L-algebras with duality and the structure group of a set-theoretic solution to the Yang-Baxter equation

2020 ◽  
Vol 224 (8) ◽  
pp. 106314 ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Structure Group ◽  
Baxter Equation ◽  
Theoretic Solution

scholarly journals Left semi-braces and solutions of the Yang–Baxter equation

2019 ◽  
Vol 31 (1) ◽  
pp. 241-263 ◽  
Author(s):  
Eric Jespers ◽  
Arne Van Antwerpen
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Baxter Equation ◽  
Polynomial Algebras ◽  
Finite Set ◽  
Theoretic Solution ◽  
Kirillov Dimension ◽  
Do So

Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

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.

Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings

2019 ◽  
Vol 18 (08) ◽  
pp. 1950145
Author(s):  
Wolfgang Rump
Keyword(s):  
Residue Field ◽  
Ring Structure ◽  
Baxter Equation ◽  
Exponential Map ◽  
Exponential Form ◽  
Degenerate Solution ◽  
Terminal Object ◽  
Theoretic Solution ◽  
Initial Object ◽  
Central Residue

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field [Formula: see text] are characterized as [Formula: see text]-braces with a compatible near-ring structure. Under this correspondence, [Formula: see text]-braces are radicals of local near-rings with radical square zero.

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.

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