= Lie algebra of a uniform pro-p group
{c}
{title2=$L_G$}
= Intrinsic Lie algebra of a uniform pro-p group
{synonym}
On the set underlying a <uniform pro-p group>, define scalar multiplication by $\lambda x=x^\lambda$, and define addition and the <Lie bracket> by
$$
x+_Ly=\lim_{n\to\infty}(x^{p^n}y^{p^n})^{p^{-n}},\qquad [x,y]_L=\lim_{n\to\infty}[x^{p^n},y^{p^n}]^{p^{-2n}}.
$$
Roots are the unique roots in the appropriate power layers. The limits give a <powerful Lie lattice>. In a suitable complete normed <group algebra>, the <p-adic logarithm> identifies these operations with ordinary addition and the associative-algebra commutator.
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