The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

Let Γ denote a 2-homogeneous bipartite distance-regular graph with diameter D⩾3 and valency k⩾3. Assume that Γ is not isomorphic to a Hamming cube. Fix a vertex x of Γ, and let T=T(x) denote the Terwilliger algebra of T with respect to x. We give three sets of generators for T, two of which satisfy the relations of the quantum universal enveloping algebra of the Lie algebra sl(2). We then describe the simple T-modules. We give a pair of canonical bases for each simple T-module, and we give the overlap function for these bases in terms of a basic hypergeometric function. Finally, we give two generators for the center of T.