Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions

Truman, Paul J.

doi:10.1080/00927872.2014.999930

Cited by 10 publications

(7 citation statements)

References 8 publications

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“…Taking N = ρ(G) gives the usual Galois action. Taking N = λ(G) produces what is called the canonical nonclassical Hopf Galois structure in [20]. This structure will be of particular importance here and we shall denote its Hopf algebra by H λ .…”

Section: Introductionmentioning

confidence: 99%

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

Koch

2021

Proc. Amer. Math. Soc. Ser. B

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Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

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Section: Introductionmentioning

confidence: 99%

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

Koch

2021

Proc. Amer. Math. Soc. Ser. B

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“…The most famous results in this area are due to Byott [3], who exhibits Galois extensions L/K of p-adic fields for which the valuation ring O L is not free over A K[G] (O L ) but is free over A H (O L ) for some other Hopf algebra H giving a Hopf-Galois structure on the extension. On the other hand, in [23] and [24], we show that a fractional ideal B of L is free over its associated order in a particular Hopf-Galois structure if and only if it is free over its associated order in the opposite Hopf-Galois structure. The main result of this section is in a similar vein.…”

Section: Integral Module Structurementioning

confidence: 85%

On ρ-conjugate Hopf–Galois structures

Truman¹

2023

Proceedings of the Edinburgh Mathematical Society

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The Hopf–Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group G correspond bijectively with certain subgroups of $ \mathrm{Perm}(G) $ . We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf–Galois structures, which we term ρ-conjugation. We study properties of this construction, with particular emphasis on the Hopf–Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct ρ-conjugates of a given Hopf–Galois structure is determined by the corresponding skew left brace, and that if $ H, H^{\prime} $ are Hopf algebras giving ρ-conjugate Hopf–Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ \mathfrak{B} $ of L is free over its associated order in H if and only if it is free over its associated order in Hʹ. We exhibit a variety of examples arising from interactions with existing constructions in the literature.

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“…The structure given by H λ(G) is called the canonical nonclassical Hopf-Galois structure in [Tru16].…”

Section: Hopf-galois Structuresmentioning

confidence: 99%

Opposite skew left braces and applications

Koch

Truman

2020

Journal of Algebra

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Given a skew left brace B, we introduce the notion of an "opposite" skew left brace B ′ , which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by B ′ is the inverse to the solution given by B; this allows us to identify the group-like elements in the Hopf algebra providing the Hopf-Galois structure using only these solutions. We also show how left ideals of B ′ correspond to the realizable intermediate fields of a certain Hopf-Galois extension of a Galois extension.

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Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions

Cited by 10 publications

References 8 publications

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

On ρ-conjugate Hopf–Galois structures

Opposite skew left braces and applications

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