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Definition

Poisson manifold

A smooth manifold whose smooth functions carry a Lie bracket that is a derivation in each argument.

Definition

Let MM be a . A Poisson structure on MM is a bilinear operation on its ,

{−,−}:C∞(M)×C∞(M)⟶C∞(M)\{-,-\}:C^\infty(M)\times C^\infty(M)\longrightarrow C^\infty(M)

that is skew-symmetric, satisfies the Jacobi identity, and obeys the Leibniz rule {f,gh}={f,g}h+g{f,h}\{f,gh\}=\{f,g\}h+g\{f,h\}. A Poisson manifold is a pair (M,{−,−})(M,\{-,-\}). Equivalently, the bracket is determined by a smooth bivector field π∈Γ(∧2TM)\pi\in\Gamma(\wedge^2 TM) through {f,g}=π(df,dg)\{f,g\}=\pi(df,dg), and the Jacobi identity is equivalent to the vanishing of the Schouten bracket [π,π][\pi,\pi].

Hamiltonian vector fields and leaves

For f∈C∞(M)f\in C^\infty(M), the derivation g↦{f,g}g\mapsto\{f,g\} is represented by a XfX_f, called the of ff. At each point, the map π♯:T∗M→TM\pi^\sharp:T^*M\to TM defined by

β(π♯α)=π(α,β)\beta(\pi^\sharp\alpha)=\pi(\alpha,\beta)

has image tangent to a canonical, generally singular foliation. Each leaf carries a symplectic form, and its dimension is the even of π\pi. The construction and its integrability are developed in Vaisman, Chapters 1–2.

Relationship to symplectic and Lie theory

Every determines a Poisson structure by inverting its nondegenerate 22-form, subject to the stated sign convention. Poisson manifolds allow the rank to vary and therefore include genuinely degenerate examples. The dual g∗\mathfrak g^* of a finite-dimensional has the linear Lie–Poisson bracket, characterized on linear functions by the of g\mathfrak g. Thus Poisson geometry simultaneously extends symplectic geometry and records infinitesimal Lie-theoretic data.

Conventions and scope
References
  1. Izu Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118, Birkhäuser, 1994. Publisher record. Relevant: Chapters 1–3.
  2. Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke, Poisson Structures, Grundlehren der mathematischen Wissenschaften 347, Springer, 2013. Publisher record. Relevant: Chapter 1.