Irreducible modules of toroidal Lie algebras arising from $\phi_\epsilon$-coordinated modules of vertex algebras

TL;DR

This paper extends 2-toroidal Lie algebras with derivations and constructs irreducible modules using $_epsilon$-coordinated modules of vertex algebras, connecting to earlier affine Lie algebra modules.

Contribution

It introduces an extension of 2-toroidal Lie algebras and provides explicit realizations of their irreducible modules via vertex algebra theory.

Findings

Explicit realization of irreducible highest weight modules

Extension of 2-toroidal Lie algebra with derivations

Connection to modules of extended affine Lie algebras

Abstract

In this paper, for every $\epsilon\in \mathbb{Z}$, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the $\phi_\epsilon$-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When $\epsilon=1$, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.