Classification of non-compact real simple generalized Jordan triple systems of the second kind
type: Item
instance of: scholarly article
title: Classification of non-compact real simple generalized Jordan triple systems of the second kind
zbMATH Open document ID: 0754.17001
author: Hiroshi Asano
published in: Hiroshima Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: A generalized Jordan triple system (GJTS) is a triple system satisfying the identity \((uv(xyz))=((uvx(yz))-(x(vuy)z)+(xy(uvz))\). Kantor constructed a \(\mathbb{Z}\)-graded Lie algebra from every GJTS; the GJTS is said to be of the \(n\)-th kind if all the subspaces with absolute values of indices greater than \(n\) are zero (and \(n\) is minimal). GJTS of the first kind are well known as Jordan triple systems. GJTS of the second kind have been discussed by Kantor and there is also a connection to the structurable algebras investigated by Allison and others. This article is about finite dimensional GJTS of the second kind over the real numbers. Such a GJTS is called nondegenerate resp. compact if Yamaguti's associative trace form is nondegenerate resp. positive definite. The ``classical'' compact simple GJTS of the second kind were classified by Kaneyuki and the author in an earlier paper. The main result of this paper states that a nondegenerate GJTS of the second kind can be obtained from a compact one by ``modification''. (Modification by an involutory automorphism is carried out by applying this automorphism to the middle element of the triple product.) It is then shown that a noncompact simple GJTS of the second kind is either modification of a simple compact one or modification of the direct sum of two copies of a simple compact GJTS by the exchange involution. Taken together, this yields a classification.
reviewed by: S Walcher
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 62997
zbMATH Keywords: Kantor construction
zbMATH Keywords: generalized Jordan triple system
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206128716
OpenAlex ID: W1608576673
Wikidata QID: Q128900035
DOI: 10.32917/HMJ/1206128716
description: scientific article; zbMATH DE number 62997