The Generalized Terwilliger Algebra of the Hypercube

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