Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3
type: Item
instance of: scholarly article
title: Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3
zbMATH Open document ID: 1300.17007
author: Eliyahu Matzri
author: Uzi Vishne
published in: Manuscripta Mathematica
publication date: 2012-02-14T00:00:00Z
review text: A composition algebra is a (non-associative) algebra \((C, \ast )\) over a field \(F\), endowed with a non-degenerate quadratic ``norm form'' \(N: C \to F\), such that \(N(a \ast b) = N(a)N(b)\), \(a,b\in C\). Two special classes of these algebras have been fully classified. In the first place, it is known that a composition \(F\)-algebra with a unit is one of the following: the field \(F\); an étale quadratic extension of \(F\); a quaternion algebra; a Cayley algebra. In these cases, the norm form is a Pfister form, whence it is either anisotropic or hyperbolic. Secondly, a composition algebra is symmetric, if the bilinear form associated to the norm is associative. These include the Okubo algebras constructed from central simple \(F\)-algebras of degree \(3\). The maximal isotropic spaces in the split Cayley algebra were described by \textit{F. van der Blij} and \textit{T. A. Springer} [Nieuw Arch. Wiskd., III. Ser. 8, 158--169 (1960; Zbl 0127.11804)]. The paper under review translates this classification to arbitrary composition algebras. The authors take as a starting point Kaplansky's theorem stating that any composition algebra can be twisted into a unital one (see Proposition 33.27 in [\textit{M.-A. Knus} et al., The book of involutions. Providence, RI: AMS (1998; Zbl 0955.16001)]). This, applied to the van der Blij-Springer description, enables them to classify the maximal isotropic spaces in an arbitrary composition algebra, and their intersections in case the algebra is symmetric. In the rest of the paper, the authors consider symmetric \(F\)-algebras \((C, \ast )\). Analyzing intersection properties of maximal isotropic spaces in \(C\), they obtain two triality results: one for \(2\)-dimensional isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. The authors bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. T…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 6006488
zbMATH Keywords: Composition algebra
zbMATH Keywords: Cayley algebra
zbMATH Keywords: norm form
zbMATH Keywords: symmetric composition algebra
zbMATH Keywords: Okubo algebra
zbMATH Keywords: isotropic subspace
zbMATH Keywords: central simple algebra
zbMATH Keywords: Kummer subspace
zbMATH Keywords: triality
zbMATH Keywords: involution of the second kind
MaRDI profile type: Publication
full work available at URL: s00229-011-0482-5
OpenAlex ID: W1980915457
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DOI: 10.1007/S00229-011-0482-5
description: scientific article; zbMATH DE number 6006488