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Ilwale, Kwalombota
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Ilwale, Kwalombota
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Algebra, Geometry and Discrete MathematicsFaculty of Science & Engineering
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Algebra and Logic
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Noncommutative Riemannian Geometry of Twisted DerivationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2023 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Linköping: Linköping University Electronic Press, 2023. , p. 63
##### Series

Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2304
##### National Category

Algebra and Logic
##### Identifiers

URN: urn:nbn:se:liu:diva-193539DOI: 10.3384/9789180751131ISBN: 9789180751124 (print)ISBN: 9789180751131 (electronic)OAI: oai:DiVA.org:liu-193539DiVA, id: diva2:1754625
##### Public defence

2023-06-02, BL32, B-building, Campus Valla, Linköping, 14:00 (English)
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt494",{id:"formSmash:j_idt494",widgetVar:"widget_formSmash_j_idt494",multiple:true}); Available from: 2023-05-04 Created: 2023-05-04 Last updated: 2023-05-04Bibliographically approved
##### List of papers

A twisted derivation is a generalized derivative satisfying a twisted version of the ordinary Leibniz rule for products. In particular, a (σ, τ )-derivation on an algebra A, is a derivation where Leibniz rule is twisted by two endomorphisms σ and τ on A. Such derivations play an important role in the theory of quantum groups, as well as in the context of discretized and deformed derivatives. In this thesis, we develop a (commutative and noncommutative) differential geometry based on (σ, τ )- derivations. To this end, we introduce the notion of (σ, τ )-algebra, consisting of an associative algebra together with a set of (σ, τ )-derivations, to construct connections satisfying a twisted Leibniz rule in analogy with (σ, τ )-derivations. We show that such connections always exist on projective modules and that it is possible to construct connections compatible with a hermitian form. To construct torsion and curvature of (σ, τ )-connections, we introduce the notion of (σ, τ )-Lie algebra and demonstrate that it is possible to construct a Levi-Civita (σ, τ )-connection. Having constructed the framework for studying (σ, τ)-connections, we demonstrate that the framework applied to commutative algebras can help to also give a good understanding of (σ, τ )-derivations on commutative algebras. In particular, we introduce a notion of symmetric (σ, τ )-derivations together with some regularity conditions. For example, we show that strongly regular (σ, τ )-derivations are always inner and there exist a symmetric (σ, τ )-connection on symmetric (σ, τ )- algebras. Finally, we introduce a (σ, τ)-Hochschild cohomology theory which in first degree captures the outer (σ, τ )-derivations of an associative algebra. Along the way, examples including both commutative and noncommutative algebras are presented to illustrate the novel concepts.

1. Levi-Civita Connections on Quantum Spheres$(function(){PrimeFaces.cw("OverlayPanel","overlay1687968",{id:"formSmash:j_idt543:0:j_idt547",widgetVar:"overlay1687968",target:"formSmash:j_idt543:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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