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Using the quantum torus to investigate the $q$-Onsager algebra

Owen Goff

TL;DR

This work embeds the $q$-Onsager algebra $O_q$ and its universal variant into the quantum torus $T_q$ via the homomorphism $p$, enabling closed-form expressions for key $O_q$-elements. By transporting Basesilhac–Kolb elements, the $B_{1,r}$ family, and the imaginary and real root vectors through $p$ (and the related $\upsilon$ map), the authors obtain explicit $T_q$-basis representations and generating-function identities. The results provide compact, computable $p$-images for the PBW-type bases and their generating structures, expressed in terms of $x,y$ and the canonical commuting combinations in $T_q$. This $T_q$-realization facilitates structural understanding of $O_q$ and its variants, with potential implications for representations and combinatorial applications in quantum groups.

Abstract

The $q$-Onsager algebra, denoted by $O_q$, is defined by generators $W_0, W_1$ and two relations called the $q$-Dolan-Grady relations. In 2017, Baseilhac and Kolb gave some elements of $O_q$ that form a Poincaré-Birkhoff-Witt basis. The quantum torus, denoted by $T_q$, is defined by generators $x, y, x^{-1}, y^{-1}$ and relations $$xx^{-1} = 1 = x^{-1}x, \qquad yy^{-1} = 1 = y^{-1}y, \qquad xy=q^2yx.$$ The set $\{x^iy^j | i,j \in \mathbb{Z} \}$ is a basis for $T_q$. It is known that there is an algebra homomorphism $p: O_q \mapsto T_q$ that sends $W_0 \mapsto x+x^{-1}$ and $W_1 \mapsto y+y^{-1}.$ In 2020, Lu and Wang displayed a variation of $O_q$, denoted by $\tilde{\mathbf{U}}^{\imath}$. Lu and Wang gave a surjective algebra homomorphism $\upsilon : \tilde{\mathbf{U}}^{\imath} \mapsto O_q.$ \medskip In their consideration of $\tilde{\mathbf{U}}^{\imath}$, Lu and Wang introduced some elements \begin{equation} \label{intrp503} \{B_{1,r}\}_{r \in \mathbb{Z}}, \qquad \{H'_n\}_{n=1}^{\infty}, \qquad \{H_n\}_{n=1}^{\infty}, \qquad \{Θ'_n\}_{n=1}^{\infty}, \qquad \{Θ_n\}_{n=1}^{\infty}. \nonumber \end{equation} These elements are defined using recursive formulas and generating functions, and it is difficult to express them in closed form. A similar problem applies to the Baseilhac-Kolb elements of $O_q$. To mitigate this difficulty, we map everything to $T_q$ using $p$ and $\upsilon$. In our main results, we express the resulting images in the basis for $T_q$ and also in an attractive closed form.

Using the quantum torus to investigate the $q$-Onsager algebra

TL;DR

This work embeds the -Onsager algebra and its universal variant into the quantum torus via the homomorphism , enabling closed-form expressions for key -elements. By transporting Basesilhac–Kolb elements, the family, and the imaginary and real root vectors through (and the related map), the authors obtain explicit -basis representations and generating-function identities. The results provide compact, computable -images for the PBW-type bases and their generating structures, expressed in terms of and the canonical commuting combinations in . This -realization facilitates structural understanding of and its variants, with potential implications for representations and combinatorial applications in quantum groups.

Abstract

The -Onsager algebra, denoted by , is defined by generators and two relations called the -Dolan-Grady relations. In 2017, Baseilhac and Kolb gave some elements of that form a Poincaré-Birkhoff-Witt basis. The quantum torus, denoted by , is defined by generators and relations The set is a basis for . It is known that there is an algebra homomorphism that sends and In 2020, Lu and Wang displayed a variation of , denoted by . Lu and Wang gave a surjective algebra homomorphism \medskip In their consideration of , Lu and Wang introduced some elements \begin{equation} \label{intrp503} \{B_{1,r}\}_{r \in \mathbb{Z}}, \qquad \{H'_n\}_{n=1}^{\infty}, \qquad \{H_n\}_{n=1}^{\infty}, \qquad \{Θ'_n\}_{n=1}^{\infty}, \qquad \{Θ_n\}_{n=1}^{\infty}. \nonumber \end{equation} These elements are defined using recursive formulas and generating functions, and it is difficult to express them in closed form. A similar problem applies to the Baseilhac-Kolb elements of . To mitigate this difficulty, we map everything to using and . In our main results, we express the resulting images in the basis for and also in an attractive closed form.

Paper Structure

This paper contains 16 sections, 53 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The $q$-Onsager algebra and the universal $q$-Onsager algebra
  4. Some elements of $\tilde{\mathbf{U}}^{\imath}$
  5. The Baseilhac-Kolb elements of $O_q$
  6. Various $\upsilon$-images in $O_q$
  7. The quantum torus $T_q$
  8. The homomorphism $p : O_q \mapsto T_q$
  9. The $p$-images of the Baseilhac-Kolb elements and $B_{1,r}$ elements
  10. The $p$-image of $\{\Theta'_n\}_{n=1}^{\infty}$
  11. The generating functions $\Theta'(t)$ and $\Theta(t)$ and their $p$-images
  12. The $p$-image of $\{\Theta_n\}_{n=1}^{\infty}$
  13. The $p$-images of $H'(t)$ and $H(t)$ in $T_q$
  14. The $p$-images of $\{H'_n\}_{n=1}^{\infty}$ and $\{H_n\}_{n=1}^{\infty}$
  15. Proposition \ref{['redgoldgreen']} revisited
  16. ...and 1 more sections

Key Result

Lemma 3.2

(See T2.) There exists an automorphism $\sigma$ of $O_q$ that sends

Theorems & Definitions (94)

  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Lemma 3.9
  • ...and 84 more