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Affine structures on groups and semi-braces

Paola Stefanelli

TL;DR

The paper introduces a novel framework of affine structures on groups and establishes a tight correspondence with semi-braces, yielding a categorical equivalence between semi-braces and affine structures. It shows that cancellative affine structures correspond to cancellative semi-braces and groupal affine structures correspond to skew braces, enabling systematic construction of bi-skew braces, including non-$\lambda$-homomorphic examples. The work further develops Zappa-product techniques to transfer affine structures to Zappa products, producing semi-braces that are not captured by traditional matched-product constructions and clarifying when these new semi-braces diverge from or align with existing ones. Overall, it provides a unified, category-theoretic approach to braces, skew braces, and set-theoretic Yang–Baxter equation solutions through affine-structure data on groups.

Abstract

We introduce affine structures on groups and show they form a category equivalent to that of semi-braces. In particular, such a new description of semi-braces includes that presented by Rump for braces. By specific affine structures, we provide several instances of bi-skew braces, including some that are not $λ$-homomorphic. Finally, we give a method for determining affine structures on the Zappa product of two groups both endowed with affine structures and prove that such a construction allows for obtaining semi-braces that are not matched product of semi-braces.

Affine structures on groups and semi-braces

TL;DR

The paper introduces a novel framework of affine structures on groups and establishes a tight correspondence with semi-braces, yielding a categorical equivalence between semi-braces and affine structures. It shows that cancellative affine structures correspond to cancellative semi-braces and groupal affine structures correspond to skew braces, enabling systematic construction of bi-skew braces, including non--homomorphic examples. The work further develops Zappa-product techniques to transfer affine structures to Zappa products, producing semi-braces that are not captured by traditional matched-product constructions and clarifying when these new semi-braces diverge from or align with existing ones. Overall, it provides a unified, category-theoretic approach to braces, skew braces, and set-theoretic Yang–Baxter equation solutions through affine-structure data on groups.

Abstract

We introduce affine structures on groups and show they form a category equivalent to that of semi-braces. In particular, such a new description of semi-braces includes that presented by Rump for braces. By specific affine structures, we provide several instances of bi-skew braces, including some that are not -homomorphic. Finally, we give a method for determining affine structures on the Zappa product of two groups both endowed with affine structures and prove that such a construction allows for obtaining semi-braces that are not matched product of semi-braces.
Paper Structure (4 sections, 15 theorems, 56 equations)

This paper contains 4 sections, 15 theorems, 56 equations.

Table of Contents

  1. Affine structures on groups
  2. Semi-braces and affine structures
  3. The special cases of skew braces
  4. Affine structures on Zappa products of groups

Key Result

Proposition 3

Let $\sigma$ be an affine structure on a group $G = \left(B,\circ\right)$. If $H = \left(C,\circ\right)$ is a group and $f:B\to C$ a group isomorphism from $G$ to $H$, then the map $\varphi:C\to C^C$ given by for every $u\in C$, is an affine structure on $H$. Moreover, if $\sigma$ is cancellative, then $\varphi$ is cancellative; if in addition $\sigma$ is groupal, then $\varphi$ is groupal.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Example 1
  • Example 2
  • Example 3
  • Example 5
  • Proposition 3
  • proof
  • Definition 4
  • Theorem 5
  • ...and 33 more