Dimension Quotients of Metabelian Lie Rings

Abstract

For a Lie ring $L$ over the ring of integers, we compare its lower central series $\{{\gamma }_n(L){\}}_{n\ge 1}$ and its dimension series $\{{\delta }_n(L){\}}_{n\ge 1}$ 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 $n\ge 1$, 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 $n\ge 1$.

arXiv comment8 pages