The compact presentation for the alternating central extension of the $q$-Onsager algebra

Abstract

The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension ${\mathcal{O}}_q$ of $O_q$. The algebra ${\mathcal{O}}_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of ${\mathcal{O}}_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of ${\mathcal{O}}_q$ that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of ${\mathcal{O}}_q$. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$.

26 pages. arXiv admin note: text overlap with arXiv:2103.03028

Cited by in corpus (1)

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