A Lazard correspondence for post-Lie rings and skew braces

TL;DR

This paper establishes a new correspondence between post-Lie rings and skew braces under certain completeness conditions, extending prior results and deepening the understanding of their algebraic structures.

Contribution

It introduces a Lazard correspondence for post-Lie rings and skew braces, expanding the algebraic framework for these structures and generalizing previous work by Smoktunowicz.

Findings

Correspondence between skew braces of order p^k and nilpotent post-Lie rings of the same order

Extension of Lazard correspondence to semi-direct sums of Lie rings

Generalization of prior results by Smoktunowicz

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.