Irreducible modules of toroidal Lie algebras arising from ${\varphi }_{\epsilon }$-coordinated modules of vertex 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 ${\varphi }_{\epsilon }$-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.