Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks

Bachiller, David

doi:10.1142/s0218216518500554

Cited by 47 publications

(60 citation statements)

References 45 publications

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“…According to the classification of skew braces of size p 2 given in [3,Proposition 2.4], there is just one non-trivial such brace with cyclic additive group and it leads to formula (4). The group (B, •) is cyclic, since according to Lemma 2.2 its p-Sylow subgroup is cyclic.…”

Section: Regular Subgroupsmentioning

confidence: 99%

Skew braces of size pq

Acri

Bonatto

2020

Communications in Algebra

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In this paper we enumerate the skew braces of size p 2 q for p, q odd primes by the classification of regular subgroups of the holomorph of the groups of size p 2 q. In particular, we provide explicit formulas for the skew braces of abelian type.Proof. The groups H, L and G c,θ are regular. If G c,β 0,r and G d,β 0,r are conjugate by some h ∈ Aut(A), then h normalizes β 0,r and so it centralizes it. So h(σ) c β 0,r = σ c β 0,r (mod ǫ, τ ). Then σ c β 0,r = σ d β 0,r (mod ǫ, τ ) which implies that c = d. The same argument applies if θ = α 1,1 β 0,r .Let G be a regular subgroup of Hol(A) such that |π 2 (G)| = p. According to Table 16 we need to discuss three cases.Assume that π 2 (G) = α 1,1 . Then the kernel has order pq and so, up to conjugation by a power of α 1,1 we can assume that G = ǫ, σ n τ m , σ a τ b α 1,1 . By condition (K) we have n = 0. So G = ǫ, τ, σ a α 1,1 and G is conjugate to H by α 0,a −1 .Assume that π 2 (G) = β 0,r . Then G has the following standard presentation:If n = 0, we conjugate G by α − m n ,1 and then by α 0,b −1 and we get L. If n = 0 we have G = ǫ, τ, σ b β 0,r = G b,β 0,r .Assume that π 2 (G) = α 1,1 β 0,r . Then

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Section: Regular Subgroupsmentioning

confidence: 99%

Skew braces of size pq

Acri

Bonatto

2020

Communications in Algebra

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“…By [, Corollary 1.10], the map

λ : (A, \circ) \to Aut (A, +)

defined by

λ false(a false) = λ_{a}

, for all

a \in A

, is a homomorphism of groups. By [, Lemma 2.4], the map

τ : (A, \circ) \to S (A)

defined by

τ false(a false) = τ_{a}

is an antihomomorphism of groups. Let

X_{1} = false{λ_{a} (x) 0pt: a \in A false}

.…”

Section: Indecomposable Solutionsmentioning

confidence: 99%

Skew left braces of nilpotent type

Cedó

Smoktunowicz

Vendramin

2018

Proc. London Math. Soc.

105

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“…The condition in the statement of Theorem 2.5 that both left ideals have to be normal in (A, +) cannot be replaced by normality in (A, •): turns G into a skew left brace such that G (2) is not a trivial skew left brace. In this example, Y is a left ideal and X and Y are normal in (G, •).…”

Section: Skew Left Braces Admitting a Factorizationmentioning

confidence: 99%

Factorizations of skew braces

et al. 2019

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We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.2010 Mathematics Subject Classification. Primary:16T25; Secondary: 81R50.

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Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks

Cited by 47 publications

References 45 publications

Skew braces of size pq

Skew braces of size pq

Skew left braces of nilpotent type

Factorizations of skew braces

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