Jordan triple systems with completely reducible derivation or structure algebras
type: Item
instance of: scholarly article
title: Jordan triple systems with completely reducible derivation or structure algebras
zbMATH Open document ID: 0558.17004
author: Erhard Neher
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Results that relate the derivations of an algebra to the structure of the algebra are extended in this article from Jordan algebras to Jordan triple systems. Specifically, let V be a finite dimensional Jordan triple system over a field k of characteristic zero. Then the structure algebra of V is completely reducible if and only if V is the direct sum of a semisimple and a trivial ideal. V is semisimple if and only if its derivation algebra is completely reducible and every derivation has trace zero. As a consequence, the author characterizes all real Jordan triple systems that have a compact automorphism group. The preceding theorems all follow from the result that, when k is algebraically closed, the derivation algebra of V is completely reducible if and only if V is the direct sum of two ideals such that one ideal is semisimple and the other is either trivial or a direct sum of ideals of the form \(X\oplus M\), where X and M are nonzero finite dimensional vector spaces over k, (\(\cdot,\cdot)\) is a nondegenerate symmetric bilinear form on X, and \(X\oplus M\) has quadratic representation \(P(x\oplus m)(y\oplus n)=(2(x,y)x-(x,x)y)\oplus ((x,x)n+2(x,y)m)\) for x,y\(\in X\) and m,n\(\in M\).
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 17A36
zbMATH DE Number: 3889753
zbMATH Keywords: Jordan triple systems
zbMATH Keywords: characteristic zero
zbMATH Keywords: structure algebra
zbMATH Keywords: derivation algebra
zbMATH Keywords: compact automorphism group
reviewed by: Robert Bix
MaRDI profile type: Publication
full work available at URL: pjm.1984.113.137
OpenAlex ID: W1975923036
DOI: 10.2140/PJM.1984.113.137
description: scientific article; zbMATH DE number 3889753