Finite irreducible modules of a class of ${\mathbb{Z}}^{+}$-graded Lie conformal algebras

Abstract

In this paper, we introduce the notion of completely non-trivial module of a Lie conformal algebra. By this notion, we classify all finite irreducible modules of a class of ${\mathbb{Z}}^{+}$-graded Lie conformal algebras $\mathcal{L}={⨁}_{i=0}^{\infty }\mathbb{C}[\partial ]L_i$ satisfying $[{L_0}_{\lambda }{L}_0]=(\partial +2\lambda )L_0,$ and $[{L_1}_{\lambda }{L}_i]\ne 0$ for any $i\in {\mathbb{Z}}^{+}$. These Lie conformal algebras include Block type Lie conformal algebra $\mathcal{B}(p)$ and map Virasoro Lie conformal algebra $\mathcal{V}(\mathbb{C}[T])=Vir\otimes \mathbb{C}[T]$. As a result, we show that all non-trivial finite irreducible modules of these algebras are free of rank one as a $\mathbb{C}[\partial ]$-module.

16 pages