Distance-regular graphs and the -tetrahedron algebra

Let Γ denote a distance-regular graph with classical parameters ( D , b , α , β ) and b ≠ 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 ⊠ q on the standard module of Γ . We describe four algebra homomorphisms into ⊠ q from the quantum affine algebra U q ( s l ̂ 2 ) ; using these we pull back the above ⊠ q -action to obtain four actions of U q ( s l ̂ 2 ) on the standard module of Γ .