Let $\{ A_j|j=0,1,...,rank(g)\}$ be the fundamental generators of the generalized $q-$Onsager algebra $\cal O_{ q}(\widehat{ g})$ introduced in \cite{ BB1}, 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 analogues of Lusztig's $r-$th 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.
Categories: math-ph cond-mat.stat-mech hep-th math.MP math.QA