The Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra $U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$

TL;DR

This paper establishes an algebraic proof of the Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra of type B, connecting Drinfeld and $RLL$ realizations via $R$-matrices and Lyndon bases.

Contribution

It provides a new algebraic proof of the Ding-Frenkel isomorphism for two-parameter quantum affine algebras, including affine $R$-matrix constructions and Lyndon basis methods.

Findings

Constructed the basic braided $R$-matrix for $U_{r,s}(so_{2n+1})$

Derived spectral parameter-dependent $R$-matrices satisfying the intertwining property

Proved the isomorphism between Drinfeld-Jimbo and $RLL$ presentations algebraically.

Abstract

From the theory of finite-dimensional weight modules, we get the basic braided $R$-matrix $\widehat R$ of $U_{r, s}(\mathfrak{so}_{2n+1})$. For its FRT presentation $U(\widehat R)$, we achieve two word-formation methods of quantum Lyndon bases (whose bracketing rules are regulated by the $RLL$-formalism) and elucidate their distribution rule within the triangular $L$-matrix. Consequently, we contribute an algebraic proof for establishing an isomorphism between the Drinfeld-Jimbo presentation and the FRT presentation. In the affine setting, we first derive two spectral parameter-dependent $R$-matrices through the Yang-Baxterization. Next, we select the only one that satisfies the intertwining property with respect to the minimal affinization. Accordingly, we obtain the $RLL$ realization of $U_{r, s}(\widehat{\mathfrak{so}_{2n+1}})$ through the Gauss decompositions of the generating matrices. Finally, we contribute an algebraic proof to the Ding-Frenkel Isomorphism Theorem between the Drinfeld realization and the $RLL$ realization.