On the R-matrix realization of the quantum loop algebra. The case of $U_q(D^{(2)}_n)$

TL;DR

This paper explores the relationship between R-matrix and Drinfeld's realizations of the quantum loop algebra $U_q(D^{(2)}_n)$, providing explicit embeddings and relations using Gaussian decomposition.

Contribution

It offers a detailed description of the embedding of $U_q(D^{(2)}_{n-1})$ into $U_q(D^{(2)}_n)$ and explicit relations between Gaussian coordinates and currents.

Findings

Explicit embedding of $U_q(D^{(2)}_{n-1})$ into $U_q(D^{(2)}_n)$

Relations between Gaussian coordinates and currents

Connection established via Gaussian decomposition approach

Abstract

The connection between the R-matrix realization and Drinfeld's realization of the quantum loop algebra $U_q(D^{(2)}_n)$ is considered using the Gaussian decomposition approach proposed by J. Ding and I. B. Frenkel. Our main result is a description of the embedding $U_q(D^{(2)}_{n-1})\hookrightarrow U_q(D^{(2)}_n)$ that underlies this connection. Explicit relations between all Gaussian coordinates of the L-operators and the currents are presented.