Finite irreducible conformal modules of rank two Lie conformal algebras

TL;DR

This paper classifies finite irreducible modules over rank two Lie conformal algebras, showing they are of rank one and explicitly describing their actions, with implications for algebras related to the Virasoro algebra.

Contribution

It proves all finite irreducible modules over rank two Lie conformal algebras are of rank one and provides explicit descriptions of their actions.

Findings

Finite irreducible modules over rank two Lie conformal algebras are of rank one.

Explicit descriptions of the actions of these modules are provided.

Results extend to modules over algebras with Virasoro as semisimple quotient.

Abstract

In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $\mathcal{H}$ is of rank one. We also describe the actions of $\mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.