining dimensions two and three, we discuss Poisson surfaces which have a Du Val singularity at the origin.

An R-matrix or an r-matrix is an additional structure on a Lie algebra, which yields a new Lie bracket on the Lie algebra or on its dual, hence leading to a Poisson structure on the dual Lie algebra or on the Lie algebra itself (in that order). Both constructions are discussed, as well as their relation when \(\mathfrak{g}\) is a quadratic Lie algebra, i.e., when \(\mathfrak{g}\) is equipped with a non-degenerate symmetric bilinear form, which is ad-invariant. We also show how R or r-matrices lead in the case of an associative Lie algebra to quadratic and cubic Poisson structures and how all these Poisson brackets are related via Lie derivatives. The connection with Poisson–Lie groups is discussed in the next chapter.

In this chapter we discuss Poisson–Lie groups and their infinitesimal counterparts Lie bialgebras. A Poisson–Lie group is a Lie group G which is equipped with a Poisson structure π, having the property that the product map on G is a morphism of Poisson manifolds. A Lie bialgebra is a Lie algebra which is at the same time a Lie coalgebra, the algebra and coalgebra structures satisfying some compatibility relation. We show that there is a functor which associates to every Poisson–Lie group a Lie bialgebra and that every finite-dimensional Lie bialgebra is the Lie bialgebra of some Poisson–Lie group, which can be chosen to be connected and simply connected. Using dressing actions, we shortly discuss the symplectic leaves of Poisson–Lie groups.

The present chapter deals with the main application of Poisson structures: the theory of integrable Hamiltonian systems. We give the basic definitions and properties of functions in involution and of the momentum map, associated to them. We also give several constructions of functions in involution: Poisson’s theorem, the Hamiltonian form of Noether’s theorem, bi-Hamiltonian vector fields, Thimm’s method, Lax equations and the Adler–Kostant–Symes theorem. Liouville’s theorem and the action-angle theorem, which are classically known for integrable systems on symplectic manifolds, are presented here in the context of general Poisson manifolds.

In this chapter we present a second application of Poisson structures: the theory of deformations of commutative associative algebras, also called deformation quantization. By definition, it consists of considering for a given Poisson manifold (M,{⋅,⋅}) all associative products ⋆ on the vector space C ^∞(M)[[ν]], which are of the form Kontsevich’s formality theorem implies that for every Poisson manifold a deformation quantization exists. Its full proof is beyond the scope of this book, but we provide the material which allows the reader to understand and appreciate Kontsevich’s formality theorem and its farreaching consequences. Thus we treat in detail the relation between deformations and cohomology (both in the context of associative products and Poisson structures), and differential graded Lie algebras, covering partly the case of L _∞-morphisms of differential graded Lie algebras.