Associated graded Lie algebra of a filtered group
= Associated graded Lie algebra of a filtered group
{title2=$\operatorname{gr}G$}
For $G_\lambda=\{g:\omega(g)\geq\lambda\}$ and $G_{\lambda+}=\{g:\omega(g)>\lambda\}$, the associated graded group
$$
\operatorname{gr}G=\bigoplus_\lambda G_\lambda/G_{\lambda+}
$$
is a graded <Lie algebra> with bracket induced by the <group commutator>. For a p-valuation, $t\operatorname{gr}_\lambda(g)=\operatorname{gr}_{\lambda+1}(g^p)$ makes it a graded $\mathbb F_p[t]$-Lie algebra.