Dimension subrings of Lie rings

Abstract

We show that, unlike the lower central series, the dimension series of a Lie ring need not stabilize when two consecutive terms coincide; in fact, arbitrarily long finite plateaux occur. We prove that $D_n(L)/\gamma_{n+1}(L)$ is central in $L/\gamma_{n+1}(L)$ for every $n\geq1$, in contrast to the group case. If $c\geq2$ and $L$ is metabelian and nilpotent of class at most $c$, we prove that $D_{2c-1}(L)=0$. In particular, our results imply that $D_5(L)\subseteq\gamma_4(L)$ for every Lie ring $L$.