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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 universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
2010 (Engelska)Ingår i: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 134, nr 3, s. 247-277Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2010. Vol. 134, nr 3, s. 247-277
Nyckelord [en]
Composition algebra, Absolute valued algebra, e-Quadratic algebra, Classification, Normal form
Nationell ämneskategori
Matematik
Identifikatorer
URN: urn:nbn:se:uu:diva-97998DOI: 10.1016/j.bulsci.2009.03.001ISI: 000276938900003OAI: oai:DiVA.org:uu-97998DiVA, id: diva2:173148
Tillgänglig från: 2009-01-29 Skapad: 2009-01-29 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
Ingår i avhandling
1. Problems in the Classification Theory of Non-Associative Simple Algebras
Öppna denna publikation i ny flik eller fönster >>Problems in the Classification Theory of Non-Associative Simple Algebras
2009 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala: Universitetsbiblioteket, 2009. s. vi, 36
Serie
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 62
Nyckelord
Division algebra, flexible algebra, normal form, composition algebra, absolute valued algebra, vector product, rotation.
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:uu:diva-9536 (URN)978-91-506-2053-5 (ISBN)
Disputation
2009-02-19, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15
Opponent
Handledare
Tillgänglig från: 2009-01-29 Skapad: 2009-01-29Bibliografiskt granskad

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