From pre-Lie rings back to braces

TL;DR

This paper explores the relationship between braces and pre-Lie rings, providing formulas to reconstruct braces from pre-Lie structures and demonstrating applications to nilpotent pre-Lie algebras.

Contribution

It introduces a formula based solely on the additive group of a brace to reverse a previous construction linking braces and pre-Lie rings.

Findings

A formula to recover braces from pre-Lie rings depending only on the additive group.

Demonstration that certain braces are groups of flows of nilpotent pre-Lie algebras.

Extension of the understanding of the structure of braces in relation to pre-Lie algebra theory.

Abstract

Let A be a brace of cardinality p^{n} where p>n+1 is prime, and ann(p^{i}) be the set of elements of additive order at most p^{i} in this brace. A pre-Lie ring related to the brace A/ann(p^{2}) was constructed in [8]. We show that there is a formula dependent only on the additive group of the brace A which reverses the construction from [8]. As an application example it is shown that the brace A/ann(p^{4}) is the group of flows of a left nilpotent pre-Lie algebra.