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 $\{{W}_{-k}\}_{k=0}^{\infty}, \{{W}_{k+1}\}_{k=0}^{\infty}, \{{G}_{k+1}\}_{k=0}^{\infty}, \{{\tilde{G}}_{k+1}\}_{k=0}^{\infty}$. 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.