Spectral Characters of a class of Integrable Representations of Toroidal Lie Algebras

TL;DR

This paper investigates the structure of finite-length integrable representations of toroidal Lie algebras, focusing on characterizing and parametrizing the blocks of the associated category, especially for simply-laced types.

Contribution

It provides a new parametrization of the blocks of the category of positive level integrable representations for simply-laced toroidal Lie algebras.

Findings

Blocks of the category are characterized and parametrized for simply-laced cases.

The study advances understanding of the representation theory of toroidal Lie algebras.

A framework for analyzing finite-length integrable modules is developed.

Abstract

In this paper we study the subcategory of finite-length objects of the category of positive level integrable representations of a toroidal Lie algebra. The main goal is to characterize the blocks of the category. In the cases when the underlying finite type Lie algebra associated with the toroidal Lie algebra is simply-laced, we are able to give a parametrization for the blocks.