uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • 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
Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity
Departamento de Algebra, Geometria y Topologia, Facultad de Ciencias, Universidad de Málaga.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2010 (English)In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 134, no 3, 247-277 p.Article in journal (Refereed) Published
Abstract [en]

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.

Place, publisher, year, edition, pages
2010. Vol. 134, no 3, 247-277 p.
Keyword [en]
Composition algebra, Absolute valued algebra, e-Quadratic algebra, Classification, Normal form
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-97998DOI: 10.1016/j.bulsci.2009.03.001ISI: 000276938900003OAI: oai:DiVA.org:uu-97998DiVA: diva2:173148
Available from: 2009-01-29 Created: 2009-01-29 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Problems in the Classification Theory of Non-Associative Simple Algebras
Open this publication in new window or tab >>Problems in the Classification Theory of Non-Associative Simple Algebras
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In spite of its 150 years history, the problem of classifying all finite-dimensional division algebras over a field k is still unsolved whenever k is not algebraically closed. The present thesis concerns some different aspects of this problem, and the related problems of classifying all composition and absolute valued algebras.

A tripartition of the class of all fields is given, based on the dimensions in which division algebras over a field exist. Moreover, all finite-dimensional flexible real division algebras are classified. This class includes in particular all finite-dimensional commutative real division algebras, of which two different classifications, along different lines, are presented.

It is shown that every vector product algebra has dimension zero, one, three or seven, and that its isomorphism type is determined by its adherent quadratic form. This yields a new and elementary proof for the corresponding, classical result for unital composition algebras.

A rotation in a Euclidean space is an orthogonal map that locally acts as a plane rotation with a fixed angle. All pairs of rotations in finite-dimensional Euclidean spaces are classified up to orthogonal similarity.

A description of all composition algebras having an LR-bijective idempotent is given. On the basis of this description, all absolute valued algebras having a one-sided unity or a non-zero central idempotent are classified.

Place, publisher, year, edition, pages
Uppsala: Universitetsbiblioteket, 2009. vi, 36 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 62
Keyword
Division algebra, flexible algebra, normal form, composition algebra, absolute valued algebra, vector product, rotation.
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-9536 (URN)978-91-506-2053-5 (ISBN)
Public defence
2009-02-19, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15
Opponent
Supervisors
Available from: 2009-01-29 Created: 2009-01-29Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text

Authority records BETA

Dieterich, Ernst

Search in DiVA

By author/editor
Dieterich, Ernst
By organisation
Algebra, Geometry and LogicAlgebra and Geometry
In the same journal
Bulletin des Sciences Mathématiques
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 567 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • 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