{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:42:57Z","timestamp":1759970577987,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,22]],"date-time":"2025-01-22T00:00:00Z","timestamp":1737504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Imam Mohammad Ibn Saud Islamic University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Hopf\u2013Galois 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\u2013Galois 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\u00d7Zn]. We conclude with an application relating calculi on groups to calculi on matrices.<\/jats:p>","DOI":"10.3390\/sym17020164","type":"journal-article","created":{"date-parts":[[2025,1,22]],"date-time":"2025-01-22T09:52:20Z","timestamp":1737539540000},"page":"164","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bratteli Diagrams, Hopf\u2013Galois Extensions and Calculi"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9146-7145","authenticated-orcid":false,"given":"Ghaliah","family":"Alhamzi","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3139-0983","authenticated-orcid":false,"given":"Edwin","family":"Beggs","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Swansea University, Swansea SA2 8PP, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF02764619","article-title":"Principal homogeneous spaces for arbitrary Hopf algebras","volume":"72","author":"Schneider","year":"1990","journal-title":"Isr. 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