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Keywords:
Derivations; iterative higher derivations; rings of differential operators; Weyl algebra
Derivations; iterative higher derivations; rings of differential operators; Weyl algebra
Summary:
We connect the theorems of Rentschler \cite {rR68} and Dixmier \cite {jD68} on locally nilpotent derivations and automorphisms of the polynomial ring $A _0$ and of the Weyl algebra $A _1$, both over a field of characteristic zero, by establishing the same type of results for the family of algebras $$A _h=\langle x, y\mid yx-xy=h(x)\rangle \,,$$ where $h$ is an arbitrary polynomial in $x$. In the second part of the paper we consider a field $\mathbb{F}$ of prime characteristic and study $\mathbb{F}[t]$\HH comodule algebra structures on $A _h$. We also compute the Makar-Limanov invariant of absolute constants of $A _h$ over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of $A _h$.
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