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

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.