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Real commutative division algebras
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2007 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 10, no 2, 179-196 p.Article in journal (Refereed) Published
Abstract [en]

The category of all two-dimensional real commutative division algebras is shown to split into two full subcategories. One of them is equivalent to the category of the natural action of the cyclic group of order 2 on the open right half plane R->0 x R. The other one is equivalent to the category of the natural action of the dihedral group of order 6 on the set of all ellipses in R-2 which are centered at the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are easily described. Together with R they classify all real commutative division algebras up to isomorphism. Moreover we describe all morphisms between the objects in this classifying set, thus obtaining a complete picture of the category of all real commutative division algebras, up to equivalence. This supplements earlier contributions of Kantor and Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Nauka, Moscow, 1973; Benkart et al., Hadronic J., 4: 497 - 529, 1981; and Althoen and Kugler, Amer. Math. Monthly, 90: 625 - 635, 1983, who achieved partial results on the classification of the real commutative division algebras.

Place, publisher, year, edition, pages
2007. Vol. 10, no 2, 179-196 p.
Keyword [en]
Real division algebra, isotopy, group action, cross-section, automorphism group, idempotent
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-97994DOI: 10.1007/s10468-006-9040-3ISI: 000244446800005OAI: oai:DiVA.org:uu-97994DiVA: diva2:173144
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

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Dieterich, Ernst

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