Composition algebras with large derivation algebras
type: Item
instance of: scholarly article
title: Composition algebras with large derivation algebras
zbMATH Open document ID: 0873.17005
author: Alberto Elduque
author: José María Pérez
published in: Journal of Algebra
publication date: 1997-09-29T00:00:00Z
review text: An algebra \(A\) over a field \(F\) is called a composition algebra if \(A\) is equipped with a strictly nondegenerate quadratic form \(q:A\rightarrow F\) verifying \(q(xy)=q(x)q(y)\). The structure of unital composition algebras is well known: they all have dimension \(1,\;2,\;4,\) or \(8\), and are either the field \(F\), if its characteristic is not \(2\), \(F\oplus F\), the quadratic separable field extensions \(K\) of \(F\), the generalized quaternion algebras over \(F\), and the Cayley-Dickson algebras over \(F\). The authors call these algebras Hurwitz algebras. It is known that any finite dimensional composition algebra can be obtained from a Hurwitz algebra \(C\), with norm \(q\), by means of a new multiplication \(x\ast y=\phi (x)\psi(y)\), where \(\phi\) and \(\psi\) are orthogonal transformations relative to \(q\). Let \(x\mapsto \bar x\) be a canonical involution in a Hurwitz algebra \(C\). The authors call an algebra \(A\) a standard composition algebra associated to \(C\) if the product in \(A\) is given by one of the terms: \(xy, \;\bar x y,\;x \bar y,\;\bar x \bar y\), where juxtaposition denotes the multiplication in \(C\). In dimension 2, any composition algebra over an algebraically closed field is standard [\textit{H. Petersson}, Abh. Math. Semin. Univ. Hamb. 35, 215-222 (1971; Zbl 0217.06701)]. One of the main results of the paper states that the same is true in dimensions 4 and 8 for composition algebras with the largest possible derivation algebras. More precisely, the authors determine all the possibilities for the Lie algebra \(\text{Der } A\) of derivations for composition algebras \(A\) of dimension 4 and all the possibilities with maximal toral rank (equal to 2) in dimension 8, and then classify the composition algebras with the largest derivation algebras. In \(\dim A=4\) case \(\dim \text{Der } A\leq 3\), and \(A\) is standard if and only if \(\text{Der } A\) is a central simple 3-dimensional algebra; and in \(\dim A=8\) case \(A\) is …
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 17A36
zbMATH DE Number: 1014003
zbMATH Keywords: composition algebra
zbMATH Keywords: derivation algebra
zbMATH Keywords: para-Hurwitz algebra
zbMATH Keywords: Okubo algebra
reviewed by: Ivan Shestakov
MaRDI profile type: Publication
full work available at URL: jabr.1996.6851
OpenAlex ID: W1982296014
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DOI: 10.1006/JABR.1996.6851
description: scientific article; zbMATH DE number 1014003