Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U({\mathfrak{s}\mathfrak{l}}_2)$

From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\mathrm{♯}$ from the universal Racah algebra $\mathrm{\Re }$ into $U({\mathfrak{s}\mathfrak{l}}_2)$. We use the Casimir elements of $\mathrm{\Re }$ to describe the kernel of $\mathrm{♯}$. By pulling back via $\mathrm{♯}$ every $U({\mathfrak{s}\mathfrak{l}}_2)$-module can be viewed as an $\mathrm{\Re }$-module. We show that for any finite-dimensional $U({\mathfrak{s}\mathfrak{l}}_2)$-module $V$, the $\mathrm{\Re }$-module $V$ is completely reducible and three generators of $\mathrm{\Re }$ act on every irreducible $\mathrm{\Re }$-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.