Rank two Artin-Schelter regular algebras and non commuting derivations
If $Δ$ and $Γ$ are two derivations of a commutative algebra $A$ such that $ΔΓ-ΓΔ=Δ$ is locally nilpotent, one can endow $A$ with a new product $\ast$ whose filtered semiclassical limit is the Poisson structure $Δ\wedgeΓ$.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 $Δ$ and $Γ$ 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 $(A,Δ\wedgeΓ)$ of rank $2$, thus explaining the title of the article.Assuming furthermore a technical condition on $Γ$, we show that the algebra $(A,\ast)$ is Calabi-Yau if and only if the trace of $Γ$ is equal to $1$ if and only if the Poisson algebra $(A,Δ\wedgeΓ)$ is unimodular.Since the trace of $Γ$ 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.