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

Rushu Zhuang, Naihong Hu and Xiao Xu

Vol. 329 (2024), No. 2, 357–395

DOI: 10.2140/pjm.2024.329.357

Abstract

We obtain the basic $R$-matrix of the two-parameter quantum group $U=U_{r,s}({\mathfrak{𝔰}\mathfrak{𝔬}}_{2n})$ via its weight representation theory and determine its $R$-matrix with spectral parameters for the two-parameter quantum affine algebra $U=U_{r,s}(\widehat{{\mathfrak{𝔰}\mathfrak{𝔬}}_{2n}})$. Using the Gauss decomposition of the $R$-matrix realization of $U=U_{r,s}({\mathfrak{𝔰}\mathfrak{𝔬}}_{2n})$, 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 $U=U_{r,s}(\widehat{{\mathfrak{𝔰}\mathfrak{𝔬}}_{2n}})$.

Keywords

basic $R$-matrix, Drinfeld realization, $RLL$ formulation, quantum affine algebra, Gauss decomposition

Mathematical Subject Classification

Primary: 16T25, 17B37, 81R50

Secondary: 17B67

Milestones

Received: 8 February 2024

Revised: 17 May 2024

Accepted: 25 May 2024

Published: 6 July 2024