Dimension Quotients of Metabelian Lie Rings

Abstract

For a Lie ring $L$ over the ring of integers, we compare its lower central series $\{\gamma_n(L)\}_{n\geq 1}$ and its dimension series $\{\delta_n(L)\}_{n\geq 1}$ defined by setting $\delta_n(L)= L\cap \varpi^n(L)$, where $\varpi(L)$ is the augmentation ideal of the universal enveloping algebra of $L$. While $\gamma_n(L)\subseteq\delta_n(L)$ for all $n\geq 1$, the two series can differ. In this paper it is proved that if $L$ is a metabelian Lie ring, then $2\delta_n(L)\subseteq\gamma_n(L)$, and $[\delta_n(L),\,L]=\gamma_{n+1}(L)$, for all $n\geq 1$.