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 into U(sl₂). We use the Casimir elements of to describe the kernel of . By pulling back via every U(sl₂)-module can be viewed as an -module. We show that for any finite-dimensional U(sl₂)-module V, the -module V is completely reducible and three generators of act on every irreducible -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.