Using the quantum torus to investigate the $q$-Onsager algebra
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2025
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| _version_ | 1866908325706203136 |
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| author | Goff, Owen |
| author_facet | Goff, Owen |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Using the quantum torus to investigate the $q$-Onsager algebra Goff, Owen Quantum Algebra Combinatorics 17B37 (Primary) 05E16, 16T20 (Secondary) 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. |
| title | Using the quantum torus to investigate the $q$-Onsager algebra |
| topic | Quantum Algebra Combinatorics 17B37 (Primary) 05E16, 16T20 (Secondary) |
| url | https://arxiv.org/abs/2504.13362 |