The Generalized Terwilliger Algebra of the Hypercube

In the year 2000, Eric Egge introduced the generalized Terwilliger algebra $\mathcal{T}$ of a distance-regular graph $\mathrm{\Gamma }$. For any vertex $x$ of $\mathrm{\Gamma },$ there is a surjective algebra homomorphism $\mathrm{♮}$ from $\mathcal{T}$ to the Terwilliger algebra $T(x)$. If $\mathrm{\Gamma }$ is complete, then $\mathrm{♮}$ is an isomorphism. If $\mathrm{\Gamma }$ is not complete, then $\mathrm{♮}$ may or may not be an isomorphism, and in general the details are unknown. We show that if $\mathrm{\Gamma }$ is a hypercube, then the algebra homomorphism $\mathrm{♮}:\mathcal{T}\to T(x)$ is an isomorphism for all vertices $x$ of $\mathrm{\Gamma }$.