Distance-regular graphs and the qq-tetrahedron algebra

Let ΓΓ denote a distance-regular graph with classical parameters (D,b,α,β)(D,b,α,β) and b≠1b≠1, α=b−1α=b−1. The condition on αα implies that ΓΓ is formally self-dual. For b=q2b=q2 we use the adjacency matrix and dual adjacency matrix to obtain an action of the qq-tetrahedron algebra ⊠q⊠q on the standard module of ΓΓ. We describe four algebra homomorphisms into ⊠q⊠q from the quantum affine algebra Uq(sl̂2); using these we pull back the above ⊠q⊠q-action to obtain four actions of Uq(sl̂2) on the standard module of ΓΓ.