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

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 $$xx^{-1} = 1 = x^{-1}x, \qquad yy^{-1} = 1 = y^{-1}y, \qquad xy=q^2yx.$$ 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 \{\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.