# Non-weight modules over the affine-Virasoro algebra of type $A_1$

## TL;DR

This paper classifies and analyzes a specific class of non-weight modules over the affine-Virasoro algebra of type A_1, focusing on their structure, simplicity, and isomorphism classes.

## Contribution

It provides the first classification and detailed analysis of non-weight modules of rank one over the affine-Virasoro algebra of type A_1.

## Key findings

- Classification of non-weight modules over the affine-Virasoro algebra of type A_1
- Determination of simplicity conditions for these modules
- Description of isomorphism classes of the modules

## Abstract

In this paper, we study a class of non-weight modules over the affine-Virasoro algebra of type $A_1$, which are free modules of rank one when restricted to the Cartan subalgebra (modulo center). We give the classification of such modules. Moreover, the simplicity and the isomorphism classes of these modules are determined.