On compact realifications of exceptional simple Kantor triple systems
type: Item
instance of: scholarly article
title: On compact realifications of exceptional simple Kantor triple systems
zbMATH Open document ID: 1168.17003
author: Daniel Mondoc
published in: Journal of Generalized Lie Theory and Applications
publication date: 2007-02-13T00:00:00Z
review text: Let \(F\) be a field of characteristic zero and let \(U\) be a finite dimensional vector space over \(F\) endowed with a trilinear map \(B: U\times U\times U \to U\). The pair \((B,U)\) is said to be a triple system. A triple system \((B,U)\) is said to be a Generalized Jordan triple system (GJTS for short) if \([L_{x,y},L_{u,v}]=L_{L_{x,y}(u),v}-L_{u,L_{y,x}(v)}\) for any \(x,y,u,v\in U\), where \(L_{x,y}(z)=B(x,y,z)\). In [Tr. Sem. Vektor. Tenzor. Analizu 16, 407--499 (1972; Zbl 0272.17001)] \textit{I. L. Kantor} associated to any GJTS \((B,U)\) a graded Lie algebra \({\mathcal L}(B)=\bigoplus_{l=-\infty}^\infty U_l\) such that \(U_{-1}=U\). A GJTS \((B,U)\) is said to be a Kantor Triple system (KTS for short) if \({\mathcal L}(B)=\bigoplus_{l=-2}^2 U_l\). B. N. Allison associated to any structurable algebra \((A,-)\) a 5-graded Lie algebra \({\mathcal K}(A)=\bigoplus_{l=-2}^2 K_l\) [Commun. Algebra 7, 1835--1875 (1979; Zbl 0422.17006)] which turns out to coincide with the Kantor's Lie algebra \({\mathcal L}(B_A)\) when we consider the triple system \(B_A(x,y,z)=(x\overline{y})z+(z\overline{y})x-(z\overline{x})y\) for any \(x,y,z\in A\) [see \textit{H. Asano} and \textit{S. Kaneyuki}, Tokyo J. Math. 11, No. 1, 105--118 (1988; Zbl 0665.17003) and Tokyo J. Math. 15, 483--484 (1992; Zbl 0774.17002)]. This paper presents a unified construction to obtain models of all realifications of complex compact exceptional simple KTS and therefore of all realifications of complex exceptional simple Lie algebras which are presented as Kantor Lie algebras of the corresponding KTS. The author first considers \({\mathcal M}(H_3(\mathbb{A^\mathbb{C}}))\) the \(2 \times 2\) matrix algebra defined by the Jordan algebra \(H_3(\mathbb{A^\mathbb{C}})\) of hermitian \(3\times 3\) matrices over a complex division algebra \(\mathbb{A^\mathbb{C}}\in\{ \mathbb{R}^\mathbb{ C }, \mathbb{C}^\mathbb{C},\;\mathbb{H}^\mathbb{C}, \;\mathbb{O}^\mathbb{C}\}\) where \(\mathbb{A^\mathbb{C}}\) denotes any…
reviewed by: Irene Paniello
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 17C20
zbMATH DE Number: 5125110
zbMATH Keywords: Kantor triple system
zbMATH Keywords: Kantor algebra
zbMATH Keywords: structurable algebra
zbMATH Keywords: realification
MaRDI profile type: Publication
full work available at URL: s060103
OpenAlex ID: W4250373256
DOI: 10.4303/JGLTA/S060103
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description: scientific article; zbMATH DE number 5125110