# Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type C

## TL;DR

This paper extends the explicit isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebras from type A to types B, C, and D, providing detailed constructions for type C.

## Contribution

It generalizes the known isomorphism to classical types B, C, D and constructs explicit maps using Gauss decomposition and the universal R-matrix.

## Key findings

- Established isomorphism for type C quantum affine algebra.
- Used Gauss decomposition to derive Drinfeld generators.
- Constructed inverse map employing the universal R-matrix.

## Abstract

An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction for type $C$ in this paper. In all classical types the Gauss decomposition of the generator matrix in the $R$-matrix presentation yields the Drinfeld generators. To prove that the resulting map is an isomorphism we follow the work of E. Frenkel and Mukhin (2002) in type $A$ and employ the universal $R$-matrix to construct the inverse map. A key role in our construction is played by a homomorphism theorem which relates the quantum affine algebra of rank $n-1$ in the $R$-matrix presentation with a subalgebra of the corresponding algebra of rank $n$ of the same type.