Idealizers in the Second Weyl Algebra

TL;DR

This paper studies idealizers in the second Weyl algebra, showing that for certain polynomials defining cuspidal curves, the idealizer of the associated right ideal is always noetherian, extending previous results.

Contribution

It proves that idealizers of specific right ideals in the second Weyl algebra are always noetherian when associated with polynomials defining cuspidal curves.

Findings

Idealizers are always noetherian for polynomials with cuspidal singularities.

Extends previous work on idealizers in Weyl algebras.

Provides new insights into the structure of differential operator rings.

Abstract

Given a right ideal $I$ in a ring $R$, the idealizer of $I$ in $R$ is the largest subring of $R$ in which $I$ becomes a two-sided ideal. In this paper we consider idealizers in the second Weyl algebra $A_2$, which is the ring of differential operators on $\mathbb{k}[x,y]$ (in characteristic $0$). Specifically, let $f$ be a polynomial in $x$ and $y$ which defines an irreducible curve whose singularities are all cusps. We show that the idealizer of the right ideal $fA_2$ in $A_2$ is always left and right noetherian, extending the work of McCaffrey.