Modules of the toroidal Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$

TL;DR

This paper investigates the structure of highest weight modules for the double affine Lie algebra , identifying singular vectors and describing modules under specific central charge conditions, advancing understanding of their representation theory.

Contribution

It introduces a new triangular decomposition for the double affine Lie algebra  and characterizes highest weight modules and singular vectors within this framework.

Findings

Singular vectors of Verma modules are explicitly determined.

Highest weight modules are classified for specific central charge conditions.

A new approach to the representation theory of double affine Lie algebras is developed.

Abstract

Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that $c_1>0$ and $c_2=0$.