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An Approach to Finite-Dimensional Real Division Composition Algebras through Reflections
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0002-0182-6205
2015 (English)In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, Vol. 139, no 4, 357-399 p.Article in journal (Refereed) Published
Abstract [en]

We consider the category of all finite-dimensional real composition algebras which are division algebras. These are precisely the finite-dimensional absolute valued algebras, and exist only in dimension 1, 2, 4 and 8. We construct three decompositions of this category, each determined by the number of reflections composing left and right multiplication by idempotents. As a consequence, we obtain new full subcategories in dimension 8, in which all morphisms are automorphisms of the octonions. This reduces considerable parts of the still open classification problem in dimension 8 to the normal form problem of an action of the automorphism group of the octonions, which is a compact Lie group of type $G_{2}$ , on pairs of orthogonal maps. We describe these subcategories further in terms of subgroups of $\mathrm{Aut}(\mathbb{O})$ and their cosets, which we express geometrically. This extends the study of finite-dimensional real division composition algebras with a one-sided unity.

Place, publisher, year, edition, pages
2015. Vol. 139, no 4, 357-399 p.
Keyword [en]
Composition algebra, division algebra, absolute valued algebra, reflection, G2-subgroup, G2-set
Mathematics
Mathematics
Identifiers
ISI: 000356200200001OAI: oai:DiVA.org:uu-224094DiVA: diva2:715312
Available from: 2014-05-03 Created: 2014-05-03 Last updated: 2015-07-06Bibliographically approved

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