Poisson CohomologyEdit
Poisson cohomology is a cohomological framework in Poisson geometry that captures the infinitesimal deformations and intrinsic invariants of a Poisson structure on a smooth manifold. It sits at the crossroads of differential geometry, algebraic topology, and mathematical physics, providing a natural tool to study how a Poisson bracket can be varied, constrained, or classified. At its core, Poisson cohomology is built from the graded algebra of multivector fields on a manifold together with a differential determined by the Poisson bivector, and it interacts richly with related notions such as deformations, symplectic geometry, and the theory of Lie algebroids.
The subject is named for its origin in Poisson geometry, a generalization of classical Hamiltonian mechanics where a bivector field induces a Poisson bracket on smooth functions. The resulting cohomology theory, denoted H^*(M, π) for a Poisson manifold (M, π), records obstructions to extending infinitesimal deformations of π and illuminates the global structure of Hamiltonian dynamics. For readers who want a precise entry point, the Poisson cohomology groups H^k(M, π) arise from a differential on the graded space of polyvector fields, and the theory interfaces with several standard constructions in differential geometry, including the Schouten–Nijenhuis bracket and Lie algebroid cohomology.
Formalism
Poisson manifolds and the Schouten–Nijenhuis bracket
A Poisson manifold is a pair (M, π) where M is a smooth manifold and π ∈ Γ(∧^2 TM) is a bivector field whose Schouten–Nijenhuis bracket with itself vanishes: [π, π] = 0. The bivector π encodes a bracket {f, g} = ⟨df ∧ dg, π⟩ on smooth functions, turning C∞(M) into a Lie algebra and giving M a rich geometric structure. The relevant algebraic gadget here is the graded Lie algebra of multivector fields ⊕_k Γ(∧^k TM) equipped with the Schouten–Nijenhuis bracket, which extends the usual Lie bracket of vector fields to higher-degree objects. For a detailed treatment, see Schouten–Nijenhuis bracket.
The Poisson differential and the cochain complex
The Poisson bivector π defines a differential d_π on the graded space of multivector fields by d_π(α) = [π, α], where α ∈ Γ(∧^k TM). Because [π, π] = 0, the operator d_π satisfies d_π^2 = 0, turning the space of multivector fields into a cochain complex. The corresponding cohomology H^k(M, π) is the k-th Poisson cohomology group. This construction can be viewed from the perspective of Lie algebroids: the cotangent bundle T^*M inherits a Lie algebroid structure induced by π, and the cohomology of this Lie algebroid recovers the same groups. See Lie algebroid for context and Poisson manifold for the base object.
Relation to deformation theory
Poisson cohomology is intimately connected to deformation theory. Infinitesimal deformations of the Poisson structure π are parametrized by H^2(M, π), while H^1(M, π) controls infinitesimal automorphisms of π and H^3(M, π) carries obstruction information. In this way, Poisson cohomology serves as the right cohomological language to address questions about how a Poisson structure can be varied within its moduli space. For broader context on deformations and their algebraic aspects, see Deformation quantization and Kontsevich formality theorem.
The symplectic (nondegenerate) case
When π is nondegenerate, i.e., M carries a symplectic form ω with π = ω^{-1}, Poisson cohomology simplifies in a striking way. In the symplectic case, Poisson cohomology is canonically related to de Rham cohomology via the natural isomorphism induced by ω. Consequently, the Poisson cohomology groups reflect the familiar topological information of the underlying manifold, reframed through the Poisson structure. See de Rham cohomology for the parallel topological viewpoint and Symplectic geometry for the broader setting.
Examples and computations
Symplectic manifolds: If (M, ω) is symplectic, the Poisson cohomology H^k(M, π) is closely tied to the de Rham cohomology of M, and in many cases there is a direct isomorphism (up to natural degree considerations) that translates Poisson-theoretic questions into differential-topological ones. See de Rham cohomology and Symplectic geometry.
Low-dimensional Poisson structures: In dimension two or three, explicit computations often illustrate the way singularities of π (where rank drops) influence the cohomology. These examples highlight that Poisson cohomology can detect global geometric features not visible from the bivector alone and connects to classical invariants in geometry.
Modular class and unimodularity: The modular class of a Poisson manifold is a distinguished element of H^1(M, π) that measures the failure of a natural volume form to be invariant under Hamiltonian flows. Unimodular Poisson structures (where the modular class vanishes) reveal a form of “volume conservation” that has both geometric and physical interpretations. See Modular class.
Connections to physics and geometry
Deformation quantization: Poisson cohomology long plays a role in deformation quantization, where Poisson brackets are deformed into noncommutative algebras. The Kontsevich formality theorem provides a deep bridge between the Poisson world and the Hochschild cohomology of associative algebras, linking cohomological invariants to the algebraic structure of quantized observables. See Kontsevich formality theorem and Deformation quantization.
Global geometry and groupoids: The global study of Poisson manifolds often involves integrating the infinitesimal data to global objects like symplectic groupoids. While Poisson cohomology is defined in terms of the infinitesimal (local) data, its interpretations and applications extend to global geometric picture through these groupoid constructions. See Symplectic groupoid.
Invariants and obstructions: Poisson cohomology serves as a repository for obstructions to deforming Poisson structures and for organizing families of equivalent Poisson structures. Its computation can be challenging in general, but it yields rich information about the geometric and dynamical content of the system.
Controversies and debates
Within the mathematical community, discussions about Poisson cohomology often focus on computability, interpretation, and the extent to which it captures global versus local data. Some researchers emphasize the extent to which H^*(M, π) determines or fails to determine deformation spaces, particularly in the presence of singularities of π. Others explore the precise relations between Poisson cohomology and other cohomology theories, such as Lie algebroid cohomology or Hochschild cohomology after deformation quantization. The symplectic case provides a clean anchor, but the general Poisson setting presents subtle phenomena where global topology and singular foliation theory play a decisive role. See Lie algebroid and Deformation quantization for broader connections.
Global realization and integration: The passage from infinitesimal data encoded in Poisson cohomology to global geometric structures such as symplectic groupoids remains an area of active development. Some questions center on when Poisson manifolds admit global integrations that reflect their cohomological invariants.
Computability and examples: Explicit computations often require sophisticated tools from homological algebra, foliation theory, and algebraic geometry. The existence of nontrivial examples with computable Poisson cohomology guides both theory and applications, illustrating both the power and the limits of the framework.