A characterization of 3-graded Lie algebras generated by a pair
type: Item
instance of: scholarly article
title: A characterization of 3-graded Lie algebras generated by a pair
zbMATH Open document ID: 1075.17010
DOI: 10.1016/S0022-4049(02)00073-7
published in: Journal of Pure and Applied Algebra
publication date: 2003-03-09T00:00:00Z
review text: A \(3\)-graded Lie algebra is a \(\mathbb{Z}\)-graded Lie algebra with non-zero elements only in degrees \(-1\), \(0\), \(1\). The authors investigate such algebras generated by a pair of elements of respective degrees \(-1\) and \(1\). The free \(3\)-graded Lie algebra \(\mathfrak{g}(x,y)\) generated by \(x\), \(y\) is defined by the natural universal property. Continuing a previous article [New York J. Math. 7, 71--86 (2001; Zbl 0985.17020)], the authors show that, if the ground field \(K\) is of characteristic zero, the free \(3\)-graded Lie algebra \(\mathfrak{g}(x,y)\) is canonically embedded in \(\mathfrak{sl}_{2}(K[t])\). They describe the image of this embedding and the ideals of \(\mathfrak{g}(x,y)\). They also prove that any \(3\)- graded Lie algebra with a pair of generators over a field \(K\) of characteristic \(0\) is isomorphic to a \(3\)-graded Lie subalgebra of \(\mathfrak{sl}_{2}(K[t]/(p))\) for some \(p\in K[t]\). The authors express the hope that \(3\)-graded Lie subalgebras generated by a pair may be a useful tool for a better understanding of more general \(3\)-graded Lie algebras, like those arising from the Kantor-Koecher-Tits construction and their unitary representations.
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 1878439
zbMATH Keywords: 3-graded Lie algebras
author: Marcelo P. De Oliveira
author: Odilon Otavio Luciano
reviewed by: Guy Roos
MaRDI profile type: Publication
Wikidata QID: Q115340152
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description: scientific article; zbMATH DE number 1878439