From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures

Caranti, A.; Stefanello, L.

doi:10.1016/j.jalgebra.2021.07.029

Cited by 14 publications

(7 citation statements)

References 16 publications

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“…(See for instance [CCDC20,CS21b] for this context.) A Rota-Baxter operator on the group (G, •) is a function B : G → G that satisfies the identity…”

Section: Introductionmentioning

confidence: 99%

Skew braces from Rota-Baxter operators: A cohomological characterisation and some examples

Caranti,

Stefanello

2022

Preprint

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Rota-Baxter operators for groups were recently introduced by L. Guo, H. Lang, and Y. Sheng.V. G. Bardakov and V. Gubarev showed that with each Rota-Baxter operator one can associate a skew brace. Skew braces on a group G can be characterised in terms of certain gamma functions from G to its automorphism group Aut(G), that are defined by a functional equation. For the skew braces obtained from a Rota-Baxter operator the corresponding gamma functions take values in the inner automorphism group Inn(G) of G.In this paper, we give a characterisation of the gamma functions on a group G, with values in Inn(G), that come from a Rota-Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group.Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota-Baxter operator.For gamma functions that can be obtained from a Rota-Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits.

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“…(See for instance [CCDC20,CS21b] for this context.) A Rota-Baxter operator on the group (G, •) is a function B : G → G that satisfies the identity…”

Section: Introductionmentioning

confidence: 99%

Skew braces from Rota-Baxter operators: A cohomological characterisation and some examples

Caranti,

Stefanello

2022

Preprint

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“…Childs in 2019 ( [15]). Alan Koch [22,23] constructed symmetric brace systems on a given group G using abelian endomorphisms of G. This construction was generalized by A. Caranti and L. Stefanello [9,10]. We show that all theses symmetric skew braces are λ-homomorphic.…”

Section: A Unification For Known Symmetric Brace Systemsmentioning

confidence: 97%

Symmetric skew braces and brace systems

Bardakov¹,

Нещадим²,

Yadav³

2022

Preprint

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For a skew left brace (G, •, •), the map λ : (G, •) → Aut (G, •), a → λa, where λa(b) = a −1 • (a • b) for all a, b ∈ G, is a group homomorphism. Then λ can also be viewed as a map from (G, •) to Aut (G, •), which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : (G, •) → Aut (G, •) is an antihomomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitute a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite sets.

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“…Define a new binary operation on N by η • π = a −1 (a(η)a(π)) for all η, π ∈ N, where the multiplication inside the brackets takes place in G. Then (N, •) is a group isomorphic to G and, since N is a G-stable subgroup of Perm(G), the brace relation (2.1) is satisfied. Therefore B N = (N, •, •) is a brace with (N, •) ∼ = G. In [11] this construction is referred to as transport of structure.…”

Section: Skew Left Bracesmentioning

confidence: 99%

Skew left braces and isomorphism problems for Hopf–Galois structures on Galois extensions

Koch

Truman

2022

J. Algebra Appl.

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Given a finite group [Formula: see text], we study certain regular subgroups of the group of permutations of [Formula: see text], which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to [Formula: see text] and Hopf–Galois structures admitted by a Galois extension of fields with Galois group isomorphic to [Formula: see text]. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf–Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf–Galois structure is determined by the corresponding skew left brace. We investigate these questions in the context of a variety of existing constructions in the literature. As an application of our results we classify the isomorphically distinct Hopf algebras that give Hopf–Galois structures on a Galois extension of degree [Formula: see text] for [Formula: see text] prime numbers.

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From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures

Cited by 14 publications

References 16 publications

Skew braces from Rota-Baxter operators: A cohomological characterisation and some examples

Skew braces from Rota-Baxter operators: A cohomological characterisation and some examples

Symmetric skew braces and brace systems

Skew left braces and isomorphism problems for Hopf–Galois structures on Galois extensions

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