The structure group of an alternative algebra
type: Item
instance of: scholarly article
title: The structure group of an alternative algebra
zbMATH Open document ID: 1014.17004
DOI: 10.1007/BF02941670
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2003-02-13T00:00:00Z
review text: Let \(A\) be an alternative algebra over a field \(F\), and let \(A'=A+F1\) be the algebra obtained by adjoining an identity element 1 to \(A\). Let \(u\) and \(v\) be two elements in \(A'\). The \((u,v)\)-homotope of \(A\) is the algebra \(\overline A\) obtained by introducing the multiplication \(x \circ y=(xu)(vy)\) into the vector space \(A\). If \(u\) and \(v\) are invertible elements of \(A'\), \(\overline A\) is said to be a \((u,v)\)-isotope of \(A\). \textit{K. McCrimmon} [Math. Ann. 191, 253-262 (1971; Zbl 0203.33802)] showed that \(\overline A\) is again alternative and also that \((u,v)\)-isotopy preserves simplicity of an algebra and invertibility of an element of the algebra. In the present paper, the structure group \(\text{Str}(A)\) of an alternative algebra \(A\) with identity element 1 is defined as the totality of triples \((\eta,u,v)\) where \(\eta\) is an isomorphism from \(A\) onto its \((u,v)\)-isotope. The author proves that for any alternative algebra, the structure group is an extension, in the group-theoretical sense, of the narrow structure group (defined as in the Jordan case) by the unit group of the nucleus. It is proved that this extension is split if \(A\) is an associative algebra, and if \(A\) is the octonion algebra over an algebraically closed field it is not split in the category of algebraic groups. The author shows that the structure group is connected with the group of Albert autotopies and also that elements of the structure group play an important role in Albert's classical approach to the construction of Albert division algebras and Loos' treatment of alternative pairs with invertible elements. Using the principle of triality, natural sets of generators for the structure group and for the group of Tits and Weiss are exhibited in the case where the underlying algebra is an octonion algebra.
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 17D05
zbMATH DE Number: 1868552
zbMATH Keywords: alternative algebra
zbMATH Keywords: isotopy
zbMATH Keywords: structure group
author: Holger P. Petersson
reviewed by: María Concepción López-Díaz
MaRDI profile type: Publication
cites work: Non-associative algebras. I: Fundamental concepts and isotopy
cites work: A construction of exceptional Jordan division algebras
cites work: Q3953947
cites work: Some groups of transformations defined by Jordan algebras. II. Groups of type F4.
cites work: Q5666699
cites work: Q4210545
cites work: The Freudenthal-Springer-Tits Constructions of Exceptional Jordan Algebras
cites work: Homotopes of alternative algebras
cites work: Nonassociative algebras with scalar involution
cites work: Composition Algebras Over Algebraic Curves of Genus Zero
cites work: Q5522244
cites work: Linear algebraic groups
cites work: Q4488076
cites work: Q4779974
description: scientific article; zbMATH DE number 1868552