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All Outputs (19)

Skew bracoids containing a skew brace (2025)
Presentation / Conference Contribution
Colazzo, I., Koch, A., Martin-Lyons, I., & Truman, P. J. (2025, April). Skew bracoids containing a skew brace. Presented at 33rd GISRUK Conference 2025, University of Bristol, England, UK

Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A consequence of this correspondence is that skew bracoids in this fami... Read More about Skew bracoids containing a skew brace.

Skew Bracoids (2023)
Journal Article
Truman, P., & Martin-Lyons, I. (2024). Skew Bracoids. Journal of Algebra, 638, 751-787. https://doi.org/10.1016/j.jalgebra.2023.10.005

Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we term a \textit{skew bracoid}. Our construction involves two groups int... Read More about Skew Bracoids.

On ρ-conjugate Hopf–Galois structures (2023)
Journal Article
Truman. (2023). On ρ-conjugate Hopf–Galois structures. Proceedings of the Edinburgh Mathematical Society, 66(1), 288-304. https://doi.org/10.1017/S0013091523000184

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... Read More about On ρ-conjugate Hopf–Galois structures.

Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions (2022)
Journal Article
Koch, A., & Truman, P. J. (2022). Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions. Journal of Algebra and Its Applications, 22(05), Article ARTN 2350118. https://doi.org/10.1142/S0219498823501189

Given a finite group G, we study certain regular subgroups of the group of permutations of G, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to G and Hopf-Galois str... Read More about Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions.

Hopf Algebras and Galois Module Theory (2021)
Book
Childs, L., Greither, C., Keating, K., Koch, A., Kohl, T., Truman, P., & Underwood, R. (2021). Hopf Algebras and Galois Module Theory. American Mathematical Society. https://doi.org/10.1090/surv/260

Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be vie... Read More about Hopf Algebras and Galois Module Theory.

Abelian fixed point free endomorphisms and the Yang-Baxter equation (2020)
Journal Article
Koch, A., Stordy, L., & Truman, P. J. (2020). Abelian fixed point free endomorphisms and the Yang-Baxter equation

We obtain a simple family of solutions to the set-theoretic Yang-Baxter equation, one which depends only on considering special endomorphisms of a finite group. We show how such an endomorphism gives rise to two non-degenerate solutions to the Yang-B... Read More about Abelian fixed point free endomorphisms and the Yang-Baxter equation.

Opposite Skew Left Braces and Applications (2020)
Journal Article
Koch, A., & Truman, P. (2020). Opposite Skew Left Braces and Applications. Journal of Algebra, 218-235. https://doi.org/10.1016/j.jalgebra.2019.10.033

Given a skew left brace B, we introduce the notion of an \opposite" skew left brace B0, 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 t... Read More about Opposite Skew Left Braces and Applications.

Hopf-Galois module structure of tamely ramified radical extensions of prime degree (2019)
Journal Article
Truman, P. J. (2020). Hopf-Galois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, 224(5), Article 106231. https://doi.org/10.1016/j.jpaa.2019.106231

Let K be a number field and let L/K be a tamely ramified radical extension of prime degree p. If K contains a primitive p th root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G = Gal(L/K)) gives the unique... Read More about Hopf-Galois module structure of tamely ramified radical extensions of prime degree.

Isomorphism problems for Hopf-Galois structures on separable field extensions (2019)
Journal Article
Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Isomorphism problems for Hopf-Galois structures on separable field extensions. Journal of Pure and Applied Algebra, 2230-2245. https://doi.org/10.1016/j.jpaa.2018.07.014

Let L=K be a finite separable extension of fields whose Galois closure E=K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L=K has the form E[N]G for some group N of order [L... Read More about Isomorphism problems for Hopf-Galois structures on separable field extensions.

The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions (2019)
Journal Article
Truman, P., & Taylor, S. (2019). The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions

Let L/F be a Galois extension of fields with Galois group isomorphic to the quaternion group of order 8. We describe all of the Hopf-Galois structures admitted by L/F, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebr... Read More about The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions.

The Structure of Hopf Algebras Acting on Dihedral Extensions (2019)
Book Chapter
Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). The Structure of Hopf Algebras Acting on Dihedral Extensions. In Advances in Algebra. SRAC 2017 (201-218)

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group Dp, p=3 prime and give explicit description... Read More about The Structure of Hopf Algebras Acting on Dihedral Extensions.

Normality and Short Exact Sequences of Hopf-Galois Structures (2019)
Journal Article
Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra, 2086-2101. https://doi.org/10.1080/00927872.2018.1529237

Every Hopf-Galois structure on a finite Galois extension K/k where G = Gal(K/k) corresponds uniquely to a regular subgroup N = B = Perm(G), normalized by ?(G) = B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra whic... Read More about Normality and Short Exact Sequences of Hopf-Galois Structures.

Commutative Hopf-Galois module structure of tame extensions (2018)
Journal Article
Truman. (2018). Commutative Hopf-Galois module structure of tame extensions. Journal of Algebra, 389-408. https://doi.org/10.1016/j.jalgebra.2018.01.047

We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of p-adic fields or number fields which is H-Galois for a commutative Hopf algebra H. Firstly, we show that if L/K is a tam... Read More about Commutative Hopf-Galois module structure of tame extensions.

Commuting Hopf-Galois structures on a separable extension (2018)
Journal Article
Truman. (2018). Commuting Hopf-Galois structures on a separable extension. Communications in Algebra, 1420-1427. https://doi.org/10.1080/00927872.2017.1346107

Let L/K be a finite separable extension of local or global fields in any characteristic, let H-1, H-2 be two Hopf algebras giving Hopf-Galois structures on the extension, and suppose that the actions of H-1, H-2 on L commute. We show that a fractiona... Read More about Commuting Hopf-Galois structures on a separable extension.

Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions (2016)
Journal Article
Truman, P. (2016). Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions. Communications in Algebra, 1119 - 1130. https://doi.org/10.1080/00927872.2014.999930

Let L/K be a finite Galois extension of local or global fields in any characteristic with nonabelian Galois group G, and let be an ambiguous ideal of L. We show that is free over its associated order in K[G] if and only if it is free over its associa... Read More about Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions.

Hopf-Galois module structure of tame Cp×Cp extensions (2016)
Journal Article
Truman, P. (2016). Hopf-Galois module structure of tame Cp×Cp extensions. https://doi.org/10.5802/jtnb.953

Let $ p $ be an odd prime number, $ K $ a number field containing a primitive $ p^{th} $ root of unity, and $ L $ a Galois extension of $ K $ with Galois group isomorphic to $ C_{p} \times C_{p} $. We study in detail the local and global structure of... Read More about Hopf-Galois module structure of tame Cp×Cp extensions.

Hopf-Galois module structure of tame biquadratic extensions (2016)
Journal Article
Truman. (2016). Hopf-Galois module structure of tame biquadratic extensions. https://doi.org/10.5802/jtnb.953

Let $ p $ be an odd prime number, $ K $ a number field containing a primitive $ p^{th} $ root of unity, and $ L $ a Galois extension of $ K $ with Galois group isomorphic to $ C_{p} \times C_{p} $. We study in detail the local and global structure of... Read More about Hopf-Galois module structure of tame biquadratic extensions.

Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois structures (2011)
Journal Article
Truman. (2011). Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois structures

We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of p-adic fields and number fields which are at most tamely ramified. We show that if L/K is an unramified extension of p-adic fields which is H-... Read More about Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois structures.