Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra

TL;DR

This paper establishes new identities within the higher rank Askey-Wilson algebra, generalizing known relations, and introduces methods that simplify calculations and extend to the $q$-Bannai-Ito algebra.

Contribution

It proves a class of identities in higher rank Askey-Wilson algebra and extends the construction algorithm with a novel coaction, simplifying complex calculations.

Findings

Generalized defining relations for higher rank Askey-Wilson algebra

New methods using a coaction to simplify algebraic calculations

Proof of relations for higher rank $q$-Bannai-Ito algebra

Abstract

The higher rank Askey-Wilson algebra was recently constructed in the $n$-fold tensor product of $U_q(\mathfrak{sl}_2)$. In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey-Wilson algebra. We extend the known construction algorithm by several equivalent methods, using a novel coaction. These allow to simplify calculations significantly. At the same time, this provides a proof of the corresponding relations for the higher rank $q$-Bannai-Ito algebra.