The \(q\)-Onsager algebra \(O_q\) is a quantized enveloping algebra for the Onsager Lie algebra \(O\). It has a presentation with two generators \(W_0, W_1\) and two relations, called the \(q\)-Dolan/Grady relations. A PBW basis for \(O_q\) was obtained by P.Baseilhac and S.Kolb. The PBW basis elements for \(O_q\) are denoted \(\{B_{n\delta+\alpha_0}\}_{n=0}^\infty\), \(\{B_{n\delta+\alpha_1}\}_{n=0}^\infty\), and \(\{B_{n\delta}\}_{n=1}^\infty\). A current algebra for the \(q\)-Onsager algebra \(O_q\) was introduced by P.Baseilhac and K.Koizumi in order to solve boundary integrable systems. This current algebra is denoted by \(\mathcal{O}_q\). The algebra \(\mathcal{O}_q\) gives a central extension of \(O_q\) and has a presentation involving a large number of generators \(\{\mathcal{W}_{-k}\}_{k=0}^\infty\), \(\{\mathcal{W}_{k+1}\}_{k=0}^\infty\), \(\{\mathcal{G}_{k+1}\}_{k=0}^{\infty}\), \(\{\tilde{\mathcal{G}}_{k+1}\}_{k=0}^\infty\), which form a PBW basis for \(\mathcal{O}_q\), and a large number of relations. It is known that there exists an algebra isomorphism \(O_q \to \langle\mathcal{W}_0, \mathcal{W}_1 \rangle\) that sends \(W_0 \to \mathcal{W}_0\) and \(W_1 \to \mathcal{W}_1\). It is known that the center \(\mathcal{Z}\) of \(\mathcal{O}_q\) is isomorphic to polynomial algebra in countably many variables. It is known that the multiplication map \(\langle \mathcal{W}_0, \mathcal{W}_1\rangle \otimes \mathcal{Z} \to \mathcal{O}_q\), \(w \otimes z\to wz\) is an isomorphism of algebras. We call this isomorphism the standard tensor product factorization of \(\mathcal{O}_q\). In the study of \(\mathcal{O}_q\) there are two point of view: we can start with the alternating generators, or we can start with the standard tensor product factorization. The goal of this paper is to describe this relationship. In this paper P.Terwilliger obtains seven main results and gives some conjectures and open problems.
P.Terwilliger shows relations between generating functions for \(O_q\) and \(\mathcal{O}_q\) as follows. \[ \frac{q+q^{-1}}{t+t^{-1}}\mathcal{W}^-\left(\frac{q+q^{-1}}{t+t^{-1}}\right) =\frac{q^{-1}t B^+(q^{-1}t)+B^-(qt)}{ (q^2-q^{-2})(t-t^{-1})}\tilde{\mathcal{G}}\left(\frac{q+q^{-1}}{t+t^{-1}}\right), \] \[ \frac{q+q^{-1}}{t+t^{-1}}\mathcal{W}^+\left(\frac{q+q^{-1}}{t+t^{-1}}\right) =\frac{B^+(q^{-1}t)+qt B^-(qt)}{ (q^2-q^{-2})(t-t^{-1})}\tilde{\mathcal{G}}\left(\frac{q+q^{-1}}{t+t^{-1}}\right), \] \[ \frac{q+q^{-1}}{t+t^{-1}}\mathcal{W}^-\left(\frac{q+q^{-1}}{t+t^{-1}}\right) = \tilde{\mathcal{G}}\left(\frac{q+q^{-1}}{t+t^{-1}}\right) \frac{qt B^+(qt)+B^-(q^{-1}t)}{ (q^2-q^{-2})(t-t^{-1})}, \] \[ \frac{q+q^{-1}}{t+t^{-1}}\mathcal{W}^+\left(\frac{q+q^{-1}}{t+t^{-1}}\right) = \tilde{\mathcal{G}}\left(\frac{q+q^{-1}}{t+t^{-1}}\right) \frac{B^+(qt)+q^{-1}t B^-(q^{-1}t)}{ (q^2-q^{-2})(t-t^{-1})}. \] Here we use \[ \mathcal{W}^-(t)=\sum_{n=0}^\infty \mathcal{W}_{-n}t^n,~~ \mathcal{W}^+(t)=\sum_{n=0}^\infty \mathcal{W}_{n+1}t^n,~~ \tilde{\mathcal G}(t)=\sum_{n=0}^\infty \tilde{\mathcal G}_{n}t^n,~~ \tilde{\mathcal G}_0=-(q-q^{-1})[2]_q^2, \] \[ {B}^-(t)=\sum_{n=0}^\infty {B}_{n\delta+\alpha_0}t^n,~~ {B}^+(t)=\sum_{n=0}^\infty {B}_{n\delta+\alpha_1}t^n. \] They are the first and the second main results of this paper. In this paper, in addition to the two main results mentioned above, the author obtains five main results. Together, the author obtains seven main results. The author introduces alternating generators of \(\mathcal{Z}\) and \(O_q\) and obtains relations between them in the similar way as the first main result.