Free Centre-by-Metabelian Lie Rings

Abstract

Abstract. We study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous and fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian 2-group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous and fine homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than 2. For that we exploit a 6-term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature.

Free centre-by-metabelian Lie algebras in characteristic 2

“…V. Kuz'min in his pioneering paper [2]. Much later it turned out (see [7], [4]) that some of the details on Lie rings given in [2] required modification. The results in [4] were conclusive in the case where K is a field of characteristic other than 2, and also for the torsion-free part of G when K = Z.…”

“…Much later it turned out (see [7], [4]) that some of the details on Lie rings given in [2] required modification. The results in [4] were conclusive in the case where K is a field of characteristic other than 2, and also for the torsion-free part of G when K = Z. Most of the present paper is devoted to calculating the dimensions of the fine homogeneous components of G under the assumption that K is a field of characteristic 2.…”

Free Centre-by-Nilpotent-by-Abelian Lie Rings of Rank 2

“…This was proved in [5] where a basis for this 2-subgroup was exhibited. However, as pointed out by Zerck [9], some of the details in [5] needed correction, and this was eventually carried out in [7] and [4]. In fact, in the case of rank 2, which turned out to be considerably easier than the case of higher ranks, a similar result on torsion in the context of the lower central quotients of the free centre-by-metabelian group was proved much earlier by Ridley [8].…”

Free centre-by-(abelian-by-exponent 2) groups

“…We refer to the proof of Lemma 3.3 in [10]. The arguments used there in the context of Lie rings are easily adapted to the group commutators in the present paper.…”

“…Following [10], we call elements of the form (4.12) Kuz'min elements. We summarize the result of our discussion so far as follows.…”