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Set-theoretic type solutions of the braid equation

Jorge A. Guccione, Juan J. Guccione, Christian Valqui

Abstract

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-braces, matched pairs of groups and invertible $1$-cocycles remain valid in our setting.

Set-theoretic type solutions of the braid equation

Abstract

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-braces, matched pairs of groups and invertible -cocycles remain valid in our setting.

Paper Structure

This paper contains 13 sections, 78 theorems, 241 equations.

Table of Contents

  1. Preliminaries
  2. Left non-degenerate coalgebra endomorphisms of X²
  3. q-magma coalgebras
  4. Coalgebra endomorphisms vs q-magma coalgebras
  5. Very strongly regular q-magma coalgebras
  6. Solutions of the braid equation and q-cycle coalgebras
  7. Weak braiding operators and Hopf q-braces
  8. Hopf Skew-braces
  9. Hopf skew-braces and invertible 1-cocycle
  10. Comparison between Hopf-skew braces and Yetter-Drinfeld braces
  11. Ideals of Hopf q-braces
  12. The socle
  13. The free Hopf q-brace over a very strongly regular q-cycle coalgebra

Key Result

Proposition 1.2

Let $X$ be a coalgebra and let $\alpha\colon X^2\to X$ and $\beta\colon X\otimes X^{\mathop{\mathrm{cop}}\nolimits}\to X$ be two maps. Set $x^y\coloneqq \alpha(x\otimes y)$ and $x\cdot y\coloneqq \beta(x\otimes y)$. If then $\alpha$ is a coalgebra map if and only if $\beta$ is.

Theorems & Definitions (263)

  • Remark 1.1
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.9
  • Remark 1.11
  • ...and 253 more