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

Agata Smoktunowicz

TL;DR

This work develops connections between braces and left-nilpotent pre-Lie rings outside the classical Lazard correspondence. By introducing pullbacks, pseudobraces, and a generalized flow framework using a $ ho^{-1}$-based construction, the authors show that, under Property 1 and 2, the quotient $A/ann(p^{2k})$ of a brace $A$ can be obtained from a pre-Lie ring via a universal formula depending only on the additive group. The main achievement is that factor braces $A/ann(p^{2k})$ are expressible as generalized group-of-flows of left-nilpotent pre-Lie rings, with the $igodot$-operation providing a consistent bridge between brace and pseudobrace realizations. This furnishes a pathway from right-nilpotent braces to broader finite braces and clarifies how pre-Lie structures orchestrate the flow-based construction in a way that is independent of the particular pullback chosen. The results enhance our understanding of radical/flow mechanisms in braces and deepen connections to pre-Lie algebras, Lie theory, and Yang-Baxter-related structures beyond Lazard’s correspondence.

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 $p^{α}$ for some $α$. 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 $i\leq α- {\frac {4α}{p-1}}$ 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.

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

TL;DR

This work develops connections between braces and left-nilpotent pre-Lie rings outside the classical Lazard correspondence. By introducing pullbacks, pseudobraces, and a generalized flow framework using a -based construction, the authors show that, under Property 1 and 2, the quotient of a brace can be obtained from a pre-Lie ring via a universal formula depending only on the additive group. The main achievement is that factor braces are expressible as generalized group-of-flows of left-nilpotent pre-Lie rings, with the -operation providing a consistent bridge between brace and pseudobrace realizations. This furnishes a pathway from right-nilpotent braces to broader finite braces and clarifies how pre-Lie structures orchestrate the flow-based construction in a way that is independent of the particular pullback chosen. The results enhance our understanding of radical/flow mechanisms in braces and deepen connections to pre-Lie algebras, Lie theory, and Yang-Baxter-related structures beyond Lazard’s correspondence.

Abstract

Let be a prime number and let be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than for some . Suppose that either (i) or that (ii) the additive group of brace has rank smaller than . It is shown that for every natural number the factor brace is obtained by a formula similar to the group of flows from a left nilpotent pre-Lie ring.

Paper Structure

This paper contains 29 sections, 49 theorems, 159 equations.

Table of Contents

  1. Background information
  2. Pullback
  3. Pseudobraces
  4. Operation $\odot$ for braces and pseudobraces
  5. Properties $1$ and $2$
  6. Theorem \ref{['71']}
  7. Proof of Theorem \ref{['71']}
  8. Elements $e_{i}(a,b)$
  9. Elements ${\tilde{e}}_{i}(a,b)$
  10. Connections between operations $\star$ and $*$
  11. Relations between $*$ and $\odot$
  12. Introduction
  13. Background information
  14. Braces
  15. Pre-Lie rings
  16. ...and 14 more sections

Key Result

Theorem 1

Let $A$ be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that for $i=1,2,\ldots$ and all $a,b\in A$ we have where $a$ appears less than $\frac{p-1}{4}$ times in this expression. Let $k$ be a natural number. Then the factor brace $A/ann(p^{2k})$ is obtained by an algebraic formula from the brace $p^{k}A$ and that this formula only depends on the additive group of brace $A$.

Theorems & Definitions (114)

  • Theorem 1
  • Lemma 2
  • Definition 3
  • Theorem 4
  • proof
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 104 more