Rank two Artin-Schelter regular algebras and non commuting derivations

If $\Delta$ and $\Gamma$ are two derivations of a commutative algebra $A$ such that $\Delta\Gamma-\Gamma\Delta=\Delta$ is locally nilpotent, one can endow $A$ with a new product $\ast$ whose filtered semiclassical limit is the Poisson structure $\Delta\wedge\Gamma$.In this article we first study theses (Poisson) algebras from an algebraic point of view, and when $A$ is a polynomial algebra, we investigate their homological properties. In particular, when the derivations $\Delta$ and $\Gamma$ are linear, the algebras $(A,\ast)$ provide, in each dimension at least four, new examples of multiparameter families of Artin-Schelter regular algebras. These algebras are deformations of Poisson algebras $(A,\Delta\wedge\Gamma)$ of rank $2$, thus explaining the title of the article.Assuming furthermore a technical condition on $\Gamma$, we show that the algebra $(A,\ast)$ is Calabi-Yau if and only if the trace of $\Gamma$ is equal to $1$ if and only if the Poisson algebra $(A,\Delta\wedge\Gamma)$ is unimodular.Since the trace of $\Gamma$ is a linear function of the parameters, the algebras $(A,\ast)$ also provide, in each dimension at least four, new examples of multiparameter families of Calabi-Yau algebras.