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 O q of O q . The algebra 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 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 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 O q . This presentation resembles the compact presentation of the alternating central extension for the positive part of U q ( sl 2 ).

The alternating central extension of the Onsager Lie algebra

“…Shortly thereafter they appeared in mathematical physics [2, Section 2]. Up to the present, O q remains an active research topic in mathematics [13,27,29,32,[43][44][45][46][47][48][49][52][53][54][55] and physics [1,2,[4][5][6][7][10][11][12]14,15].…”

The $q$-Onsager algebra and its alternating central extension

“…This paper is part of a sequence [14], [15], [16] concerning the q-Onsager algebra and its alternating central extension. We refer to those papers for background information and historical remarks.…”

“…In [3] Baseilhac and Koizumi introduce a current algebra for O q , in order to solve boundary integrable systems with hidden symmetries. Following [16] we denote this current algebra by O q . In [5, Definition 3.1] Baseilhac and Shigechi give a presentation of O q by generators and relations.…”