Higher order relations for ADE-type generalized q-Onsager algebras

TL;DR

This paper explores higher order relations in generalized q-Onsager algebras related to affine Lie algebras, proposing new conjectured relations that generalize Lusztig's higher order q-Serre relations, with proofs for cases up to r=5.

Contribution

It introduces new conjectured relations between monomials of fundamental generators in generalized q-Onsager algebras, extending Lusztig's higher order q-Serre relations, and provides proofs for specific cases.

Findings

Relations proven for r ≤ 5

Conjectured relations for all r

Supporting evidence for generic r

Abstract

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.