The $q$-Onsager Algebra and the Quantum Torus

The $q$-Onsager algebra, denoted $O_q$, is defined by two generators $W_0,W_1$ and two relations called the $q$-Dolan-Grady relations. Recently, Terwilliger introduced some elements of $O_q$, said to be alternating. These elements are denoted $\{W_{-k}{\}}_{k=0}^{\mathrm{\infty }},\{W_{k+1}{\}}_{k=0}^{\mathrm{\infty }},\{G_{k+1}{\}}_{k=0}^{\mathrm{\infty }},\{{\tilde{G}}_{k+1}{\}}_{k=0}^{\mathrm{\infty }}$. The alternating elements of $O_q$ are defined recursively. By construction, they are polynomials in $W_0$ and $W_1$. It is currently unknown how to express these polynomials in closed form. In this paper, we consider an algebra $T_q$, called the quantum torus. We present a basis for $T_q$ and define an algebra homomorphism $p:O_q\mapsto T_q$. In our main result, we express the $p$-images of the alternating elements of $O_q$ in the basis for $T_q$. These expressions are in a closed form that we find attractive.