Matrices, Bratteli Diagrams and Hopf-Galois Extensions

TL;DR

This paper explores the structure of matrix embeddings in Bratteli diagrams, revealing their relation to Hopf-Galois extensions and quantum principal bundles, and computes universal connections with applications to group calculi.

Contribution

It demonstrates that matrix embeddings are iterated direct sums of Hopf-Galois extensions and computes associated universal connections, linking group calculi to matrix calculi.

Findings

Matrix embeddings are iterated direct sums of Hopf-Galois extensions.

The algebra M_n(C) forms a trivial quantum principal bundle for C[Z_n x Z_n].

Connections on matrices relate to known group calculi.

Abstract

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.