Distance-regular graphs and the $q$-tetrahedron algebra

TL;DR

This paper explores the algebraic structure of certain distance-regular graphs with specific parameters, establishing connections with the $q$-tetrahedron algebra and quantum affine algebra to understand their symmetries and representations.

Contribution

It introduces a novel action of the $q$-tetrahedron algebra on these graphs and constructs four related actions of the quantum affine algebra $U_q( ext{sl}_2)$, linking graph theory with quantum algebra.

Findings

Established an action of $oxtimes_q$ on the standard module of $ ext{distance-regular graphs}$

Constructed four algebra homomorphisms from $U_q( ext{sl}_2)$ to $oxtimes_q$

Derived four $U_q( ext{sl}_2)$-actions on the standard module of the graphs

Abstract

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$ and $b\not=1$, $\alpha=b-1$. The condition on $\alpha$ implies that $\Gamma$ is formally self-dual. For $b=q^2$ we use the adjacency matrix and dual adjacency matrix to obtain an action of the $q$-tetrahedron algebra $\boxtimes_q$ on the standard module of $\Gamma$. We describe four algebra homomorphisms into $\boxtimes_q$ from the quantum affine algebra $U_q({\hat{\mathfrak{sl}}_2})$; using these we pull back the above $\boxtimes_q$-action to obtain four actions of $U_q({\hat{\mathfrak{sl}}_2})$ on the standard module of $\Gamma$.