Rank two Artin–Schelter regular algebras from noncommuting derivations

Abstract

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 these (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 two, 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.