Dimension quotients of metabelian Lie rings

Abstract

For a Lie ring L over the ring of integers, we compare its lower central series {γn(L)}n≥1 and its dimension series {δn (L)}n≥1 defined by setting δn (L) = L ∩ ωn(L), where ω(L) is the augmentation ideal of the universal enveloping algebra of L. While γn (L) ⊆ δn (L) for all n ≥ 1, the two series can differ. In this paper, it is proved that if L is a metabelian Lie ring, then 2δn (L) ⊆ γn (L), and [δn (L),L] = γn+1(L), for all n ≥ 1. © 2017 World Scientific Publishing Company.