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 Poincaré-Birkhoff-Witt basis. The quantum torus, denoted by $T_q$, is defined by generators $x,y,x^{-1},y^{-1}$ and relations $$x{x}^{-1}=1=x^{-1}x,y{y}^{-1}=1=y^{-1}y,xy=q^{2}yx\mathrm{.}$$ The set $\{x^i{y}^j\mathrm{\mid }i,j\in \mathbb{Z}\}$ 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 $W_1\mapsto y+y^{-1}\mathrm{.}$ In 2020, Lu and Wang displayed a variation of $O_q$, denoted by ${\tilde{\mathbf{U}}}^{\mathrm{ı}}$. Lu and Wang gave a surjective algebra homomorphism $\upsilon :{\tilde{\mathbf{U}}}^{\mathrm{ı}}\mapsto O_q\mathrm{.}$ \medskip In their consideration of ${\tilde{\mathbf{U}}}^{\mathrm{ı}}$, 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.