Identity classification in triple systems
type: Item
instance of: scholarly article
title: Identity classification in triple systems
zbMATH Open document ID: 0596.17002
DOI: 10.1016/0021-8693(85)90189-9
author: John R. Faulkner
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: The author classifies nontrivial triple systems \(\mathcal M\) defined on a finite-dimensional vector space over an algebraically closed field of characteristic zero such that: (i) every left multiplication \(L(x,y)\), \(x,y\in\mathcal M\), is a derivation of \(U\), where \(L(x,y)z=xyz\) is the triple product of \(\mathcal M\), (ii) \(\mathcal M\) is irreducible under \(\operatorname{Der}(\mathcal M)\) and (iii) \(\mathcal M\) satisfies an (unspecified) 3-linear identity. After studying all possible 3-linear identities in general triple systems (via the left ideal structure in the group algebra of \(S_3)\) the author shows that \(\mathcal M\) lies in one of 6 classes each defined by identities. Calling left-skew: \(xyz=-yxz\), left-symmetric: \(xyz=yxz\), Jacobi: \(xyz+yzx+zxy=0\), cyclic: \(xyz=yzx\), these classes are (a) left-skew, (b) left-symmetric, (c) left-skew and Jacobi (Lie triple systems), (d) left-symmetric and Jacobi, (e) Jacobi, (f) left skew and cyclic. For each of these six classes the author obtains a precise classification using the representation theory of Lie algebras. This can be done since for example an immediate consequence of (i) and (ii) is that \(\operatorname{Der}(\mathcal M)\) is semisimple and \(\mathcal M\) is a self-dual module. As a by-product of his methods he gives a quick proof of the well-known classification of flexible Lie-admissible algebras \(A\) with \(A^-\) simple.
reviewed by: Erhard Neher
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17D25
Mathematics Subject Classification ID: 17A20
zbMATH DE Number: 3959714
zbMATH Keywords: inner-derivation triple systems
zbMATH Keywords: anti-Lie triple systems
zbMATH Keywords: alternating triple systems
zbMATH Keywords: Lie triple systems
zbMATH Keywords: representation theory of Lie algebras
zbMATH Keywords: flexible Lie-admissible algebras
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90189-9
OpenAlex ID: W1986025473
cites work: Flexible Lie-admissible algebras
cites work: Dynkin diagrams for Lie triple systems
cites work: Simple anti-jordan pairs1
cites work: Q3897665
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Lie superalgebras
cites work: Adjoint Operators in Lie Algebras and the Classification of Simple Flexible Lie-Admissible Algebras
cites work: Lectures on Chevalley Groups
cites work: Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen
description: scientific article; zbMATH DE number 3959714