The classification of $\bf Z$-graded modules of the intermediate series over the $q$-analog Virasoro-like algebra

TL;DR

This paper completes the classification of Z-graded modules of the intermediate series over the q-analog Virasoro-like algebra by constructing four classes of irreducible modules and proving their completeness.

Contribution

It provides a complete classification of Z-graded modules of the intermediate series over the q-analog Virasoro-like algebra, including explicit constructions and a proof of their exhaustiveness.

Findings

Constructed four classes of irreducible Z-graded modules

Proved all such modules are direct sums of trivial or constructed modules

Established the classification as complete

Abstract

In this paper, we complete the classification of the {\bf Z}-graded modules of the intermediate series over the $q$-analog Virasoro-like algebra $L$. We first construct four classes of irreducible {\bf Z}-graded $L$-modules of the intermediate series. Then we prove that any {\bf Z}-graded $L$-modules of the intermediate series must be the direct sum of some trivial $L$-modules or one of the modules constructed by us.