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

Abstract

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\mathcal{A}}_{q,p}({\widehat{sl}}_2)$, 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 ${\Psi }^{\ast }$ 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.

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