Generalised Witt algebras and idealizers

Susan J. Sierra, Špela Špenko

Research output: Contribution to journalArticlepeer-review


Let k be an algebraically closed field of characteristic zero, and let Γ be an additive subgroup of k. Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the generalised Witt algebra WΓ in terms of three families, one parameterised by A2 and two by P1. In this note, we use the first family to construct a homomorphism Φ from the enveloping algebra U(WΓ) to a skew extension of k[a,b]. We show that the image of Φ 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(WΓ) under Φ is not left or right noetherian, giving a new proof that U(WΓ) is not noetherian.
We construct Φ 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 Mg is one-dimensional for all g∈G. Given a shift-invariant family of intermediate series A-modules parametrised by a scheme X, we construct a homomorphism Φ from A to a skew-extension of k[X]. The kernel of Φ consists of those elements which annihilate all modules in X.
Original languageEnglish
Pages (from-to)415-428
Number of pages9
JournalJournal of Algebra
Early online date12 Apr 2017
Publication statusPublished - 1 Aug 2017


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