Quotients in graded Lie algebras. Martindale-like quotients for Kantor pairs and Lie triple systems
type: Item
instance of: scholarly article
title: Quotients in graded Lie algebras. Martindale-like quotients for Kantor pairs and Lie triple systems
zbMATH Open document ID: 1331.17025
author: E. Garcia
author: Miguel Gómez Lozano
published in: Algebras and Representation Theory
publication date: 2013-04-15T00:00:00Z
review text: In 2004 the authors [J. Pure Appl. Algebra 194, 127--145 (2004; Zbl 1127.17027)] introduced Martindale-like systems of quotients of a Jordan system with respect to a filter of ideals. Using the construction of maximal Lie algebras of quotients by \textit{M. Siles Molina} [J. Pure Appl. Algebra 188, 175--188 (2004; Zbl 1036.17006)], the authors gave explicit constructions of maximal Jordan pairs, triple systems and algebras with respect to power filters of ideals. As pointed out by the authors, a fundamental aspect in their construction is the fact that the maximal Lie algebra of quotients of a nondegenerate \(\mathbb Z\)-graded Lie algebra \(L=L_{-1}\oplus L_0\oplus L_1\) is \(\mathbb Z\)-graded with the same support. In the present work it is proved that if \(G\) is an abelian group and \(L\) is a \(G\)-graded Lie algebra with finite support, then its maximal algebra of quotients with respect to a power filter of sturdy graded ideals is \(G\)-graded (in the particular case of a \(\mathbb Z\)-graded Lie algebra with finite support, the maximal graded algebra of quotients has the same support as the original Lie algebra). From these results, the authors introduce a notion of Martindale-like quotient for Kantor pairs and Lie triple systems and construct their maximal systems of quotients.
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17A40
zbMATH DE Number: 6154028
zbMATH Keywords: Quotient in graded Lie algebra
zbMATH Keywords: Martidale-like quotient
zbMATH Keywords: Kantor pairs
zbMATH Keywords: Lie triple systems
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: s10468-011-9303-5
OpenAlex ID: W2084213779
cites work: Elementary groups and invertibility for kantor pairs
cites work: Q4795284
cites work: On simple split Lie triple systems
cites work: Nondegeneracy for Lie Triple Systems and Kantor Pairs
cites work: Jordan systems of Martindale-like quotients
cites work: Q5622399
cites work: Prime rings satisfying a generalized polynomial identity
cites work: The ring of fractions of a Jordan algebra
cites work: Q3474784
cites work: Algebras of quotients of Lie algebras.
DOI: 10.1007/S10468-011-9303-5
description: scientific article; zbMATH DE number 6154028