Double Poisson Cohomology

• Double Poisson cohomology is a cohomological theory capturing deformation, representation, and geometric aspects of noncommutative Poisson brackets.
• Its bivector-free formulation and functorial design enable universal compatibility with classical Poisson geometry via representation functors.
• Extensions to vertex algebras and gauged structures broaden its applications in Hamiltonian PDEs, moduli space studies, and deformation quantization.

Double Poisson cohomology is the principal cohomological theory associated to double Poisson brackets—noncommutative generalizations of classical Poisson brackets on associative algebras—capturing deformation, representation-theoretic, and geometric features of noncommutative spaces. The theory has rapidly evolved to include robust, functorial formulations independent of auxiliary choices such as bivector presentations, connections to vertex algebra settings, and adaptations to quasi-Poisson, gauged, and representation-theoretic settings, thereby providing a unifying cohomological structure for noncommutative Poisson geometry (Fairon et al., 25 Sep 2025).

1. Foundations: Double Poisson Brackets and Representation Functors

At the algebraic core, a double Poisson bracket on an associative algebra $A$ over a field of characteristic zero is a bilinear map

$$\{\,\cdot\,,\,\cdot\,\} : A \otimes A \to A \otimes A$$

satisfying:

• cyclic skewsymmetry: $\{a, b\} = -\{b, a\}^\sigma$ (where $\sigma$ permutes the tensor factors),
• double derivation properties (Leibniz in both entries),
• a double Jacobi identity expressing a noncommutative analog of the Jacobi identity via cyclic permutations of tensor factors.

The fundamental property of this structure, as articulated by Van den Bergh, is its functorial compatibility: under the representation functor $A \mapsto \mathcal{O}(A, d)$ that encodes the coordinate ring of the $d$-dimensional representation scheme, the double bracket "pushes down" to an ordinary Poisson bracket on the commutative algebra $\mathcal{O}(A, d)$. This "Kontsevich–Rosenberg principle" ensures that noncommutative Poisson structures consistently induce commutative Poisson geometry across all representation spaces, making double Poisson geometry a natural language for studying moduli of representations, deformation theory, and noncommutative symplectic structures (Fairon et al., 25 Sep 2025, Olshanski et al., 2023).

2. The Cohomology Theory: Classical Approach and Generalization

The first construction of double Poisson cohomology, by Pichereau and Van de Weyer, began with the existence of a noncommutative bivector $P \in (T^*A)_2$ such that $\{P, P\} = 0$, where $T^*A$ is the free tensor algebra on the space of double derivations $D_B(A)$. In this approach, the double Poisson complex is the graded vector space $((T^*A)_\sharp, d_P)$ with $\deg(d_P) = 1$ and $d_P = \{P, -\}$, and its cohomology is denoted $dPH(A)$. This theory, however, was only defined for brackets admitting such a bivector presentation and did not universally apply to all double Poisson brackets (Fairon et al., 25 Sep 2025).

3. Completed Double Poisson Cohomology: Bivector-Free Formulation

A foundational advance presented in (Fairon et al., 25 Sep 2025) is the definition of completed double Poisson cohomology, which removes the necessity of a global bivector. Instead, for any double Poisson bracket, one works intrinsically with the graded space: $$\mathbb{A} := A / [A, A] \,\oplus\, \bigoplus_{n\geq1} \mathrm{BR}(A)_n,$$ where $\mathrm{BR}(A)_n$ is the space of noncommutative $n$-brackets, and defines a differential $d$ by extending the bracket via a cyclic formula: $$d(\bar{a})(b) = -m(\{a, b\}), \quad d(\omega)(a_1, \dots, a_{n+1}) = \sum_{i=0}^{n} (-1)^i m_i(\{a_i, \omega(a_0, \ldots, \widehat{a_i}, \ldots, a_n)\}),$$ with suitable signs and permutations, $m$ and $m_i$ representing multiplication and cyclic shifts. A central result is that $d^2 = 0$ for any double Poisson bracket, making $(\mathbb{A}, d)$ a cochain complex whose cohomology $\,\widehat{dPH}(A)$ is universally defined [(Fairon et al., 25 Sep 2025), Theorem 3.1]. This approach is shown to generalize the previous theory and, under suitable smoothness or "mu-map" isomorphism conditions, is isomorphic to the earlier $dPH(A)$ [(Fairon et al., 25 Sep 2025), Theorem 3.2].

Further, the construction extends to weaker structures:

• Quasi-Poisson cohomology: for quasi-double brackets, suitable modifications of $d$ yield "quasi-double Poisson" cohomology.
• Gauged Poisson cohomology: after quotienting by the differentials associated to gauge elements (one for each idempotent in the base ring).

This shows formal robustness of the cohomology theory under relaxations and generalizations of the structure.

