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Deformed solutions of the Yang-Baxter equation associated to dual weak left $\star$-braces

Shoufeng Wang

Abstract

As generalizations of dual weak left braces and skew left braces, in this paper, dual weak left $\star$-braces and square skew left braces are introduced, respectively. We firstly show that a dual weak left $\star$-brace is exactly a strong semilattice of a family of square skew left braces. Then we introduce distributors for dual weak left $\star$-braces and prove that the map deformed by each distributor is always a solution of the Yang-Baxter equation. Our work may be regarded as extending and enriching some related results on skew left braces and weak left braces in literature.

Deformed solutions of the Yang-Baxter equation associated to dual weak left $\star$-braces

Abstract

As generalizations of dual weak left braces and skew left braces, in this paper, dual weak left -braces and square skew left braces are introduced, respectively. We firstly show that a dual weak left -brace is exactly a strong semilattice of a family of square skew left braces. Then we introduce distributors for dual weak left -braces and prove that the map deformed by each distributor is always a solution of the Yang-Baxter equation. Our work may be regarded as extending and enriching some related results on skew left braces and weak left braces in literature.
Paper Structure (4 sections, 22 theorems, 114 equations)

This paper contains 4 sections, 22 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual weak left $\star$-braces
  4. Deformed solutions on dual weak left $\star$-braces

Key Result

Lemma 2.1

Let $(S,\cdot,\, \ast)$ be a regular $\star$-semigroup, $e,f\in P(S,\cdot)$ and $a, b\in S$.

Theorems & Definitions (38)

  • Lemma 2.1: Nordahl-ScheiblichYamada2
  • Lemma 2.2: Theorem 4.1 in Petrich1
  • Lemma 2.3: Theorem 4.1 in Auinger
  • Lemma 2.4: Lemma 2.7 in Liu-Wang
  • Lemma 2.5
  • proof
  • Lemma 2.6: Lemma 2.8 in Liu-Wang
  • Definition 2.7: Definition 3.1 in Liu-Wang
  • Lemma 2.8: Lemma 3.6 in Liu-Wang
  • Lemma 2.9
  • ...and 28 more