Coordinatization of triangulated Jordan systems
type: Item
instance of: scholarly article
title: Coordinatization of triangulated Jordan systems
zbMATH Open document ID: 0651.17001
DOI: 10.1016/0021-8693(88)90302-X
author: Kevin McCrimmon
author: Erhard Neher
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let J be a Jordan triple system. J is called triangulated if there is a Peirce decomposition \(J=J_{11}\oplus J_{12}\oplus J_{22}\) relative to an orthogonal tripotent \((e_ 1,e_ 2)\) and if there is \(u\in J_{12}\) such that \(P(e)u=u\) and \(P(u)e=e\) for \(e=e_ 1+e_ 2\). J is called faithfully triangulated if it is triangulated and if \(\{x_ 1e_ 1u\}=0\) implies that \(x_ 1=0\) for \(x_ 1\in J_ 1\). If J is a faithfully triangulated triple system, the authors determine two subspaces of J, one isomorphic to a diagonal system of \(2\times 2\) Hermitian matrices, and one isomorphic to an ample Clifford system determined by a quadratic form. When J is simple, it equals at least one of these subspaces, and this yields a classification of simple triangulated Jordan triple systems. By coordinatizing non-semisimple systems, the authors describe unital bimodules for diagonal Hermitian matrix systems.
Mathematics Subject Classification ID: 17A40
zbMATH DE Number: 4061451
zbMATH Keywords: coordinatization
zbMATH Keywords: Jordan triple system
zbMATH Keywords: Peirce decomposition
zbMATH Keywords: simple triangulated Jordan triple systems
zbMATH Keywords: unital bimodules
zbMATH Keywords: diagonal Hermitian matrix systems
reviewed by: Robert Bix
MaRDI profile type: Publication
cites work: Q5186181
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cites work: Lectures on quadratic Jordan algebras
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description: scientific article; zbMATH DE number 4061451