Dimension Quotients of Metabelian Lie Rings

TL;DR

This paper investigates the relationship between the lower central series and the dimension series in metabelian Lie rings, establishing bounds and equalities that clarify their differences and similarities.

Contribution

It proves that for metabelian Lie rings, doubling the dimension series is contained in the lower central series, and their commutator relations are explicitly characterized.

Findings

2δ_n(L) ⊆ γ_n(L) for all n≥1

[ ext{δ}_n(L), L] = γ_{n+1}(L) for all n≥1

The two series can differ but are closely related in 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 $n\geq 1$.