Actions of the additive group $\mathsf{G}_a$ on Ore extensions

We connect the theorems of Rentschler and Dixmier 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 $\mathsf A_h$.