Pseudo-composition algebras
type: Item
instance of: scholarly article
title: Pseudo-composition algebras
zbMATH Open document ID: 0788.17003
DOI: 10.1007/BF02572391
author: Kurt Meyberg
author: J. Marshall Osborn
published in: Mathematische Zeitschrift
publication date: 1994-06-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174555
review text: Let \(F\) be an algebraically closed field of characteristic not 2 or 3, and let \(A\) be a commutative algebra over \(F\). Then \(A\) is called a pseudo-composition algebra if there exists a symmetric bilinear form \(\varphi\neq 0\) such that \(x^ 3= \varphi(x,x)x\), for all \(x\in A\). The pseudo-composition algebra \(A\) with identity element \(e\) is of quadratic type if every element \(x\in A\) satisfies an equation of the form \(x^ 2+\beta(x)x+ \gamma(x,x)e=0\). In this paper it is shown that in any pseudo-composition algebra \(A\) the radical \(\text{Rad }\varphi\) is an ideal, and that either (i) \(A\) is of quadratic type, (ii) \(A/\text{Rad }\varphi\) is of quadratic type, or (iii) \(A\) may be constructed from an alternative quadratic algebra.
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 17A65
zbMATH DE Number: 561286
zbMATH Keywords: pseudo-composition algebra
zbMATH Keywords: symmetric bilinear form
zbMATH Keywords: quadratic type
zbMATH Keywords: radical
MaRDI profile type: Publication
cites work: Q3129444
cites work: Composition algebras and their automorphisms
cites work: Q3822356
cites work: Q3808268
cites work: Q3267480
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description: scientific article; zbMATH DE number 561286