4. Computational Results and Notable Examples

A significant application is the explicit computation for the path algebra $kQ$ of a quiver, where the completed double Poisson cohomology is acyclic in degrees $>0$: $\widehat{dPH}^0(kQ) = kQ/[kQ, kQ]$, and $\widehat{dPH}^n(kQ) = 0$ for $n\geq 1$ [(Fairon et al., 25 Sep 2025), §5]. This demonstrates the resolution of the "double Poisson acyclicity" conjecture in this archetypal case relevant to moduli of quiver representations.

Other explicit calculations include:

• $A = k[x]$ and its truncations, with $dPH^0$ spanned by central polynomials, $dPH^1$ by Euler and higher derivations, and vanishing in higher degrees.
• Algebras of two generators, such as $k\langle u,v \rangle$ with quartic brackets, showing richer, non-trivial cohomology.
• The theory naturally accommodates "iterated" or "gauged" Poisson extensions, as in the context of double Ore versus Poisson double extensions (Lou et al., 2016).

5. Extension to Double Poisson Vertex Algebras

At the level of differential (noncommutative) geometry and Hamiltonian PDEs, double Poisson vertex algebras (dPVAs) extend the theory to associative algebras $V$ endowed with a derivation $\partial$ and a 2-fold $\lambda$-bracket: $$\llbracket a_\lambda b \rrbracket : V \otimes V \to V \otimes V[\lambda]$$ satisfying sesquilinearity, skewsymmetry, and variants of the Leibniz rule. Representation functors $V \mapsto V_N$ (respecting $\partial$) again produce classical PVAs on representation spaces, thus generalizing the Kontsevich–Rosenberg principle to the vertex setting.

Three families of cohomology complexes are constructed:

• Basic cohomology: via cochains $$V^{\otimes n} \to V^{\otimes (n+1)}[\lambda_1, ..., \lambda_n]$$ compatible with the differential and $\lambda$-bracket,
• Reduced cohomology: after quotienting by image of $\partial$,
• Variational cohomology: analogous to the de Rham-to-variational reduction in the commutative setting.

A key theorem is the functorial compatibility: under the representation functor, these double Poisson vertex (co)homology theories descend to the corresponding commutative Poisson vertex algebra cohomologies [(Fairon et al., 25 Sep 2025), Chapters 6–8].

6. Relations and Functorial Bridges Between Cohomologies

An important conceptual result is the intertwining of double Poisson algebra and double Poisson vertex algebra cohomologies via "jet" and "quotient" functors. Namely, for a (noncommutative) algebra $A$ one forms its (noncommutative) jet algebra $J_\infty A$, which is a double Poisson vertex algebra, and there exist natural cohomology maps: $$\widehat{dPH}(A) \to dP_vH(J_\infty A), \qquad dP_vH(V) \to \widehat{dPH}(V / \langle \partial V \rangle)$$ These functorial maps commute with differentials and specialize, under representations, to corresponding maps in the commutative setting, ensuring that developed theories are universal and detect all classical invariants [(Fairon et al., 25 Sep 2025), Chapters 10–11].

7. Impact and Outlook: Universality, Deformation Theory, and Noncommutative Geometry

Double Poisson cohomology is a universal tool for measuring deformations, obstructions, and symmetry properties of double Poisson structures. It generalizes Hochschild cohomology for noncommutative Poisson algebras, controls the deformation/quantization theory of noncommutative spaces, and underpins structures on moduli spaces of representations, quiver varieties, and Fukaya categories (Chen et al., 2015, Olshanski et al., 2023). Its extension to the vertex setting (dPVAs) incorporates the geometry underlying Hamiltonian PDEs and classical field theories, with functorial bridges ensuring compatibility and passage to classical/commutative limits.

The formalism accommodates quasi- and gauged Poisson structures, promotes robust spectral sequence arguments, and allows for explicit computations in key examples, notably resolving acyclicity and calculating non-trivial cohomology classes. As such, double Poisson (vertex) cohomology paves the way for new invariants in noncommutative symplectic and derived geometry, with implications for deformation quantization, Calabi–Yau algebras, and higher representation theory.

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Aspect	Key Property/Result	Reference
Bivector-free definition	Completed double Poisson cohomology defined for all double Poisson brackets	(Fairon et al., 25 Sep 2025)
Path algebra acyclicity	$\widehat{dPH}^n(kQ) = 0$ for $n \geq 1$ (path algebra/quiver case)	(Fairon et al., 25 Sep 2025)
Compatibility with rep	Under representation functors, double Poisson (vertex) cohomologies push forward to classical Poisson cohomologies	(Fairon et al., 25 Sep 2025)
Cohomology in dPVAs	Basic/reduced/variational complexes for noncommutative differential algebras given and functorially related	(Fairon et al., 25 Sep 2025)
Relation to deformation	Double Poisson cohomology governs infinitesimal deformations and obstructions in the double Poisson context	(Fairon et al., 25 Sep 2025)

In conclusion, double Poisson cohomology (including its vertex and gauged generalizations) provides a comprehensive, functorial, and computationally tractable invariant that strengthens and unifies approaches to noncommutative Poisson geometry, representation functor compatibility, and deformation theory.