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Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms$T_0$, $T_1$ of the $q$-Onsager algebra $\mathcal O_q$, that are roughlyanalogous to the Lusztig automorphisms of $U_q(\widehat{\mathfrak{sl}}_2)$. Weuse $T_0, T_1$ and a certain antiautomorphism of $\mathcal O_q$ to obtain anaction of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on$\mathcal O_q$ as a group of (auto/antiauto)-morphisms. The action forms apattern much more symmetric than expected. We show that a similar phenomenonoccurs for the associated current algebra $\mathcal A_q$. We give someconjectures and problems concerning $\mathcal O_q$ and $\mathcal A_q$.

An action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on the $q$-Onsager algebra and its current algebra