Ideally constrained Lie algebras
type: Item
instance of: scholarly article
title: Ideally constrained Lie algebras
zbMATH Open document ID: 1048.17016
DOI: 10.1016/S0021-8693(02)00057-1
author: Norberto Gavioli
author: Valerio Monti
published in: Journal of Algebra
publication date: 2002-11-28T00:00:00Z
review text: The authors study a class of graded Lie algebras satisfying a `narrowness' condition on their lattice of graded ideals. Motivated by analogies with (pro-)\(p\)-groups, they naturally restrict their attention to graded Lie algebras \(L=\bigoplus_{i=1}^{\infty}L_i\) over a field which are generated by \(L_1\). This is shortened to \textit{Lie algebras} in the paper and this review. In particular, for each \(i\) the \(i\)th term \(L^i\) of the lower central series of \(L\) coincides with \(\bigoplus_{j\geq i}^{\infty}L_j\). A Lie algebra is \textit{ideally \(r\)-constrained} if for every graded ideal \(I\) of \(L\) there is a positive integer \(i\) such that \(L^i\supseteq I\supseteq L^{i+r}\). Requiring that \(L\) is \(r\)-constrained for some \(r\) is equivalent to \(L\) having \textit{finite obliquity,} a natural translation of the meaning given to these terms for (pro-)\-\(p\)-groups by \textit{G. Klaas, C. R. Leedham-Green} and \textit{W. Plesken} [Linear pro-\(p\)-groups of finite width. Lect. Notes Math. 1674, Springer, Berlin (1997; Zbl 0901.20013)]. Two-generated, ideally 1-constrained pro-\(p\) groups and Lie algebras have been called \textit{thin} and investigated by several authors including the reviewer. Most interest lies in the case of base field of positive characteristic, because the authors show that soluble ideally \(r\)-constrained Lie algebras over fields of characteristic zero have finite dimension. In various contexts a solubility assumption on a pro-\(p\) group or Lie algebra imposes some kind of periodicity on the structure. In fact, the authors prove that the dimensions of the homogeneous components of a soluble ideally \(r\)-constrained Lie algebra of (nilpotency) class large enough are periodic from some point on. Moreover, a soluble ideally \(r\)-constrained Lie algebra of class large enough is a quotient of a unique largest infinite-dimensional soluble ideally \(r\)-constrained Lie algebra. Finally, the authors prove that a finitely genera…
reviewed by: Sandro Mattarei
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 1836719
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: ideally constrained
Wikidata QID: Q115340542
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836719