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

Abstract

The q-Onsager algebra Oq is defined by two generators and two relations, called the q-Dolan/Grady relations. We investigate the alternating central extension Oq of Oq. The algebra Oq was introduced by Baseilhac and Koizumi, who called it the current algebra of Oq. Recently Baseilhac and Shigechi gave a presentation of Oq 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 Oq 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 Oq. This presentation resembles the compact presentation of the alternating central extension for the positive part of Uq(slˆ2).