Double Poisson Vertex Algebras

• Double Poisson vertex algebras are noncommutative differential algebras equipped with a 2-fold λ-bracket, defined by sesquilinearity, cyclic skewsymmetry, and double Leibniz rules.
• They induce Poisson vertex algebra structures on representation spaces via the Kontsevich–Rosenberg principle, linking noncommutative and commutative geometries.
• Their cohomology theory extends classical deformation and integrable system analysis, enabling new applications in noncommutative Hamiltonian PDEs.

A double Poisson vertex algebra (dPVA) is a noncommutative analogue of a Poisson vertex algebra, synthesizing Van den Bergh’s double Poisson bracket formalism on associative algebras with the λ-bracket structure central to the theory of Poisson vertex algebras in Hamiltonian PDE. A dPVA is a unital, generally noncommutative, differential algebra V equipped with a 2-fold λ-bracket $\{a_\lambda b\}\in(V\otimes V)[\lambda]$, subject to axioms of noncommutative sesquilinearity, cyclic skewsymmetry, and double Leibniz rules. The salient feature of this structure is compatibility with the representation functor: for each finite-dimensional representation space, the 2-fold λ-bracket of the dPVA induces a standard λ-bracket making the coordinate ring into an (ordinary, commutative) Poisson vertex algebra. The development of dPVA cohomology establishes the noncommutative counterparts of the basic, reduced, and variational Poisson vertex algebra cohomologies, and these new invariants descend under representations to the established commutative variants (Fairon et al., 25 Sep 2025).

1. Formalism of Double Poisson Vertex Algebras

Let $V$ be a unital associative (possibly noncommutative) differential algebra endowed with derivation $\partial$. A double Poisson vertex algebra structure consists of a linear map: $$\{-_\lambda-\} : V \otimes V \to (V \otimes V)[\lambda]$$ termed a 2-fold λ-bracket. The defining axioms are:

• Sesquilinearity

$$\{\partial a_\lambda b\} = -\lambda \{a_\lambda b\}, \quad \{a_\lambda \partial b\} = (\lambda+\partial)\{a_\lambda b\}$$

• Double Leibniz rules (left/right)

$$\{a_\lambda\,bc\} = \{a_\lambda b\}\,c + b\,\{a_\lambda c\}$$

and for the left slot, a noncommutative rule involving shifted arguments.

• Cyclic skewsymmetry

$$\{a_\lambda b\} = -\{b_{-\lambda-\partial} a\}^\sigma$$

where $\sigma$ is the flip map on $V \otimes V$.

• Double Jacobi identity

$$\{a_\lambda\{b_\mu c\}_L\} - \{b_\mu\{a_\lambda c\}_R\} = \{\{a_\lambda b\}_{\lambda+\mu} c\}_L$$

with subscripts $L,R$ indicating the way the inner and outer brackets interact in the tensor product structure (Sole et al., 2014, Sole et al., 2023).

Crucially, composing the 2-fold λ-bracket with multiplication $m: V\otimes V \to V$ yields an induced bracket on $V$, which, although typically not a derivation in the first slot, descends to a Lie conformal structure on the quotient $V / ([V, V] + \partial V)$.

2. Representation Functor and Kontsevich–Rosenberg Principle

A defining property of double Poisson (vertex) algebras is their compatibility with representation functors; this is often called the Kontsevich–Rosenberg principle (Álvarez-Cónsul et al., 2021, Sole et al., 2023, Olshanski et al., 2023, Fairon et al., 25 Sep 2025). For each $n\geq1$, the representation space of $V$ is $\mathrm{Rep}_n(V)$, with commutative coordinate ring $$V_n = \mathbb{K}[ a_{ij} \mid a\in V, 1\leq i,j\leq n ]$$. A double λ-bracket induces a $\lambda$-bracket on $V_n$ by

$$\{a_{ij\ \lambda} b_{kl}\} = \{a_\lambda b\}'_{kj} \{a_\lambda b\}''_{il}$$

in Sweedler notation $$\{a_\lambda b\} = \{a_\lambda b\}' \otimes \{a_\lambda b\}''$$ (Olshanski et al., 2023, Fairon et al., 2021). The induced bracket equips $V_n$ with the structure of an ordinary (multiplicative or additive) Poisson vertex algebra. This behavior assures that every noncommutative double bracket structure has a canonical commutative shadow on finite-dimensional representation spaces, and the noncommutative cohomology functorially descends to the standard PVA cohomology (Fairon et al., 25 Sep 2025).

3. Double Poisson Vertex Algebra Cohomology

To extend the deformation theory and obstruction calculus from the commutative to the noncommutative vertex setting, several noncommutative cohomology theories are introduced (Fairon et al., 25 Sep 2025):

• Basic double PVA cohomology: defined on complexes whose 0-th term is the abelianization $A_\natural = A / [A, A]$, and, in positive degrees, consists of multilinear $n$-brackets on $A$ with cyclic skewsymmetry and appropriate Leibniz-type rules;
• Reduced double PVA cohomology: further quotients the basic cohomology by actions of the derivation and bracket commutators;
• Variational double PVA cohomology: tailored to handle (bi-)Hamiltonian deformations in the context of noncommutative integrable hierarchies, directly paralleling the variational cohomology in commutative PVAs.

