Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and Uq (sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let 0 be a distance-regular graph with diameter d ≥ 3 and valency k ≥ 3, and assume0 is not isomorphic to the d-cube. Fix a vertex x of0, and let T = T (x) denote the Terwilliger algebra of 0 with respect to x . Fix any complex number q 6∈ {0, 1,−1}. Then T is generated by certain matrices satisfying the defining relations of Uq (sl(2)) if and only if 0 is bipartite and 2-homogeneous.