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

Abstract

Let $Γ$ denote a distance-regular graph with classical parameters $(D,b,α,β)$ and $b\not=1$, $α=b-1$. The condition on $α$ implies that $Γ$ 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 $Γ$. 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 $Γ$.