Left Braces and the Quantum Yang–Baxter Equation

Abstract

Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

“…In Section 5, using the techniques of [41] and skew left braces of nilpotent type we will generalize the results of this section to non-involutive solutions.…”

“…By Lemma 1.7, this is equivalent to R m (X, A) ⊆ Soc n+1 (A), which is equivalent to X ⊆ Soc m+n+1 (A) by the inductive hypothesis. Recall that a finite group G is said to be p-nilpotent if there exists a normal Hall p ′ -subgroup of G. One proves that this subgroup is characteristic in G. Following [41] we define right p-nilpotent skew left braces of nilpotent type:…”

“…To extend some of our results to non-involutive solutions different methods are needed. Following the ideas of Meng, Ballester-Bolinches and Romero [41], we introduce right p-nilpotency of skew left braces of nilpotent type and use this concept to explore retractable non-involutive solutions in Section 5. Finally, in Section 6, again following [41], we study left p-nilpotent skew left braces.…”

“…Each A n is a left ideal of A. We refer to [11,15,41,45,48,49,51] for results on left nilpotent skew left braces.…”

Factorizations of skew braces

“…It is natural to ask which grouptheoretical properties of permutation groups detect multipermutation solutions. There are strong results in this directions [7,14,15,22,23,25]. However, to understand multipermutation solutions it is not enough to know the group structure of their permutation groups.…”