# Finite irreducible modules of Lie conformal algebras $\mathcal{W}(a,b)$ and some Schr\"{o}dinger-Virasoro type Lie conformal algebras

## TL;DR

This paper classifies all finite nontrivial irreducible conformal modules of Lie conformal algebras $
W(a,b)$ and Schr"odinger-Virasoro type algebras, showing they are all of rank one, advancing understanding of their module structures.

## Contribution

It provides a complete classification of finite irreducible modules for these Lie conformal algebras, revealing they are all rank one modules, which is a new structural insight.

## Key findings

- All finite irreducible modules of $
W(a,b)$ are of rank one.
- Finite irreducible modules of Schr"odinger-Virasoro type algebras are characterized.
- The classification method applies to multiple related Lie conformal algebras.

## Abstract

Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of $\mathcal{W}(a,b)$. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schr\"{o}dinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$ are characterized.