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Morphisms of real calculi from a geometric and algebraic perspective
Linköping University, Department of Mathematics, Algebra, Geometry and Discrete Mathematics. Linköping University, Faculty of Science & Engineering.
2021 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting.

This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere.

Abstract [sv]

Ickekommutativ geometri har under de senaste fyra decennierna blivit ett etablerat forskningsområde inom matematiken. Nya idéer och koncept utvecklas i snabb takt, och en viktig fysikalisk tillämpning av teorin är inom kvantteorin. Denna avhandling kommer att fokusera på ett derivationsbaserat tillvägagångssätt inom ickekommutativ geometri där ramverket real calculi används, vilket är ett relativt direkt sätt att studera ämnet på. Eftersom analogin mellan real calculi och klassisk Riemanngeometri är intuitivt klar så är real calculi användbara när man undersöker hur klassiska koncept inom Riemanngeometri kan generaliseras till en ickekommutativ kontext.

Denna avhandling ämnar att klargöra vissa algebraiska aspekter av real calculi genom att introducera morfismer för dessa, vilket möjliggör studiet av real calculi på en strukturell nivå. I synnerhet diskuteras real calculi över matrisalgebror från både ett algebraiskt och ett geometriskt perspektiv. Morfismer tolkas även geometriskt, vilket leder till en ickekommutativ teori för inbäddningar. Som ett exempel blir den ickekommutativa torusen minimalt inbäddad i den ickekommutativa 3-sfären.  

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2021. , p. 46
Series
Linköping Studies in Science and Technology. Licentiate Thesis, ISSN 0280-7971 ; 1913
Keywords [en]
noncommutative geometry, embeddings, matrix algebras, real calculi, morphisms, free module, projective module
National Category
Algebra and Logic Geometry
Identifiers
URN: urn:nbn:se:liu:diva-175740DOI: 10.3384/lic.diva-175740ISBN: 9789179296162 (print)OAI: oai:DiVA.org:liu-175740DiVA, id: diva2:1555580
Presentation
2021-06-10, BL32, B-building, Linköpings universitet, Linköping, 09:00 (English)
Opponent
Supervisors
Available from: 2021-05-20 Created: 2021-05-19 Last updated: 2021-05-20Bibliographically approved
List of papers
1. Noncommutative minimal embeddings and morphisms of pseudo-Riemannian calculi
Open this publication in new window or tab >>Noncommutative minimal embeddings and morphisms of pseudo-Riemannian calculi
2021 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 159, article id 103898Article in journal (Refereed) Published
Abstract [en]

In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together with noncommutative embeddings. We show that basic concepts, such as the second fundamental form and the Weingarten map, translate into the noncommutative setting and, in particular, we prove a noncommutative analogue of Gauss equations for the curvature of a submanifold. Moreover, the mean curvature of an embedding is readily introduced, giving a natural definition of a noncommutative minimal embedding, and we illustrate the novel concepts by considering the noncommutative torus as a minimal surface in the noncommutative 3-sphere. (c) 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Place, publisher, year, edition, pages
ELSEVIER, 2021
Keywords
Noncommutative minimal submanifold; Noncommutative embedding; Noncommutative Levi-Civita connection
National Category
Other Physics Topics
Identifiers
urn:nbn:se:liu:diva-172402 (URN)10.1016/j.geomphys.2020.103898 (DOI)000596080700016 ()
Note

Funding Agencies|Swedish Research CouncilSwedish Research Council [2017-03710]

Available from: 2021-01-10 Created: 2021-01-10 Last updated: 2023-10-16
2. Projective real calculi over matrix algebras
Open this publication in new window or tab >>Projective real calculi over matrix algebras
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In analogy with the geometric situation, we study real calculi over projective modules and describe how they are related to free real calculi using real calculus homomorphisms. Moreover, we study real calculi over matrix algebras and discuss several aspects of the classification problem for real calculi in this case. We also use matrix algebras to give concrete examples of real calculi where the module is projective, and how this affects the existence of a Levi-Civita connection. 

Keywords
noncommutative geometry, matrix algebras, real calculi, morphisms, free module, projective module
National Category
Algebra and Logic
Identifiers
urn:nbn:se:liu:diva-175742 (URN)10.48550/arXiv.2107.04627 (DOI)
Available from: 2021-05-17 Created: 2021-05-17 Last updated: 2023-10-16Bibliographically approved

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Tiger Norkvist, Axel
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