Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses

Colazzo, Ilaria; Jespers, Eric; Antwerpen, Arne Van; Verwimp, Charlotte

doi:10.1016/j.jalgebra.2022.07.019

Cited by 8 publications

(3 citation statements)

References 99 publications

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“…In 2017, Guarnieri and Vendramin introduced the concept of skew left braces [8] and proved that every skew left brace provides a non-involutive solution to the Yang-Baxter equation. Skew left braces have been widely applied in various branches of mathematics, including connections to regular subgroups [9], Hopf-Galois extensions [10], triply factorized groups [11], Garside theory [12,13], ring theory [14,15], flat manifolds [16], and pre-Lie algebras [17].…”

Section: Introductionmentioning

confidence: 99%

Rota–Baxter Operators on Skew Braces

Wang,

Zhang,

Zhang

2024

Mathematics

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In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as (Z,+,∘) and (Z/(4),+,∘). We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace.

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Section: Introductionmentioning

confidence: 99%

Rota–Baxter Operators on Skew Braces

Wang,

Zhang,

Zhang

2024

Mathematics

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“…Recently, idempotent left non-degenerate solutions of the Yang-Baxter equation have been completely described (see [7,16,23] and the classifications therein). A similar study can also be done for the pentagon equation, motivated by the fact that already for semigroups of small order there are several of idempotent type, i.e., s 2 = s. The question of studying this class of solutions arose explicitly during the talk held in the occasion of "First UMI meeting of PhD students" [17].…”

Section: Introductionmentioning

confidence: 99%

Idempotent set-theoretical solutions of the pentagon equation

Mazzotta

2023

Boll Unione Mat Ital

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A set-theoretical solution of the pentagon equation on a non-empty set X is a function $$s:X\times X\rightarrow X\times X$$ s : X × X → X × X satisfying the relation $$s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$$ s 23 s 13 s 12 = s 12 s 23 , with $$s_{12}=s\times \,{{\,\textrm{id}\,}}_X$$ s 12 = s × id X , $$s_{23}={{\,\textrm{id}\,}}_X \times \, s$$ s 23 = id X × s and $$s_{13}=({{\,\textrm{id}\,}}_X\times \, \tau )s_{12}({{\,\textrm{id}\,}}_X\times \,\tau )$$ s 13 = ( id X × τ ) s 12 ( id X × τ ) , where $$\tau :X\times X\rightarrow X\times X$$ τ : X × X → X × X is the flip map given by $$\tau (x,y)=(y,x)$$ τ ( x , y ) = ( y , x ) , for all $$x,y\in X$$ x , y ∈ X . Writing a solution as $$s(x,y)=(xy,\theta _x(y))$$ s ( x , y ) = ( x y , θ x ( y ) ) , where $$\theta _x: X \rightarrow X$$ θ x : X → X is a map, for every $$x\in X$$ x ∈ X , one has that X is a semigroup. In this paper, we study idempotent solutions, i.e., $$s^2=s$$ s 2 = s , by showing that the idempotents of X have a crucial role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $$\theta _1$$ θ 1 is a monoid homomorphism.

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“…This work established a new area of mathematical research that is still growing. Furthermore, the machinery developed by Etingof et al has been extended over the last years to solutions that are non-involutive or even degenerate; see, for example, [6,7,10,11,13].…”

Section: Introductionmentioning

confidence: 99%