A new formula for Lazard's correspondence for finite braces and pre-Lie algebras

TL;DR

This paper introduces a simple algebraic formula linking finite right nilpotent Fp-braces with finite nilpotent pre-Lie algebras, extending Lazard's correspondence and providing classifications and structural insights.

Contribution

It presents a new algebraic formula for the Lazard correspondence between finite braces and pre-Lie algebras, confirming and extending previous work by Rump.

Findings

Established a new formula for the correspondence

Classified all right nilpotent Fp-braces generated by one element of size p^4

Proved the sum of left nilpotent ideals is left nilpotent

Abstract

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite Fp-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent Fp-braces generated by one element of cardinality p^4 is obtained, answering a question posed by Leandro Vendramin. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left nilpotent ideal, therefore every finite brace contains the largest left nilpotent ideal.