liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Kahler-Poisson algebras
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-8727-2169
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2019 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 136, p. 156-172Article in journal (Refereed) Published
Abstract [en]

We introduce Kahler-Poisson algebras as analogues of algebras of smooth functions on Kahler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a Kahler-Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated Kahler-Poisson algebra constructed as a localization. At the end, detailed examples are provided in order to illustrate the novel concepts. (C) 2018 Elsevier B.V. All rights reserved.

Place, publisher, year, edition, pages
ELSEVIER SCIENCE BV , 2019. Vol. 136, p. 156-172
Keywords [en]
Lie-Rinehart algebra; Kahler manifold; Levi-Civita connection; Curvature
National Category
Other Physics Topics
Identifiers
URN: urn:nbn:se:liu:diva-154675DOI: 10.1016/j.geomphys.2018.11.001ISI: 000456763300013OAI: oai:DiVA.org:liu-154675DiVA, id: diva2:1292869
Note

Funding Agencies|Swedish Research Council [621-2013-4538]

Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2020-01-20
In thesis
1. Kähler-Poisson Algebras
Open this publication in new window or tab >>Kähler-Poisson Algebras
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we introduce Kähler-Poisson algebras and study their basic properties. The motivation comes from differential geometry, where one can show that the Riemannian geometry of an almost Kähler manifold can be formulated in terms of the Poisson algebra of smooth functions on the manifold. It turns out that one can identify an algebraic condition in the Poisson algebra (together with a metric) implying that most geometric objects can be given a purely algebraic formulation. This leads to the definition of a Kähler-Poisson algebra, which consists of a Poisson algebra and a metric fulfilling an algebraic condition. We show that every Kähler- Poisson algebra admits a unique Levi-Civita connection on its module of inner derivations and, furthermore, that the corresponding curvature operator has all the classical symmetries. Moreover, we present a construction procedure which allows one to associate a Kähler-Poisson algebra to a large class of Poisson algebras. From a more algebraic perspective, we introduce basic notions, such as morphisms and subalgebras, as well as direct sums and tensor products. Finally, we initiate a study of the moduli space of Kähler-Poisson algebras; i.e for a given Poisson algebra, one considers classes of metrics giving rise to non-isomorphic Kähler-Poisson algebras. As it turns out, even the simple case of a Poisson algebra generated by two variables gives rise to a nontrivial classification problem.

Abstract [sv]

I denna avhandling introduceras Kähler-Poisson algebror och deras grundläggande egenskaper studeras. Motivationen till detta kommer från differentialgeometri där man kan visa att den metriska geometrin för en Kählermångfald kan formuleras i termer av Poisson algebran av släta funktioner på mångfalden. Det visar sig att man kan identifiera ett algebraiskt villkor i en Poissonalgebra (med en metrik) som gör det möjligt att formulera de flesta geometriska objekt på ett algebraiskt vis. Detta leder till definitionen av en Kähler-Poisson algebra, vilken utgörs av en Poissonalgebra och en metrik som tillsammans uppfyller ett kompatibilitetsvillkor. Vi visar att för varje Kähler-Poisson algebra så existerar det en Levi-Civita förbindelse på modulen som utgörs av de inre derivationerna, och att den tillhörande krökningsoperatorn har alla de klassiska symmetrierna. Vidare presenteras en konstruktion som associerar en Kähler-Poisson algebra till varje algebra i en stor klass av Poissonalgebror. Ur ett mer algebraiskt perspektiv så introduceras flera grundläggande begrepp, såsom morfier, delalgebror, direkta summor och tensorprodukter. Slutligen påbörjas en studie av modulirum för Kähler-Poisson algebror, det vill säga ekvivalensklasser av metriker som ger upphov till isomorfa Kähler-Poisson strukturer. Det visar sig att även i det enkla fallet med en Poisson algebra genererad av två variabler, så leder detta till ett icke-trivialt klassificeringsproblem.  

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. p. 38
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2044
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-163164 (URN)10.3384/diss.diva-163164 (DOI)9789179299095 (ISBN)
Public defence
2020-02-27, Planck, Fysikhuset, entrance 57, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2020-02-18 Created: 2020-01-20 Last updated: 2020-02-18Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full text

Search in DiVA

By author/editor
Arnlind, JoakimAl-Shujary, Ahmed
By organisation
Mathematics and Applied MathematicsFaculty of Science & Engineering
In the same journal
Journal of Geometry and Physics
Other Physics Topics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 746 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf