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