Poisson Pseudoalgebras

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