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- W4299524114 abstract "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 $X$." @default.
- W4299524114 created "2022-10-02" @default.
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- W4299524114 date "2016-10-03" @default.
- W4299524114 modified "2023-10-16" @default.
- W4299524114 title "Generalised Witt algebras and idealizers" @default.
- W4299524114 doi "https://doi.org/10.48550/arxiv.1610.00776" @default.
- W4299524114 hasPublicationYear "2016" @default.
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