Abstract
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.
Highlights
We will be discussing the q-Onsager algebra Oq [2, 20]
The algebra Oq is a q-deformation of the Onsager algebra from mathematical physics [17], [21, Remark 9.1] and is currently being used to investigate statistical mechanical models such as the XXZ open spin chain [1, 2, 4, 6,7,8]
The algebra Oq appears in the theory of tridiagonal pairs; this is a pair of diagonalizable linear transformations on a finitedimensional vector space, each acting on the eigenspaces of the other in a block-tridiagonal fashion [13,19]
Summary
We will be discussing the q-Onsager algebra Oq [2, 20]. This algebra is infinite-dimensional and noncommutative, with a presentation involving two generators and two relations called the q-Dolan/Grady relations. The algebra Oq comes up in algebraic combinatorics, in connection with the subconstituent algebra of a Q-polynomial distance-regular graph [13, 18]. This topic is where Oq originated; to our knowledge the q-Dolan/Grady relations first appeared in [18, Lemma 5.4]. At the end of the paper we give some conjectures and problems concerning Oq and Aq
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