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${\left\{W_{-k}\right\}}_{k=0}^{\infty },{\left\{W_{k+1}\right\}}_{k=0}^{\infty },{\left\{G_{k+1}\right\}}_{k=0}^{\infty },{\left\{{\tilde{G}}_{k+1}\right\}}_{k=0}^{\infty }.$

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.