The aim of this paper is to show a close relationship between the factoralgebras by the terms of the upper central series ζ n (L) of a Leibniz algebra L and the terms of its lower central series γ n (L). Specifically we show that finiteness of codimension of some ζ k (L) implies finiteness of dimension of γ k+1 (L) and give explicit bounds for this dimension. We also improve this in the case k = 1, which corresponds to the center and commutator subalgebra of the algebra, respectively. These results are analogous to the results that have been obtained for groups and Lie algebras.

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Let V be a vector space over a field F . If G ≤ GL V F , the central dimension of G is the F -dimension of the vector space V/C V G . In Dixon et al. (2004) and Kurdachenko and Subbotin (2006), soluble linear groups in which the set icd G of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. In this article we study periodic locally radical linear groups, in which the set icd G satisfies one of the weak chain conditions: the weak minimal condition or the weak maximal condition.

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