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Near braces and p-deformed braided groups

Anastasia Doikou, Bernard Rybolowicz

Abstract

Motivated by recent findings on the derivation of parametric non-involutive solutions of the Yang-Baxter equation we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi-parametric, non-degenerate, non-involutive solutions of the set-theoretic Yang-Baxter equation. These solutions are generalisations of the known ones coming from braces and skew braces. Bijective maps associated to the inverse solutions are also constructed. Furthermore, we introduce the generalized notion of p-deformed braided groups and p-braidings and we show that every p-braiding is a solution of the braid equation. We also show that certain multi-parametric maps within the near braces provide special cases of p-braidings.

Near braces and p-deformed braided groups

Abstract

Motivated by recent findings on the derivation of parametric non-involutive solutions of the Yang-Baxter equation we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi-parametric, non-degenerate, non-involutive solutions of the set-theoretic Yang-Baxter equation. These solutions are generalisations of the known ones coming from braces and skew braces. Bijective maps associated to the inverse solutions are also constructed. Furthermore, we introduce the generalized notion of p-deformed braided groups and p-braidings and we show that every p-braiding is a solution of the braid equation. We also show that certain multi-parametric maps within the near braces provide special cases of p-braidings.
Paper Structure (4 sections, 16 theorems, 42 equations)

This paper contains 4 sections, 16 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Set-theoretic solutions of the YBE and near braces
  3. Generalized bijective maps $\&$ solutions of the braid equation
  4. $p$-deformed braided groups and near braces

Key Result

Theorem 1.3

(Rump's theorem, [25][26]). Assume $(B, +, \circ)$ is a left brace. If the map $\check r_B: B\times B \to B \times B$ is defined as ${\check r}_B(x,y)=(\sigma _{x}(y), \tau _{y}(x))$, where $\sigma _{x}(y)=x\circ y-x$, $\tau _{y}(x)=t\circ x-t$, and $t$ is the inverse of $\sigma _{x}(y)$ in the circ

Theorems & Definitions (44)

  • Definition 1.1: [25][26][6]
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4: Rump
  • Theorem 1.5: Theorem GV
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 34 more