An action of the free product Z 2 ? Z 2 ? Z 2 on the q-Onsager algebra and its current algebra

摘要

Recently Pascal Baseilhac and Stefan Kolb introduced some automorphismsT0,T1of theq-Onsager algebraOq, that are roughly analogous to the Lusztig automorphisms ofUq(sl?2). We useT0,T1and a certain antiautomorphism ofOqto obtain an action of the free productZ2?Z2?Z2onOqas 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 algebraAq. We give some conjectures and problems concerningOqandAq.