Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra

TL;DR

This paper classifies all finite irreducible conformal modules over certain Lie conformal algebras related to the Virasoro algebra, providing explicit module descriptions and extending results to related algebras.

Contribution

It provides a complete classification of finite irreducible modules over a class of Lie conformal algebras related to Virasoro, including explicit module structures and extensions to other algebras.

Findings

All finite irreducible modules over $ ext{W}(b)$ are classified.

Explicit module isomorphisms are provided for different cases.

Results include classifications over Heisenberg-Virasoro and Schrödinger-Virasoro conformal algebras.

Abstract

In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras $\mathcal{W}(b)$ with $b\in\mathbb{C}$ related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over $\mathcal{W}(b)$ is proved to be isomorphic to $M_{\Delta,\alpha,\beta}$ with $\Delta\neq 0$ or $\beta\neq 0$ if $b=0$, or $M_{\Delta,\alpha}$ with $\Delta\neq 0$ if $b\neq0$. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the Lie conformal algebra of $\mathcal{W}(2,2)$-type are classified. Finally, the same thing is done for the Schr\"odinger-Virasoro conformal algebra.