Generalised Witt algebras and idealizers

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $\Gamma$ be an additive subgroup of $\Bbbk$. Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the generalised Witt algebra $W_\Gamma$ in terms of three families, one parameterised by ${\mathbb A}^2$ and two by ${\mathbb P}^1$. In this note, we use the first family to construct a homomorphism $\Phi$ from the enveloping algebra $U(W_\Gamma)$ to a skew extension of ${\Bbbk}[a,b]$. We show that the image of $\Phi$ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of $U(W_\Gamma)$ under $\Phi$ is not left or right noetherian, giving a new proof that $U(W_\Gamma)$ is not noetherian. We construct $\Phi$ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let $G$ be an arbitrary group and let $A$ be a $G$-graded ring. A graded $A$-module $M$ is an intermediate series module if $M_g$ is one-dimensional for all $g \in G$. Given a shift-invariant family of intermediate series $A$-modules parametrised by a scheme $X$, we construct a homomorphism $\Phi$ from $A$ to a skew-extension of ${\Bbbk}[X]$. The kernel of $\Phi$ consists of those elements which annihilate all modules in $X$.