A Lazard correspondence for post-Lie rings and skew braces

Abstract

We develop a Lazard correspondence between post-Lie rings and skew braces that satisfy a natural completeness condition. This is done through a thorough study of how the Lazard correspondence behaves on semi-direct sums of Lie rings. In particular, for a prime $p$ and $k<p$, we obtain a correspondence between skew braces of order $p^k$ and left nilpotent post-Lie rings of order $p^k$ on a nilpotent Lie ring. This therefore extends results by Smoktunowicz.