Some categories of modules for toroidal Lie algebras

TL;DR

This paper explores specific module categories for toroidal Lie algebras using formal variable techniques, classifying irreducible integrable modules within these categories.

Contribution

It introduces and analyzes two module categories for toroidal Lie algebras, providing classification results for irreducible integrable modules.

Findings

Evaluation modules are contained in category _{ au}

Restricted modules and their tensor products are in _{ au}

Classification of irreducible integrable modules in both categories

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 $\mathcal{C}_{\tau}$.