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

2019 ◽  
Vol 30 (01) ◽  
pp. 91-115
Author(s):  
E. Acri ◽  
R. Lutowski ◽  
L. Vendramin
Keyword(s):  
Short Proof ◽  
Linear Representation ◽  
Structure Group ◽  
Baxter Equation ◽  
Product Property ◽  
Unique Product ◽  
Bieberbach Group

Using Bieberbach groups, we study multipermutation involutive solutions to the Yang–Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right [Formula: see text]-nilpotent skew braces. The theory of left [Formula: see text]-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.

Left orders in Garside groups

2016 ◽  
Vol 26 (07) ◽  
pp. 1349-1359 ◽  
Author(s):  
Fabienne Chouraqui
Keyword(s):  
Cantor Set ◽  
Total Order ◽  
Theoretical Solution ◽  
Structure Group ◽  
Baxter Equation ◽  
Unique Product ◽  
Bieberbach Group ◽  
Left Order ◽  
Left And Right

We consider the structure group of a non-degenerate symmetric (non-trivial) set-theoretical solution of the quantum Yang–Baxter equation. This is a Bieberbach group and also a Garside group. We show this group is not bi-orderable, that is it does not admit a total order which is invariant under left and right multiplications. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with a recurrent left order and with space of left orders homeomorphic to the Cantor set, while there exist others that are even not unique product groups.

scholarly journals Torsion-free group without unique product property

1987 ◽  
Vol 108 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Eliyahu Rips ◽  
Yoav Segev
Keyword(s):  
Free Group ◽  
Torsion Free Group ◽  
Product Property ◽  
Unique Product ◽  
Torsion Free

scholarly journals On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

2019 ◽  
Vol 62 (3) ◽  
pp. 683-717 ◽  
Author(s):  
Victoria Lebed ◽  
Leandro Vendramin
Keyword(s):  
Structure Group ◽  
Baxter Equation ◽  
Finite Quotient ◽  
Torsion Free

AbstractThis paper explores the structure groups G(X,r) of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient $\overline {G}_{(X,r)}$ of G(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G(X,r), then it also injects into $\overline {G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.

scholarly journals New examples of torsion-free non-unique product groups

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
William Carter
Keyword(s):  
Free Groups ◽  
Infinite Family ◽  
Product Property ◽  
Unique Product ◽  
Large Sets ◽  
Torsion Free

Abstract.We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.

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