An action of the free product Z2⋆Z2⋆Z2 on the q-Onsager algebra and its current algebra

Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms T0, T1 of the q-Onsager algebra Oq, that are roughly analogous to the Lusztig automorphisms of Uq(slˆ2). We use T0,T1 and a certain antiautomorphism of Oq to obtain an action of the free product Z2⋆Z2⋆Z2 on Oq as a group of (auto/antiauto)-morphisms. The action forms a pattern much more symmetric than expected. We show that a similar phenomenon occurs for the associated current algebra Aq. We give some conjectures and problems concerning Oq and Aq.