Isomorphism between the $R$-Matrix and Drinfeld Presentations of Quantum Affine Algebra: Types $B$ and $D$

TL;DR

This paper establishes an isomorphism between the $R$-matrix and Drinfeld presentations of quantum affine algebras specifically for types $B$ and $D$, extending previous work on type $A$ and $C$.

Contribution

It provides a detailed proof of the isomorphism for types $B$ and $D$, completing the classical types in the framework linking $R$-matrix and Drinfeld presentations.

Findings

Gauss decomposition yields Drinfeld generators for types $B$ and $D$

The arguments are similar across all classical types with specific adaptations for orthogonal Lie algebras

The paper confirms the isomorphism between two fundamental presentations of quantum affine algebras

Abstract

Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type $C$ were given therein, while the present paper deals with types $B$ and $D$. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie algebras.