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