On finite dimensional absolute valued algebras satisfying \((x,x,x)=0\)
type: Item
instance of: scholarly article
title: On finite dimensional absolute valued algebras satisfying \((x,x,x)=0\)
zbMATH Open document ID: 0623.17002
DOI: 10.1007/BF01200223
author: Mohamed Lamei El-Mallah
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: The author proves that the para-quaternion algebra \(H^*\) is the only (up to isomorphism) 4-dimensional absolute valued real algebra which satisfies the identity \((x,x,x)=0\) and is not isomorphic to the quaternion algebra H. In \(H^*\) we show that there are infinitely many idempotents; only one of them e is central and each other idempotent has the form: \[ -e+(\sqrt{3}/2)(\alpha i+\beta j+\gamma k)\quad or\quad - e- (\sqrt{3}/2)(\alpha i+\beta j+\gamma k) \] where the real numbers \(\alpha\), \(\beta\) and \(\gamma\) satisfy \(\alpha^ 2+\beta^ 2+\gamma^ 2=1\). It is also shown that the pseudo-octonion algebra is the only 8- dimensional absolute valued algebra which satisfies the identity \((x,x,x)=0\) and does not contain a central idempotent.
Mathematics Subject Classification ID: 17A60
Mathematics Subject Classification ID: 17A20
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 4010722
zbMATH Keywords: flexible algebra
zbMATH Keywords: Cayley algebra
zbMATH Keywords: para-octonion algebra
zbMATH Keywords: para-quaternion algebra
zbMATH Keywords: 4-dimensional absolute valued real algebra
zbMATH Keywords: idempotents
zbMATH Keywords: pseudo- octonion algebra
zbMATH Keywords: 8-dimensional absolute valued algebra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4010722