Double Poisson (vertex) algebra cohomology
Maxime Fairon, Daniele Valeri
TL;DR
This work introduces and analyzes completed double Poisson cohomology $\,\widehat{dPH}(\mathcal A)$ to remove the bivector requirement and extend cohomology theory to quasi- and gauged contexts, establishing its compatibility with representation functors via the Kontsevich–Rosenberg principle. It builds a robust framework linking the noncommutative DP and dPVA formalisms, via Chemla-type formulas and jet/quotient functors, and proves square-zero differentials in both the standard PVW setting and its completed generalization. The authors develop gauged and quasi-Poisson variants, prove foundational squaring properties, and provide extensive computations for algebras with one or two generators (notably $\Bbbk[x]$, $\Bbbk[x]/(x^r)$, and $\Bbbk\langle u,v\rangle$), illustrating how cohomology groups behave under quotients and gauge reductions. They also connect the noncommutative theory with its commutative counterparts through representation-based reductions and discuss fusion as a cohomology-preserving operation. Overall, the paper lays a comprehensive noncommutative cohomology framework with concrete calculations and structural links to NC Poisson geometry and PVA theory, opening avenues for NC integrable systems and moduli space applications.
Abstract
A noncommutative (NC) version of Poisson geometry was initiated by Van den Bergh by introducing at the level of associative algebras the formalism of double Poisson brackets. Their key property is to induce (standard) Poisson brackets under each representation functor. Then, Pichereau and Van de Weyer developed and studied the corresponding cohomology theory under the assumption that there exists a NC bivector defining the double Poisson bracket. Our first main result is that one can remove this assumption by constructing a completed double Poisson cohomology valid in any situation, hence generalizing the approach of Pichereau-Van de Weyer. As an application, we show that the double Poisson cohomology complex associated to the path algebra of a quiver is acyclic. Furthermore, we show that this new double Poisson cohomology theory can be adapted to weaker forms of double Poisson brackets (called quasi-Poisson and gauged Poisson), and that it is compatible with representation functors. A second focus of this memoir concerns the formalism of double Poisson vertex algebras. These were introduced by De Sole, Kac and the second author, as NC versions of Poisson vertex algebras, which induce the latter structures under each representation functor. Our second main result is the development of cohomology theories for double Poisson vertex algebras. These are NC analogues of the basic, reduced and variational Poisson vertex algebra cohomologies. More importantly, we prove that under each representation functor these cohomology theories are compatible with their commutative counterparts. As an application, we compute the double Poisson vertex algebra cohomology of the generalized NC de Rham complex and of the generalized NC variational complex. Finally, we describe the relation between the double Poisson algebra and double Poisson vertex algebra cohomologies using jet and quotient functors.