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The $q$-Onsager algebra and its alternating central extension

The $q$-Onsager algebra $O_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. The alternating central extension $\mathcal O_q$ has a presentation involving the alternating generators $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, $\lbrace \mathcal W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1}\rbrace_{k=0}^\infty$, $\lbrace \mathcal {\tilde G}_{k+1}\rbrace_{k=0}^\infty$ and a large number of relations. Let $\langle \mathcal W_0, \mathcal W_1 \rangle$ denote the subalgebra of $\mathcal O_q$ generated by $\mathcal W_0$, $\mathcal W_1$. It is known that there exists an algebra isomorphism $O_q \to \langle \mathcal W_0, \mathcal W_1 \rangle$ that sends $W_0\mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. It is known that the center $\mathcal Z$ of $\mathcal O_q$ is isomorphic to a 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 \mapsto 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 natural points of view: we can start with the alternating generators, or we can start with the standard tensor product factorization. It is not obvious how these two points of view are related. The goal of the paper is to describe this relationship. We give seven main results; the principal one is an attractive factorization of the generating function for some algebraically independent elements that generate $\mathcal Z$.