Notes on highest weight modules of the elliptic algebra ${\cal A}_{q,p}\left(\widehat{sl}_2\right)$

TL;DR

This paper constructs and analyzes highest weight modules for the elliptic algebra ${ m A}_{q,p}(\\widehat{sl}_2)$, introducing vertex operators and establishing bounds consistent with known characters.

Contribution

It provides a new construction of modules for the elliptic algebra and conjectures their properties, including ordering rules and dimension bounds.

Findings

Introduces vertex operators with specific commutation relations.

Establishes ordering rules for operators.

Finds an upper bound for independent vectors matching known characters.

Abstract

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $\Phi$ and $\Psi^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $\Phi$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules.