Notes on Highest Weight Modules of the Elliptic Algebra $\mathfrak{A}_{q, p}(\hat{sl_2})$

We discuss a construction of highest weight modules for the recently definedelliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjecturesconcerning them. The modules are generated by the action of the components ofthe operator $L$ on the highest weight vectors. We introduce the vertexoperators $\Phi$ and $\Psi^*$ through their commutation relations with the$L$-operator. We present ordering rules for the $L$- and $\Phi$-operators andfind an upper bound for the number of linearly independent vectors generated bythem, which agrees with the known characters of $\widehat{sl}_2$-modules.Comment: Nonstandard macro package eliminate