Matrices, Bratteli Diagrams and Hopf-Galois Extensions

We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf-Galois extensions (quantum principle bundles) for certain abelian groups. The corresponding strong universal connections are computed. We show that $M_{n}(\mathbb{C})$ is a trivial quantum principle bundle for the Hopf algebra $\mathbb{C}[\mathbb{Z}_{n} \times \mathbb{Z}_{n}]$. We conclude with an application relating known calculi on groups to calculi on matrices.

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