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Bratteli Diagrams, Hopf–Galois Extensions and Calculi

Ghaliah Alhamzi Orcid Logo, Edwin Beggs Orcid Logo

Symmetry, Volume: 17, Issue: 2, Start page: 164

Swansea University Author: Edwin Beggs Orcid Logo

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DOI (Published version): 10.3390/sym17020164

Abstract

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 cor...

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Published in: Symmetry
ISSN: 2073-8994
Published: MDPI AG 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa68985
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spelling 2025-03-11T14:29:15.3397869 v2 68985 2025-02-27 Bratteli Diagrams, Hopf–Galois Extensions and Calculi a0062e7cf6d68f05151560cdf9d14e75 0000-0002-3139-0983 Edwin Beggs Edwin Beggs true false 2025-02-27 MACS 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 (ℂ) is a trivial quantum principle bundle for the Hopf algebra ℂ[ℤ×ℤ]. We conclude with an application relating calculi on groups to calculi on matrices. Journal Article Symmetry 17 2 164 MDPI AG 2073-8994 Hopf–Galois extensions; Bratteli diagrams; differential calculi 22 1 2025 2025-01-22 10.3390/sym17020164 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Another institution paid the OA fee The research is funded by Imam Mohammad Ibn Saud Islamic University for covering the open access fee. 2025-03-11T14:29:15.3397869 2025-02-27T14:40:46.9146671 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Ghaliah Alhamzi 0000-0002-9146-7145 1 Edwin Beggs 0000-0002-3139-0983 2 68985__33698__22112d6077be4d62ab0cf4f5f07c52ee.pdf symmetry-17-00164-v2.pdf 2025-02-27T14:45:16.4276827 Output 274648 application/pdf Accepted Manuscript true © 2025 by the authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. true eng https://creativecommons.org/ licenses/by/4.0/
title Bratteli Diagrams, Hopf–Galois Extensions and Calculi
spellingShingle Bratteli Diagrams, Hopf–Galois Extensions and Calculi
Edwin Beggs
title_short Bratteli Diagrams, Hopf–Galois Extensions and Calculi
title_full Bratteli Diagrams, Hopf–Galois Extensions and Calculi
title_fullStr Bratteli Diagrams, Hopf–Galois Extensions and Calculi
title_full_unstemmed Bratteli Diagrams, Hopf–Galois Extensions and Calculi
title_sort Bratteli Diagrams, Hopf–Galois Extensions and Calculi
author_id_str_mv a0062e7cf6d68f05151560cdf9d14e75
author_id_fullname_str_mv a0062e7cf6d68f05151560cdf9d14e75_***_Edwin Beggs
author Edwin Beggs
author2 Ghaliah Alhamzi
Edwin Beggs
format Journal article
container_title Symmetry
container_volume 17
container_issue 2
container_start_page 164
publishDate 2025
institution Swansea University
issn 2073-8994
doi_str_mv 10.3390/sym17020164
publisher MDPI AG
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
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description 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 (ℂ) is a trivial quantum principle bundle for the Hopf algebra ℂ[ℤ×ℤ]. We conclude with an application relating calculi on groups to calculi on matrices.
published_date 2025-01-22T08:18:47Z
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