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

TL;DR

This paper classifies all finite irreducible modules of a specific class of $Z^+$-graded Lie conformal algebras, including Block type and map Virasoro algebras, showing they are all free of rank one as $C[partial]$-modules.

Contribution

It introduces the notion of completely non-trivial modules and provides a classification of finite irreducible modules for certain $Z^+$-graded Lie conformal algebras, including Block type and map Virasoro algebras.

Findings

All non-trivial finite irreducible modules are free of rank one as $C[partial]$-modules.

Classified modules for a class of $Z^+$-graded Lie conformal algebras including Block type and map Virasoro.

Established the structure of irreducible modules for these 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 $\mathcal{L}=\bigoplus_{i=0}^{\infty} \mathbb{C}[\partial]L_i$ satisfying $ [{L_0}_\lambda L_0]=(\partial+2\lambda)L_0,$ and $[{L_1}_\lambda L_i]\neq 0$ 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.