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

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 ${\mathrm{A}}<em>1<sup>{(1)}$. For $N$ with $q</em>{00}q_{01}$ 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.