Leonard triples and hypercubes

Let V denote a vector space over C with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear operators on V such that for each of these operators there exists a basis of V with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let D denote a positive integer and let QD denote the graph of the D-dimensional hypercube. Let X denote the vertex set of QD and let A ∈ MatX(C) denote the adjacency matrix of QD . Fix x ∈X and let A∗ ∈ MatX(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of MatX(C) generated by A,A∗. We refer to T as the Terwilliger algebra of QD with respect to x. The matrices A and A∗ are related by the fact that 2iA=A∗Aε −AεA∗ and 2iA∗ =AεA−AAε, where 2iA =AA∗ −A∗A and i2 =−1. We show that the triple A, A∗, A acts on each irreducible T -module as a Leonard triple. We give a detailed description of these Leonard triples.