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

Abstract

Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms $T_0$, $T_1$ of the $q$-Onsager algebra ${\mathcal{O}}_q$, that are roughly analogous to the Lusztig automorphisms of $U_q({\widehat{\mathfrak{sl}}}_2)$. We use $T_0,T_1$ and a certain antiautomorphism of ${\mathcal{O}}_q$ to obtain an action 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 a pattern much more symmetric than expected. We show that a similar phenomenon occurs for the associated current algebra ${\mathcal{A}}_q$. We give some conjectures and problems concerning ${\mathcal{O}}_q$ and ${\mathcal{A}}_q$.

15 pages

References in corpus (6)

• The Universal Askey-Wilson Algebra
• Some algebra related to $P$-and $Q$-polynomial association schemes
• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• An attractive basis for the $q-$Onsager algebra
• The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• How to sharpen a tridiagonal pair

Cited by in corpus (7)

• The alternating central extension of the $q$-Onsager algebra
• The alternating PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A conjecture concerning the $q$-Onsager algebra
• The $q$-Onsager algebra and its alternating central extension
• A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A new PBW basis for the alternating central extension of the $q$-Onsager algebra