The differential on these complexes is constructed as a Chemla-type operator, involving cyclic permutations and insertion of double brackets: $$d(Q)(a_1,\dots,a_{n+1}) = \sum_{s=1}^{n+1} (-1)^{n s} \sigma^s \circ (Q \otimes \mathrm{id}) \circ (\mathrm{id}^{\otimes(n-1)} \otimes \{-, -\}) \circ \sigma^{-s}(a_1,\dots,a_{n+1})$$ where $\sigma$ is the cyclic permutation (Fairon et al., 25 Sep 2025).

An essential property is compatibility with representation functors: under the induced map to each commutative representation algebra, the double cohomology complex maps to the standard PVA cohomology complex: $dP_vH(V) \longrightarrow P_vH(V_n)$ for any finite $n$, with all structure maps compatible with differentials.

4. Applications and Example Computations

Applications are multifold:

• The basic theory recovers the classical cohomology of Poisson vertex algebras upon reduction to commutative algebras via representations.
• Explicit computations in free and truncated polynomial algebras show the vanishing or presence of certain noncommutative cohomologies and clarify the extension to higher-degree brackets (Fairon et al., 25 Sep 2025).
• The double PVA cohomology of the generalized noncommutative de Rham and variational complexes is discussed, showing new phenomena such as infinite-dimensionality in the noncommutative setting, unlike in the commutative theory (Sole et al., 2014, Fairon et al., 25 Sep 2025).

Key results include generalizations beyond bivector-defined structures (lifting the restriction found in earlier works such as Pichereau–Van de Weyer), explicit computation of cohomologies for path algebras of quivers (which are shown to be acyclic), and demonstration that paired double brackets automatically yield compatible Poisson structures on all representation spaces.

5. Jet and Quotient Functors: Relation to Double Poisson Algebra Cohomology

The theory incorporates two key functors paralleling those in the classical algebraic setting:

• Jet functor: sends a double Poisson algebra $A$ to its jet algebra $J_\infty A$ (the universal differential envelope), which then acquires a double Poisson vertex algebra structure. Cohomology maps as $\hat{dPH}(A) \to dP_vH(J_\infty A)$.
• Quotient functor: for a double Poisson vertex algebra $V$, forms the quotient $q(V) := V / \partial V$, which is a double Poisson algebra; the induced map is $dP_vH(V) \to \hat{dPH}(q(V))$.

Diagrams of these functors mediate between the double Poisson and double Poisson vertex algebra cohomology worlds, establishing commutative diagrams that interlink the noncommutative and commutative theories (Fairon et al., 25 Sep 2025). The resulting framework clarifies the passage from associative, noncommutative geometry to noncommutative Hamiltonian PDE and their associated moduli.

6. Relation to Integrable Systems and Noncommutative Geometry

The double Poisson vertex algebra framework is specifically adapted to describing nonabelian and noncommutative integrable systems. Via the trace bracket on $V/([V,V]+\partial V)$ induced from the double λ-bracket, one recovers a Lie conformal algebra structure, which supports the construction of noncommutative hierarchies of integrable Hamiltonian PDE through noncommutative versions of the Lenard-Magri recursion (Sole et al., 2014, Sole et al., 2023). All Hamiltonian evolutionary equations, functorial reductions, and symmetry arguments descend compatibly to their commutative forms on representation spaces, providing a conceptual and concrete link between noncommutative Poisson geometry and the classical theory of integrable PDE.

7. Outlook and Future Directions

The double Poisson vertex algebra framework sets the stage for several further developments:

• Extending to the paper of quasi-Poisson and gauged Poisson vertex algebras, for which modified cohomology theories are shown to remain compatible with representations (Fairon et al., 25 Sep 2025).
• Examining higher homotopical and dg enhancements, as signaled by connections to pre-Calabi–Yau and double $P_\infty$-algebra frameworks (Fernández et al., 2019, Iyudu et al., 2019).
• Applying jet and Hamiltonian reduction constructions for quiver and involutive representation spaces, as well as understanding modular behaviors under double extensions and deformation quantization (Bozec et al., 2023, Lou et al., 2016).
• Analyzing explicit classification and construction problems for double PVAs and their modules, especially in connection to noncommutative deformation quantization and supersymmetric field theories (Ekstrand et al., 2011, Oh et al., 2019).

This infrastructure achieves a robust noncommutative generalization of Poisson vertex algebra theory, maintaining functorial links to commutative geometry, and unlocking new perspectives on integrable models, representation invariants, and the structure of noncommutative moduli spaces.