The elliptic quantum algebra $A_{q,p}(\hat {sl_n})$ and its bosonization at level one

TL;DR

This paper introduces an elliptic quantum algebra $A_{q,p}( hat {sl_n})$, extending previous work, and provides a bosonization at level one using $Z_n$ symmetric R-matrices and free bosonic fields.

Contribution

It extends the elliptic quantum algebra framework to $A_{q,p}( hat {sl_n})$ and develops a bosonization at level one, generalizing prior results for $A_{q,p}( sh_2)$.

Findings

Defined the algebra $A_{q,p}( hat {sl_n})$ via $RLL=LLR^*$ relations.

Constructed bosonic free field realizations of vertex operators.

Provided a bosonization at level one for the algebra.

Abstract

We extend the work of Foda et al and propose an elliptic quantum algebra $A_{q,p}(\hat {sl_n})$. Similar to the case of $A_{q,p}(\hat {sl_2})$, our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are $Z_n$ symmetric R-matrices with the elliptic moduli chosen differently and a factor is also involved. With the help of the results obtained by Asai et al, we realize type I and type II vertex operators in terms of bosonic free fields for $Z_n$ symmetric Belavin model. We also give a bosonization for the elliptic quantum algebra $A_{q,p}(\hat {sl_n})$ at level one.