Algebraic approach to Rump's results on relations between braces and pre-Lie algebras

TL;DR

This paper provides an algebraic reformulation of Rump's correspondence between left nilpotent right R-braces and pre-Lie algebras, extending the understanding beyond geometric methods to algebraic formulae over various fields.

Contribution

It offers an algebraic interpretation of Rump's results, applicable to fields of large prime characteristic and characteristic zero, simplifying and broadening the original geometric approach.

Findings

Algebraic reformulation of Rump's correspondence

Applicability to fields of large prime characteristic

Extension to fields of characteristic zero

Abstract

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right R-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper we explain Rump's correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.