lower central series

Given a group $G$, its **lower central series** is the inductively defined descending sequence

$$G={G}_{0}\supset {G}_{1}\supset {G}_{2}\supset \dots $$

in which ${G}_{k}=[G,{G}_{k-1}]$ is the subgroup generated by all commutators ${\mathrm{ghg}}^{-1}{h}^{-1}$ where $g\in G$ and $h\in {G}_{k-1}$.

Similarly, given a Lie algebra $L$, its lower central series is the inductively defined descending sequence of Lie subalgebras $L={L}_{0}\supset {L}_{1}\supset {L}_{2}\supset \dots $ in which ${L}_{k}=[L,{L}_{k-1}]$ is the Lie subalgebra generated bt all commutators $[l,h]$ where $l\in L$ and $h\in {L}_{k-1}$.

Created on June 16, 2011 17:58:35
by Zoran Škoda
(161.53.130.104)