Rings of differential operators on curves

TL;DR

This paper investigates the structure of certain finitely generated algebras over algebraically closed fields, showing that those with nonzero locally nilpotent derivations have quadratic growth and are closely related to rings of differential operators on curves.

Contribution

It establishes a connection between algebras with locally nilpotent derivations and rings of differential operators on curves, characterizing their growth and algebraic properties.

Findings

Algebras with nonzero locally nilpotent derivations have quadratic growth.

Such algebras either satisfy a polynomial identity or are subalgebras of differential operator rings.

These algebras are birationally isomorphic to rings of differential operators on smooth affine curves.

Abstract

Let $k$ be an algebraically closed field of characteristic 0 and let $A$ be a finitely generated $k$-algebra that is a domain whose Gelfand-Kirillov dimension is in $[2,3)$. We show that if $A$ has a nonzero locally nilpotent derivation then $A$ has quadratic growth. In addition to this, we show that $A$ either satisfies a polynomial identity or $A$ is isomorphic to a subalgebra of $\mathcal{D}(X)$, the ring of differential operators on an irreducible smooth affine curve $X$, and $A$ is birationally isomorphic to $\mathcal{D}(X)$.