$RLL$-realization of two-parameter quantum affine algebra in type $D_n^{(1)}$

TL;DR

This paper constructs the $RLL$-formalism for the two-parameter quantum affine algebra of type $D_n^{(1)}$, providing explicit $R$-matrices and relations using representation theory and Gauss decomposition.

Contribution

It introduces the $RLL$-formalism for the two-parameter quantum affine algebra of type $D_n^{(1)}$, including explicit $R$-matrices and commutation relations.

Findings

Explicit $R$-matrix with spectral parameters derived.

Gauss decomposition used to study generator relations.

Established $RLL$-formalism for the algebra.

Abstract

We obtain the basic $R$-matrix of the two-parameter Quantum group $U=U_{r,s}\mathcal(\mathfrak{so}_{2n})$ via its weight representation theory and determine its $R$-matrix with spectral parameters for the two-parameter quantum affine algebra $U=U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n}})$. Using the Gauss decomposition of the $R$-matrix realization of $U=U_{r,s}\mathcal(\mathfrak{so}_{2n})$, we study the commutation relations of the Gaussian generators and finally arrive at its $RLL$-formalism of the Drinfeld realization of two-parameter quantum affine algebra $U=U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n}})$.