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Fused K-operators and the $q$-Onsager algebra

Guillaume Lemarthe, Pascal Baseilhac, Azat M. Gainutdinov

Abstract

We study universal solutions to reflection equations with a spectral parameter, so-called K-operators, within a general framework of universal K-matrices - an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra $H$, its comodule algebra $B$ and a pair of consistent twists. In our setting, the universal K-matrix is an element of $B\otimes H$ satisfying certain axioms, and we consider the case $H$ is the quantum loop algebra for $sl_2$, and $B={\cal A}_q$ is the alternating central extension of the $q$-Onsager algebra. Considering tensor products of evaluation representations of $H$ in "non-semisimple" cases, the new set of axioms allows us to introduce and study fused K-operators of spin-$j$; in particular, to prove that for all $j\in\frac{1}{2}\mathbb{N}$ they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra ${\cal A}_q$ for small values of spin-$j$. The precise relation between the fused K-operators of spin-$j$ and evaluations of a universal K-matrix for ${\cal A}_q$ is conjectured based on supporting evidences. We finally discuss implications of our results on the K-operators for quantum integrable systems.

Fused K-operators and the $q$-Onsager algebra

Abstract

We study universal solutions to reflection equations with a spectral parameter, so-called K-operators, within a general framework of universal K-matrices - an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra , its comodule algebra and a pair of consistent twists. In our setting, the universal K-matrix is an element of satisfying certain axioms, and we consider the case is the quantum loop algebra for , and is the alternating central extension of the -Onsager algebra. Considering tensor products of evaluation representations of in "non-semisimple" cases, the new set of axioms allows us to introduce and study fused K-operators of spin-; in particular, to prove that for all they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra for small values of spin-. The precise relation between the fused K-operators of spin- and evaluations of a universal K-matrix for is conjectured based on supporting evidences. We finally discuss implications of our results on the K-operators for quantum integrable systems.
Paper Structure (57 sections, 45 theorems, 356 equations, 5 figures)

This paper contains 57 sections, 45 theorems, 356 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Universal K-matrix and comodule algebras
  4. Goal and main results
  5. K-operators for ${\mathcal{A}}_q$ and universal TT-relations
  6. Universal R and K matrices
  7. Universal R-matrix
  8. Twist pairs
  9. Universal K-matrix
  10. Example of (H,B)
  11. Evaluation map
  12. Comodule algebra $\mathcal{A}_q$
  13. Tensor product representations of LUqsl2 and sub-representations
  14. Analysis of the tensor product representation of LUqsl2
  15. Intertwining map E(j+1/2) and its pseudo-inverse F(j+1/2)
  16. ...and 42 more sections

Key Result

Lemma 2.3

The pair $(H^{cop,\psi},\mathfrak{R}_{21}^{\psi\psi})$ is a quasi-triangular Hopf algebra.

Figures (5)

  • Figure 2: Quantum loop module $\mathbb{C}^{3}_u$ of spin-1 where the red and blue diagrams correspond to irreducible components of the $\mathcal{L} U_q sl_2$ action.
  • Figure 3: Supporting evidence for Conjecture \ref{['conj1']}, where the double-sided arrows connecting two equations signify their equality under the assumption that Conjecture \ref{['conj1']} is true.
  • Figure : $u_1/u_2=\pm q^{-j-\frac{1}{2}}$.
  • Figure : $u_1/u_2=\pm q^{-j-\frac{1}{2}}$.
  • Figure : $u_1/u_2= \pm q^{j+\frac{1}{2}}$.

Theorems & Definitions (112)

  • Definition 2.1: Dr0
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4: Dr0Dr89
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 102 more