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On dynamical skew braces and skew bracoids

Davide Ferri

TL;DR

The paper develops a comprehensive framework connecting dynamical and quiver-theoretic versions of the Yang–Baxter equation via skew braces, groupoids, and heaps. It proves that under a technical hypothesis, solutions from dynamical skew braces are braided groupoids, and that every connected braided groupoid can be parallelised to a dynamical skew brace, establishing a tight correspondence between these structures. It then analyzes the maximal zero-symmetric dynamical skew brace of a group, giving explicit counts and invariants that govern the combinatorial shape of the associated quiver, and shows how these invariants encode the number of brace structures on a given group. A central theme is the Matsumoto–Shimizu correspondence, which provides a labelling mechanism converting skew-bracoid data into dynamical skew braces and vice versa, yielding an equivalence between connected skew bracoids and connected zero-symmetric dynamical skew braces. The results unify several strands (quiver-theoretic ybe, braided groupoids, and dynamical braces) and illuminate fundamental geometric and combinatorial aspects of these algebraic objects with potential implications for qutries and categorical approaches to the Yang–Baxter equation.

Abstract

Dynamical skew braces are known to produce solutions to the quiver-theoretic Yang--Baxter equation. Under a technical hypothesis, we prove that these solutions are braided groupoids (and hence skew bracoids in the sense of Sheng, Tang and Zhu). Conversely, every connected braided groupoid can be parallelised, making it isomorphic to a dynamical skew brace. We study the combinatorics of these objects, depending on some strings of integer invariants.

On dynamical skew braces and skew bracoids

TL;DR

The paper develops a comprehensive framework connecting dynamical and quiver-theoretic versions of the Yang–Baxter equation via skew braces, groupoids, and heaps. It proves that under a technical hypothesis, solutions from dynamical skew braces are braided groupoids, and that every connected braided groupoid can be parallelised to a dynamical skew brace, establishing a tight correspondence between these structures. It then analyzes the maximal zero-symmetric dynamical skew brace of a group, giving explicit counts and invariants that govern the combinatorial shape of the associated quiver, and shows how these invariants encode the number of brace structures on a given group. A central theme is the Matsumoto–Shimizu correspondence, which provides a labelling mechanism converting skew-bracoid data into dynamical skew braces and vice versa, yielding an equivalence between connected skew bracoids and connected zero-symmetric dynamical skew braces. The results unify several strands (quiver-theoretic ybe, braided groupoids, and dynamical braces) and illuminate fundamental geometric and combinatorial aspects of these algebraic objects with potential implications for qutries and categorical approaches to the Yang–Baxter equation.

Abstract

Dynamical skew braces are known to produce solutions to the quiver-theoretic Yang--Baxter equation. Under a technical hypothesis, we prove that these solutions are braided groupoids (and hence skew bracoids in the sense of Sheng, Tang and Zhu). Conversely, every connected braided groupoid can be parallelised, making it isomorphic to a dynamical skew brace. We study the combinatorics of these objects, depending on some strings of integer invariants.

Paper Structure

This paper contains 20 sections, 24 theorems, 60 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quivers
  4. Groupoids and left semiloopoids
  5. Yang--Baxter maps and braidings
  6. Groupoids of pairs
  7. Heaps
  8. Quiver-theoretic skew braces
  9. Post-semiloopoids
  10. Dynamical skew braces
  11. Dynamical sets and the dynamical ybe
  12. Dynamical skew braces
  13. Dynamical skew braces and braidings
  14. The maximal (zero-symmetric) dynamical skew brace of a group
  15. On the combinatorics of dynamical skew braces
  16. ...and 5 more sections

Key Result

Lemma 2.17

Let $\mathscr{G}$ be a left unital associative semiloopoid. Then for all $\lambda\in \Lambda_{\mathrm{in}}$, and for all $\mu, \nu\in \Lambda_{\mathrm{un}}$ in the same connected component, such that $\mathscr{G}(\lambda,\nu)\neq \emptyset$, there are bijections $\mathscr{G}(\lambda,\mu)\cong \maths

Figures (3)

  • Figure 1: The ybe for $\sigma$ can be seen as the closure of cubes, where each oriented face represents the application of $\sigma$. If $\sigma$ is moreover involutive, the orientation of the faces is superfluous.
  • Figure 2: Some examples of admissible and nonadmissible quivers for dynamical skew braces over $A = \mathbb{Z}/3\mathbb{Z}$.
  • Figure 3: Although this quiver has an admissible number of vertices, and technically satisfies conditions (ii) and (iii) of Theorem \ref{['thm:maximal_zerosymm_dbrace']}, it does not correspond to any dynamical skew brace. Indeed, it is not a subquiver of the one in Example \ref{['ex:Z3_initials']}.

Theorems & Definitions (106)

  • Definition 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Definition 2.13
  • Remark 2.14
  • ...and 96 more