Quotients in graded Lie algebras. Martindale-like quotients for Kantor pairs and Lie triple systems
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.
scientific article; zbMATH DE number 6154028