= Lazard Lie algebra
{c}
{title2=$L^*(G)$}
In the setting of a <uniform pro-p group> $G$, its integral <Lazard Lie algebra> is a finite free $\mathbb Z_p$-<module> whose addition and <Lie bracket> recover infinitesimal versions of group multiplication and the <group commutator>. It carries more integral information than the associated $\mathbb Q_p$-<Lie algebra>. The construction requires hypotheses on the group and its filtration; it is not available unchanged for every <profinite group>.<Some complements to the Lazard isomorphism by Annette Huber and others>[Definition 2.6.6, Example 2.6.7 and section 3.3]{ref}
For odd $p$, a concrete example is $G=1+pM_n(\mathbb Z_p)$. The convergent <matrix logarithm> and <matrix exponential> identify its Lie algebra with $pM_n(\mathbb Z_p)$, using ordinary matrix addition and bracket $[A,B]=AB-BA$. Multiplication corresponds to the <Baker--Campbell--Hausdorff formula>, beginning
$$
\log(\exp A\exp B)=A+B+\tfrac12[A,B]+\cdots.
$$
For $p=2$, the analogous safe neighborhood is $1+4M_n(\mathbb Z_2)$ with Lie algebra $4M_n(\mathbb Z_2)$. The smaller neighborhood $1+2\mathbb Z_2$ contains $-1$, so torsion already prevents it from being uniform.<Some complements to the Lazard isomorphism by Annette Huber and others>[Examples 2.2.3 and 2.6.7; Definition 3.3.2 and following paragraph]{ref}
For the additive group $\mathbb Z_p$, the associated Lie algebra is rank one with zero bracket. Under suitable coefficient hypotheses, the <Lazard isomorphism> compares its <Lie algebra cohomology> with <continuous group cohomology>.
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