Distance-regular graphs of $q$-Racah type and the $q$-tetrahedron algebra

TL;DR

This paper explores the connection between distance-regular graphs of $q$-Racah type and the $q$-tetrahedron algebra, establishing a homomorphism and generating properties under certain conditions.

Contribution

It introduces a homomorphism from the $q$-tetrahedron algebra to the subconstituent algebra of a $q$-Racah type graph and shows how the latter is generated.

Findings

Established a homomorphism from $oxtimes_q$ to $T$.

Proved $T$ is generated by the image and the center $Z(T)$.

Assumed all irreducible $T$-modules are thin.

Abstract

In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra $T$ of a distance-regular graph that has $q$-Racah type; (ii) the $q$-tetrahedron algebra $\boxtimes_q$ which is a $q$-deformation of the three-point $sl_2$ loop algebra. Assuming that every irreducible $T$-module is thin, we display an algebra homomorphism from $\boxtimes_q$ into $T$ and show that $T$ is generated by the image together with the center $Z(T)$.