 Lie algebra of a uniform pro-p group

On the set underlying a uniform pro-p group, define scalar multiplication by $\lambda x=x^{\lambda }$, and define addition and the Lie bracket byRoots 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.