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

A. Liashyk, S. Pakuliak

TL;DR

This work establishes a precise bridge between the R-matrix (RTT) realization and the Drinfeld current realization of the twisted quantum loop algebra ${U_q(D^{(2)}_n)}$ by exploiting a Gaussian decomposition of L-operators. It proves an embedding ${U_q(D^{(2)}_{n-1})}\hookrightarrow U_q(D^{(2)}_n)}$ that underpins the connection and derives explicit relations between Gaussian coordinates and currents, including central and Nazarov-type central elements and composed currents. The authors develop Cartan–Weyl realizations for small ranks ($n=2,3$), uncover isomorphisms with twisted $A$-type algebras, and outline the general ${U_q(D^{(2)}_n)}$ structure via currents and projections, enabling transfer of RTT data to the Drinfeld current framework and paving the way for current-based Bethe-vector constructions. The results provide a robust, rank-agnostic method to translate between L-operator data and Drinfeld currents for the ${D^{(2)}_n}$ series and guidance for extending to other twisted and exceptional cases.

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.

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

TL;DR

This work establishes a precise bridge between the R-matrix (RTT) realization and the Drinfeld current realization of the twisted quantum loop algebra by exploiting a Gaussian decomposition of L-operators. It proves an embedding that underpins the connection and derives explicit relations between Gaussian coordinates and currents, including central and Nazarov-type central elements and composed currents. The authors develop Cartan–Weyl realizations for small ranks (), uncover isomorphisms with twisted -type algebras, and outline the general structure via currents and projections, enabling transfer of RTT data to the Drinfeld current framework and paving the way for current-based Bethe-vector constructions. The results provide a robust, rank-agnostic method to translate between L-operator data and Drinfeld currents for the series and guidance for extending to other twisted and exceptional cases.

Abstract

The connection between the R-matrix realization and Drinfeld's realization of the quantum loop algebra is considered using the Gaussian decomposition approach proposed by J. Ding and I. B. Frenkel. Our main result is a description of the embedding that underlies this connection. Explicit relations between all Gaussian coordinates of the L-operators and the currents are presented.
Paper Structure (22 sections, 14 theorems, 269 equations)

This paper contains 22 sections, 14 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. ${\rm R}$-matrix
  3. Properties of R-matrix
  4. Algebra $U_q(D^{(2)}_{n})$ in ${\rm R}$-matrix form
  5. Centers in $U_q(D^{(2)}_{n})$
  6. Gaussian coordinates
  7. Normal ordering of the Gaussian coordinates
  8. Embedding $U_q(D^{(2)}_{n-1})\hookrightarrow U_q(D^{(2)}_{n})$
  9. Embedding relations for ${\rm R}(u,v)$
  10. The embedding theorem
  11. Cartan-Weyl realization of $U_q(D^{(2)}_{n})$
  12. Algebra $U_q(D^{(2)}_{2})$
  13. Algebra $U_q(D^{(2)}_{3})$
  14. Isomorphism of $U_q(D^{(2)}_{3})$ and $U_{q^{1/2}}(A^{(2)}_3)$
  15. Algebra $U_q(D^{(2)}_{n})$
  16. ...and 7 more sections

Key Result

Proposition 2.1

${\rm R}$-matrix for $U_q(D^{(2)}_{n})$ has the form where with and

Theorems & Definitions (16)

  • Proposition 2.1
  • Proposition 3.1
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.1
  • Proposition 5.1
  • ...and 6 more