Higher Order Relations for ADE-Type Generalized q-Onsager Algebras

Abstract

Let $${\left\{\mathsf{A}_j|j=0,1,\ldots,rank(g)\right\}}$$ be the fundamental generators of the generalized q-Onsager algebra $${{\cal O}_{q}({\widehat{g}})}$$ introduced in Baseilhac and Belliard (Lett Math Phys 93:213–228, 2010), where $${\widehat{g}}$$ is a simply laced affine Lie algebra. New relations between certain monomials of the fundamental generators—indexed by the integer $${r\in\mathbb{Z}^{+}}$$ —are conjectured. These relations can be seen as deformed analogs of Lusztig’s rth higher order q-Serre relations associated with $${{\cal U}_q({\widehat g})}$$ , which are recovered as special cases. The relations are proven for $${r\leq 5}$$ . For r generic, several supporting evidences are presented.