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On left nilpotent skew braces of class 2

A. Ballester-Bolinches, L. A. Kurdachenko, V. Pérez-Calabuig

TL;DR

This work studies left nilpotent skew braces of class $2$, i.e., skew braces with $B^3=0$, and derives quantitative bounds linking additive and brace nilpotency. The authors introduce a filtration via $S_n= ext{Ker}\, ext{λ}^{(n)} olinebreak \\cap B^2$ and prove that $B^{(2+mk)}\subseteq S_{r-k}$, where $m$ and $r$ are the nilpotency classes of $(B,+)$ and $(B^2,+)$, yielding $B^{(2+mr+1)}=0$; hence $B$ is right nilpotent of class at most $2+mr$, and, in the nilpotent-type setting, centrally nilpotent. In the abelian-type case, the right nilpotency class is at most $3$ ($B^{(4)}=0$), with sharp examples showing the bounds are best possible. These results have implications for Yang–Baxter equation solutions, as they imply multipermutation behavior for the associated skew braces, and the paper outlines future work on finitely generated left nilpotent skew braces of class $2$. The methods center on the star product, the $ ext{λ}$-action, and structural filtrations that connect additive nilpotency to brace nilpotency.

Abstract

The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible.

On left nilpotent skew braces of class 2

TL;DR

This work studies left nilpotent skew braces of class , i.e., skew braces with , and derives quantitative bounds linking additive and brace nilpotency. The authors introduce a filtration via and prove that , where and are the nilpotency classes of and , yielding ; hence is right nilpotent of class at most , and, in the nilpotent-type setting, centrally nilpotent. In the abelian-type case, the right nilpotency class is at most (), with sharp examples showing the bounds are best possible. These results have implications for Yang–Baxter equation solutions, as they imply multipermutation behavior for the associated skew braces, and the paper outlines future work on finitely generated left nilpotent skew braces of class . The methods center on the star product, the -action, and structural filtrations that connect additive nilpotency to brace nilpotency.

Abstract

The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces of class , i.e. skew braces with . We prove that if is of nilpotent type, then is centrally nilpotent. In fact, we show that is right nilpotent of class at most , i.e. , where and are the nilpotency classes of the additive group of and , respectively. If is of abelian type, then is actually right nilpotent of class , i.e. , and this bound is best possible.
Paper Structure (3 sections, 6 theorems, 19 equations)

This paper contains 3 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof and consequences of Theorem \ref{['teo:A']}

Key Result

Theorem A

Let $B$ be a skew brace of nilpotent type such that $B$ is left nilpotent of class $2$. Then, $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. In particular, $B$ is centrally ni

Theorems & Definitions (12)

  • Theorem A
  • Corollary 1
  • Corollary 2
  • Remark 3
  • Remark 4
  • Lemma 5
  • proof : Proof of Theorem \ref{['teo:A']}
  • Example 6
  • Example 7
  • Proposition 9
  • ...and 2 more