Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into U(sl2)

From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism ♯ from the universal Racah algebra ℜ into U(sl2). We use the Casimir elements of ℜ to describe the kernel of ♯. By pulling back via ♯ every U(sl2)-module can be viewed as an ℜ-module. We show that for any finite-dimensional U(sl2)-module V, the ℜ-module V is completely reducible and three generators of ℜ act on every irreducible ℜ-submodule of V as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube H(D, 2) and a decomposition of the second distance operator of H(D, 2) induced by Johnson graphs.