# Modules for loop Affine-Virasoro Algebras

## TL;DR

This paper investigates the representation theory of loop Affine-Virasoro Algebras, defining Verma modules, classifying irreducible modules, and constructing central operators, thus advancing understanding of their algebraic structure.

## Contribution

It introduces a classification of irreducible modules, conditions for finite-dimensional weight spaces, and constructs commuting central operators for loop Affine-Virasoro Algebras.

## Key findings

- Irreducible highest weight modules with finite-dimensional weight spaces characterized.
- Irreducible integrable modules are either highest or lowest weight modules.
- Construction of affine central operators commuting with the algebra action.

## Abstract

In this paper we study the representations of loop Affine-Virasoro Algebras. As they have canonical triangular decomposition, we define Verma modules and its irreducible quotients. We give necessary and sufficient condition for an irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either an highest weight module or a lowest weight module whenever the canonical central element acts non trivially. At the end we construct Affine central operators for each integer and they commute with the action of the Affine Lie Algebra.