Classification of irreducible weight modules over $W$-algebra W(2,2)

TL;DR

This paper classifies irreducible weight modules over the $W$-algebra $W(2,2)$, showing their support covers the entire weight lattice and characterizing modules with finite dimensional weight spaces as Harish-Chandra modules.

Contribution

It provides a complete classification of irreducible weight modules over $W(2,2)$, including their support and structure, extending understanding of these modules.

Findings

Support of modules coincides with the weight lattice

All nontrivial weight spaces are infinite dimensional

Modules with finite dimensional weight spaces are Harish-Chandra modules

Abstract

We show that the support of an irreducible weight module over the $W$-algebra $W(2, 2)$, which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the the $W$-algebra $W(2, 2)$, having a nontrivial finite dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module of the intermediate series).