Poisson pseudoalgebras

For any cocommutative Hopf algebra H and a left H-module V, we construct an operad $P_H^{cl}\left(V\right)$, which in the special case when H is the algebra of polynomials in one variable reduces to the classical operad $P^{cl}\left(V\right)$ of [5]. Morphisms from the Lie operad to $P^{cl}\left(V\right)$ correspond to Poisson vertex algebra structures on V. Likewise, our operad $P_H^{cl}\left(V\right)$ gives rise to the notion of a Poisson pseudoalgebra; thus extending the notion of a Lie pseudoalgebra from [1]. As a byproduct of our construction, we introduce two cohomology theories for Poisson pseudoalgebras, generalizing the variational and classical cohomology of Poisson vertex algebras.