On some connections between braces and pre-Lie rings outside of the context of Lazard's correspondence

TL;DR

This paper explores the relationship between braces and pre-Lie rings outside Lazard's correspondence, providing conditions under which certain factor braces relate to nilpotent pre-Lie rings.

Contribution

It establishes a connection between braces and pre-Lie rings for specific classes of braces beyond Lazard's framework.

Findings

Factor braces $A/p^{i}A$ relate to nilpotent pre-Lie rings.

Conditions on the additive group influence the brace's structure.

Formulas similar to flows in pre-Lie rings describe these factor braces.

Abstract

Let $p>3$ be a prime number and let $A$ be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than $p^{\alpha }$ for some $\alpha $. Suppose that either (i) $A^{\lfloor{\frac {p-1}4}\rfloor}\subseteq pA$ or that (ii) the additive group of brace $A$ has rank smaller than ${\lfloor{\frac {p-1}4}\rfloor}$. It is shown that for every natural number $i\leq \alpha- {\frac {4\alpha }{p-1}}$ the factor brace $A/p^{i}A$ is obtained by a formula similar to the group of flows from a left nilpotent pre-Lie ring.