Lazard Lie algebra ($L^{\ast }(G)$)

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.^[ref]

For odd $p$, a concrete example is $G=1+p{M}_n({\mathbb{Z}}_p)$. The convergent matrix logarithm and matrix exponential identify its Lie algebra with $p{M}_n({\mathbb{Z}}_p)$, using ordinary matrix addition and bracket $[A,B]=AB-BA$. Multiplication corresponds to the Baker--Campbell--Hausdorff formula, beginningFor $p=2$, 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.^[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.