Graded modules for Virasoro-like algebra

TL;DR

This paper classifies irreducible graded modules over the Virasoro-like algebra, identifying their types and characterizing modules with nonzero center, while proving the nonexistence of certain intermediate series modules.

Contribution

It provides a complete classification of irreducible graded modules over the Virasoro-like algebra, including modules with nonzero center and ruling out intermediate series modules.

Findings

Irreducible modules are either uniformly bounded or generalized highest weight.

All generalized highest weight irreducible modules are explicitly determined.

No nontrivial Z-graded modules of intermediate series exist.

Abstract

In this paper, we consider the classification of irreducible ${\bf Z}$- and ${\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonzero center. Finally, we prove that there does not exist any nontrivial ${\bf Z}$-graded modules of intermediate series.