The ring of differential operators on a monomial curve is a Hopf algebroid
arXiv:2405.08490
Abstract
This article considers cuspidal curves whose coordinate rings are numerical semigroup algebras. Using a general result about descent of Hopf algebroid structures, their rings of differential operators are shown to be cocommutative and conilpotent left Hopf algebroids. If the semigroups are symmetric so that the curves are Gorenstein, they are full Hopf algebroids (admit an antipode).
New section 4.8 giving an overview of previous results by Sweedler, and by Ben-Zvi and Nevins on Hopf algebroid structures on rings of differential operators