Distance-regular graphs and the q-tetrahedron algebra

Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and b 6= 1, α = b − 1. The condition on α implies that Γ is formally self-dual. For b = q we use the adjacency matrix and dual adjacency matrix to obtain an action of the q-tetrahedron algebra ⊠q on the standard module of Γ. We describe four algebra homomorphisms into ⊠q from the quantum affine algebra Uq(ŝl2); using these we pull back the above ⊠q-action to obtain four actions of Uq(ŝl2) on the standard module of Γ.