Ideally constrained Lie algebras
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 generated ideally \(1\)-constrained Lie algebra of nilpotency class at least four over a finite field can be regarded as a central quotient of a two-generated, ideally \(1\)-constrained Lie algebra over an extension field. This Lie-theoretic analogue of a result of \textit{C. Bonmassar} and \textit{C. M. Scoppola} about \(p\)-groups [Boll. Un. Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 2, No. 1, 161--168 (1999; Zbl 0920.20016)] essentially reduces the study of ideally \(1\)-constrained Lie algebras over finite fields to the case of thin Lie algebras.
scientific article; zbMATH DE number 1836719