Cyclic compositions and trisotopies
type: Item
instance of: scholarly article
title: Cyclic compositions and trisotopies
zbMATH Open document ID: 1172.17004
author: Holger P. Petersson
published in: Journal of Algebra
publication date: 2007-01-26T00:00:00Z
review text: \textit{T. A. Springer} introduced the notion of cyclic composition in a work published at Universität Göttingen, mimeographed notes (1963) entitled ``Oktaven, Jordan-Algebren und Ausnahmegruppen'' [see the revised English version ``Octonions, Jordan algebras and exceptional groups.'' Berlin: Springer (2000; Zbl 1087.17001)]. The original name of the notion ``twisted composition'' appears also in ``The Book of Involutions'' by \textit{M.-A. Knus, A. S. Merkurjev, M. Rost} and \textit{J.-P. Tignol} (1998; Zbl 0955.16001). A quintuple \(S=(E,\rho,M,Q,*)\) is a cyclic composition if (1) \((E,\rho)\) is a cyclic cubic étale \(k\)-algebra, (2) \((M,Q)\) is a quadratic space over \(E\), so \(Q: M\to E\) is non-singular, (3) \((M,*)\) is a nonassociative algebra, (4) \(Q\) permits twisted composition: \(Q(x*y)=\rho(Q(x))\rho^2(Q(y))\), and (5) \(Q\) is twisted associative: \(Q(x*y,z)=\rho(Q(y*z,x))\). The concept was a tool for the study of Albert algebras (i.e. exceptional simple Jordan algebras). It was \textit{A. A. Albert} himself who in a work in Ann. Math. (2) 67, 1--28 (1958; Zbl 0079.04701) had introduced a different tool called ``cyclic trisotopies'' in the terminology of the paper under review. These ``cyclic trisotopies'' are a kind of composition algebras over étale cubic algebras. To be more precise a quintuple \(A=(E,\rho,C,g,b)\) is called a cyclic trisotopy when (1) \((E,\rho)\) is a cyclic cubic étale \(k\)-algebra, (2) \(C\) is a composition algebra over \(E\), (3) \(b\) is an invertible element of \(C\), (4) \(g: C\to C\) is a \(\rho\)-semilinear map satisfying the identity \(g(xy)=[g(x)b][b^{-1}g(y)]\) and (5) \(g\) stabilizes the line through \(b\) and \(g^3(x)=bxb^{-1}\). The work under review studies both concepts relating them by means of a quadrupling of composition algebras. This allows the author to enumerate cyclic trisotopies and compositions in a rational way (without scalar extensions of the ground field). The author relates cyclic trisotopie…
reviewed by: Candido Martín González
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 17C40
Mathematics Subject Classification ID: 17C50
zbMATH DE Number: 5118805
zbMATH Keywords: Cyclic compositions
zbMATH Keywords: cyclic trisotopy
zbMATH Keywords: composition algebra
zbMATH Keywords: cubic norm structures
zbMATH Keywords: Tits process
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.06.045
OpenAlex ID: W2012315755
cites work: A construction of exceptional Jordan division algebras
cites work: Q3953947
cites work: Q4210545
cites work: Q2770573
cites work: The Freudenthal-Springer-Tits Constructions of Exceptional Jordan Algebras
cites work: Homotopes of alternative algebras
cites work: A taste of Jordan algebras
cites work: A classification theorem for Albert algebras
cites work: Composition Algebras Over Algebraic Curves of Genus Zero
cites work: The structure group of an alternative algebra
cites work: Structure Theorems for Jordan Algebras of Degree Three Over Fields of Arbitrary Characteristic
cites work: The toral Tits process of Jordan algebras
cites work: Jordan algebras of degree 3 and the Tits process
cites work: The étale Tits process of Jordan algebras revisited.
cites work: Lois polynomes et lois formelles en théorie des modules
cites work: Q4488076
cites work: Cayley algebra bundles onA2K
DOI: 10.1016/J.JALGEBRA.2006.06.045
Recommended article: Q4488076
Recommended article: Q4316253
Recommended article: A survey on Albert algebras
Recommended article: Symmetric composition algebras over algebraic varieties
Recommended article: Octonion algebras obtained from associative algebras with involution
Recommended article: Bivariant twisted cyclic theory and spectral sequences of crossed products
Recommended article: Q4313491
Recommended article: Q3130277
Recommended article: Isotopy and invariants of Albert algebras
Recommended article: The cayley-dickson doubling process revisited
description: scientific article; zbMATH DE number 5118805