$RLL$-realization of two-parameter quantum affine algebra of type $B_n^{(1)}$ and normalized quantum Lyndon bases

Abstract: We use the theory of finite-dimensional weight modules to derive the basic braided $R$-matrix of $U_{r, s}(\mathfrak{so}{2n+1})$, and establish the isomorphism between the FRT formalism and the Drinfeld-Jimbo presentation. As an application, we acquire the distribution rule of two normalized qauntum Lyndon bases (regulated by the $RLL$ relations) in the $L$-matrix. In the affine case $U{r, s}(\widehat{\mathfrak{so}_{2n+1}})$, we obtain the corresponding spectral parameter dependent $R$-matrix, as well as its $RLL$ realization via the Gauss decomposition. In this way, we obtain intrinsically its Drinfeld realization. We also provide another affinization and the corresponding new affine quantum affine algebra.