Dimension quotients of metabelian Lie rings

Abstract

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].