On pre-Lie rings related to some non-Lazard braces
Agata Smoktunowicz
TL;DR
The paper addresses finite braces of order $p^{n}$ by imposing conditions like $A^{\frac{p-1}{2}}\subseteq pA$ or $p^{i}A$ being ideals, and constructs left nilpotent pre-Lie rings associated to these braces. It develops a framework with properties $1'$ and $1''$, a pullback map, and a $f$-map to connect braces to pre-Lie structures, proving left nilpotency and showing how to recover factor braces from the pre-Lie data via flow-type formulas dependent only on the additive group. The main contributions include a general method to pass from certain braces to left nilpotent pre-Lie rings and back to factor braces $A/ann(p^{4k})$, providing a Lazard-type correspondence in a finite, highly nilpotent setting and broadening the scope beyond classical Lazard conditions. This work offers new structural links between finite braces and nonassociative algebras, with potential applications to understanding Yang–Baxter solutions and related algebraic systems in finite contexts.
Abstract
Let A be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that either (i) the additive group of brace $A$ has rank smaller than $p-3$, or (ii) $A^{\frac {p-1}2}\subseteq pA$ or (iii) $p^{i}A$ is an ideal in in $A$ for each $i$. It is shown that there is a pre-Lie ring associated to brace $A$. The left nilpotency index of this pre-Lie ring can be arbitrarily large. Let $A$ be a brace of cardinality $p^{n}$ for some prime number $p$. Denote $ann(p^{i})=\{a\in A: p^{i}a=0\}$. Suppose that for $i=1,2,\ldots $ and all $a,b\in A$ we have \[a*(a*(\cdots *a*b))\in pA, a*(a*(\cdots *a*ann(p^{i})))\in ann(p^{i-1})\] where $a$ appears less than $\frac {p-1}4$ times in this expression. Let $k$ be such that $p^{k(p-1)}A=0$. It is shown that the brace $A/ann(p^{4k})$ is obtained from a left nilpotent pre-Lie ring by a formula which depends only on the additive group of brace $A$. We also obtain some applications of this result.