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Finiteness conditions on skew braces and solutions of the Yang-Baxter equation

Rosa Cascella, Silvia Properzi, Arne Van Antwerpen

Abstract

A finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation gives rise to a structure skew brace $B(X,r)$ that is a $λ_f$-skew brace, i.e. every element has finitely many $λ$-images, and whose additive group is $FC$. This motivates the study of finiteness conditions on skew braces. We first study the general class of $λ_f$ skew braces and the subclass where the additive group is $FC$, showing that these properties share a resemblance to finite conjugacy, having an analog of the $FC$-center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index.

Finiteness conditions on skew braces and solutions of the Yang-Baxter equation

Abstract

A finite non-degenerate set-theoretic solution of the Yang-Baxter equation gives rise to a structure skew brace that is a -skew brace, i.e. every element has finitely many -images, and whose additive group is . This motivates the study of finiteness conditions on skew braces. We first study the general class of skew braces and the subclass where the additive group is , showing that these properties share a resemblance to finite conjugacy, having an analog of the -center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index.
Paper Structure (6 sections, 49 theorems, 110 equations)

This paper contains 6 sections, 49 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on skew braces and solutions
  3. The index of a sub skew brace
  4. Skew braces with finite lambda-classes
  5. Skew braces with finite lambda-classes
  6. Applications to solutions

Key Result

Proposition 2.2

Let $B$ be a skew brace and $H$ be a subgroup of $(B,+)$. Then $\operatorname{PStab}_B(H)$ is a normal subgroup of $(\operatorname{Stab}_\lambda(H),\circ)$ with $(\operatorname{Stab}_\lambda(H)/\operatorname{PStab}_B(H),\circ)$ isomorphic to a subgroup of $\operatorname{Aut}(H,+)$.

Theorems & Definitions (125)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • ...and 115 more