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Using the quantum torus to investigate the $q$-Onsager algebra (2504.13362v1)

Published 17 Apr 2025 in math.QA and math.CO

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 Poincar\'e-Birkhoff-Witt basis. The quantum torus, denoted by $T_q$, is defined by generators $x, y, x{-1}, y{-1}$ and relations $$xx{-1} = 1 = x{-1}x, \qquad yy{-1} = 1 = y{-1}y, \qquad xy=q2yx.$$ The set ${xiyj | i,j \in \mathbb{Z} }$ 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 {\Theta'_n}{n=1}{\infty}, \qquad {\Theta_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.

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