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

TL;DR

This paper introduces dual weak left $igstar$-braces and square skew left braces, showing that certain deformations produce solutions to the Yang-Baxter equation, extending previous algebraic structures.

Contribution

It defines dual weak left $igstar$-braces and square skew left braces, and proves that deformations via distributors yield solutions to the Yang-Baxter equation.

Findings

Dual weak left $igstar$-braces are strong semilattices of square skew left braces.

Deformations by distributors always produce solutions to the Yang-Baxter equation.

The work extends results on skew left braces and weak left braces.

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.