On the invariants mod 2 of Albert algebras
type: Item
instance of: scholarly article
title: On the invariants mod 2 of Albert algebras
zbMATH Open document ID: 0830.17016
published in: Journal of Algebra
publication date: 1996-01-30T00:00:00Z
review text: For a field \(\mathbb{K}\) of characteristic different from 2, Jordan algebras over \(\mathbb{K}\) are defined as those commutative algebras over \(\mathbb{K}\) satisfying the ``Jordan identity'' \(x^2 \circ (y \circ x) = (x^2 \circ y) \circ x\). Jordan algebras which cannot be seen as subspaces of associative algebras closed under the ``Jordan product'' \(x \circ y := {1\over 2} (xy + yx)\) are called exceptional. It is known that central simple exceptional Jordan algebras over \(\mathbb{K}\) (in short, Albert algebras over \(\mathbb{K}\)) are finite-dimensional. More precisely, the dimensions of Albert algebras are always equal to 27, and they are either division Jordan algebras (in the sense of Jacobson) or reduced. Moreover, if an Albert algebra \(J\) over \(\mathbb{K}\) is reduced, then it is isomorphic to the Jordan algebra of all \(3 \times 3\) matrices with entries in an octonion algebra over \(\mathbb{K}\) (uniquely determined by \(J\) and called the coordinate algebra of \(J\)) which are hermitian relative to a canonical involution. Therefore the problem of classifying Albert algebras centers in those that are division algebras. Since, up to a suitable scalar extension if necessary, every Albert algebra over \(\mathbb{K}\) becomes reduced (over the extended field), a tool to attack the above problem is to consider, for a given Albert algebra \(J\) over \(\mathbb{K}\), a ``reducing field'' of \(J\), namely a field extension \(\mathbb{F}/\mathbb{K}\) such that \(J \otimes_\mathbb{K} \mathbb{F}\) is reduced. In the present paper the authors prove that, if \(J\) is an Albert algebra over \(\mathbb{K}\), then there exists a unique octonion algebra \(C\) over \(\mathbb{K}\) such that, for every reducing field \(\mathbb{F}\) of \(J\), \(C \otimes_\mathbb{K} \mathbb{F}\) is the coordinate algebra of \(J \otimes_\mathbb{K} \mathbb{F}\). This nice result, showing that division Albert algebras also have a ``coordinate octonion algebra'', improves and simplifies the p…
Mathematics Subject Classification ID: 17C40
zbMATH DE Number: 783903
zbMATH Keywords: Albert algebra
zbMATH Keywords: octonion algebra
zbMATH Keywords: coordinate algebra
zbMATH Keywords: division Albert algebras
author: Holger P. Petersson
author: Michel L. Racine
reviewed by: Ángel Rodríguez Palacios
MaRDI profile type: Publication
full work available at URL: jabr.1995.1166
OpenAlex ID: W2089819070
DOI: 10.1006/JABR.1995.1166
description: scientific article; zbMATH DE number 783903