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Vector product algebras
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 41, no Part 5, 898-902 p.Article in journal (Refereed) Published
Abstract [en]

Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.

Place, publisher, year, edition, pages
2009. Vol. 41, no Part 5, 898-902 p.
National Category
URN: urn:nbn:se:uu:diva-97996DOI: 10.1112/blms/bdp066ISI: 000271378900014OAI: oai:DiVA.org:uu-97996DiVA: diva2:173146
Available from: 2009-01-29 Created: 2009-01-29 Last updated: 2010-06-30Bibliographically 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.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 62
Division algebra, flexible algebra, normal form, composition algebra, absolute valued algebra, vector product, rotation.
National Category
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
Available from: 2009-01-29 Created: 2009-01-29Bibliographically approved

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