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

Abstract

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D,b,\alpha ,\beta )$ and $b\ne 1$, $\alpha =b-1$. 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 ${⊠}_q$ on the standard module of $\Gamma$. We describe four algebra homomorphisms into ${⊠}_q$ from the quantum affine algebra $U_q({\widehat{\mathfrak{sl}}}_2)$; using these we pull back the above ${⊠}_q$-action to obtain four actions of $U_q({\widehat{\mathfrak{sl}}}_2)$ on the standard module of $\Gamma$.

22 pages