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Normal forms for the G2-action on the real symmetric 7 × 7-matrices by conjugation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2007 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 312, no 2, 668-688 p.Article in journal (Refereed) Published
Abstract [en]

The exceptional Lie group G(2) subset of O-7 (R) acts on the set of real symmetric 7 x 7-matrices by conjugation. We solve the normal form problem for this group action. In view of the earlier results [G.M. Benkart, D.J. Britten, J.M. Osbom, Real flexible division algebras, Canad. J. Math. 34 (1982) 550-588; J.A. Cuenca Mira, R. De Los Santos Villodres, A. Kaidi, A. Rochdi, Real quadratic flexible division algebras, Linear Algebra Appl. 290 (1999) 1-22; E. Darpb, On the classification of the real flexible division algebras, Colloq. Math. 105 (1) (2006) 1-17], this gives rise to a classification of all finite-dimensional real flexible division algebras. By a classification is meant a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. We also give a parametrisation of the set of all real symmetric matrices, based on eigen values.

Place, publisher, year, edition, pages
2007. Vol. 312, no 2, 668-688 p.
Keyword [en]
Normal form, Group action, Vector product, Octonion, Automorphism, Real division algebra, Flexible algebra
National Category
URN: urn:nbn:se:uu:diva-97995DOI: 10.1016/j.jalgebra.2007.03.007ISI: 000247409100009OAI: oai:DiVA.org:uu-97995DiVA: diva2:173145
Available from: 2009-01-29 Created: 2009-01-29 Last updated: 2011-02-03Bibliographically 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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