Universal $R$-matrix of double parameter quantum affine algebra $U_{q,Q}({\hat {sl_2}})$

TL;DR

This paper explicitly constructs the universal R-matrix for a double parameter quantum affine algebra of type A_1^{(1)} and computes its representation-specific R-matrix at roots of unity.

Contribution

It provides an explicit formula for the universal R-matrix of a double parameter quantum affine algebra and calculates its representation at roots of unity.

Findings

Explicit formula for the universal R-matrix of the algebra.

Construction of a 2N-dimensional representation at roots of unity.

Explicit calculation of the R-matrix for the representation.

Abstract

We give the explicit formula of the universal $R$-matrix of a double parameter (or two-parameter, or multi-parameter) quantum affine algebra of type ${\mathrm{A}}_1^{(1)}$. For $N$ with $q_{00}q_{01}$ being a primitive $N$-th root of unity, we introduce its $2N$-dimensional representation and explicitly calculate the $R$-matrix associated with it via the universal $R$-matrix.