Actions of the additive group $\mathsf{G}_a$ on a class of noncommutative deformations of the plane arising as Ore extensions

Preprint

We connect the theorems of Rentschler [68] and Dixmier [68] on locally nilpotent derivations and automorphisms of the polynomial ring $\mathsf{A}_0$ and of the Weyl algebra $\mathsf{A}_1$, both over a field of characteristic zero, by establishing the same type of results for the family of algebras $\mathsf{A}_h=\langle x, y\mid yx-xy=h(x)\rangle$, where $h$ is an arbitrary polynomial in $x$. On the second part of the paper we consider a field $\mathbb{F}$ of prime characteristic and study $\mathbb{F}[t]$-comodule algebra structures on $\mathsf{A}_h$. We also compute the Makar-Limanov invariant of absolute constants of $\mathsf{A}_h$ over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of $\mathsf{A}_h$.

Publication

Year of publication: 2020