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

TL;DR

This paper introduces a simplified, compact presentation for the alternating central extension of the $q$-Onsager algebra, making its structure more manageable and easier to work with.

Contribution

It provides a new, concise presentation of the algebra that reduces complexity compared to previous descriptions.

Findings

The compact presentation involves fewer generators and relations.

It simplifies the algebra's structure, facilitating further research.

The presentation resembles that of a related quantum 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 $U_q(\widehat{\mathfrak{sl}}_2)$.