Bratteli Diagrams, Hopf–Galois Extensions and Calculi

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Hopf–Galois extensions extend the idea of principal bundles to noncommutative geometry, using Hopf algebras as symmetries. We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf–Galois extensions (quantum principal bundles) for certain finite abelian groups. The corresponding strong universal connections are computed. We show that Mn(C) is a trivial quantum principle bundle for the Hopf algebra C[Zn×Zn]. We conclude with an application relating calculi on groups to calculi on matrices.