Solutions of the Yang–Baxter equation associated with a left brace

Abstract

Given a left brace G, a method to construct all the involutive, nondegenerate set-theoretic solutions (Y, s) of the YBE, such that G(Y, s) ∼ = G is given. This method depends entirely on the brace structure of G.

Skew braces and the Yang–Baxter equation

“…He proved that if a left brace A has additive group isomorphic to Z/p k , where p > 2 is a prime number, then (A, ·) ≃ Z/p k . According to [3], the converse holds for all p. In [1], Bachiller classified left braces of size p 2 and p 3 , where p is a prime number. The techniques used in these papers might prove useful to address the questions, problems and conjectures in this section.…”

“…Brute force seems not to be good enough. In [3], Bachiller, Cedó and Jespers, give a method to construct all finite solutions of a given size. For it to work, one needs the classification of left braces.…”

On skew braces and their ideals

“…This is the only classical brace of size eight with additive group is isomorphic to C 8 and multiplicative group isomorphic to C 4 × C 2 . It has four ideals which are isomorphic to 0, B 2,1 , B 4,1 and B 8,5 . The quotients of A are then isomorphic to B 8,5 , B 4,2 , B 2,1 and 0.…”

Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition

“…The investigations have intensified since the discovery of new algebraic structures that are related to a set-theoretic solution and that determine all non-degenerate bijective (involutive) solutions on a finite set (see for example [4,5,25,36]). Namely, to deal with involutive non-degenerate solutions Rump [36] introduced the algebraic structure called a (left) brace and to deal with arbitrary nondegenerate solutions Guarnieri and Vendramin [25] extended this notion to that of a skew brace.…”