Some categories of modules for toroidal Lie algebras

Abstract

In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $\tau$. More specifically, we define and study two categories $\mathcal{E}_{\tau}$ and $\mathcal{C}_{\tau}$ of $\tau$-modules using generating functions, where $\mathcal{E}_{\tau}$ is proved to contain the evaluation modules while $\mathcal{C}_{\tau}$ contains certain restricted $\tau$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $\mathcal{E}_{\tau}$ and $\mathcal{C}_{\tau}$.