Truman, PJ, Kohl, T, Koch, A and Underwood, R (2018) Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra. ISSN 0092-7872 (In Press)

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Abstract

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 λ(G) ≤ B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is HN = (K[N])λ(G). For a given such N we consider the Hopf-Galois structure arising from a subgroup P ⊳ N that is also normalized by λ(G). This subgroup gives rise to a Hopf sub-algebra HP ⊆ HN with fixed field F = KHP . By the work of Chase and Sweedler, this yields a Hopf-Galois structure
on the extension K/F where the action arises by base changing HP to F ⊗k HP which is an F-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on K/F relates to that on K/k. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those HP acting on K/F relates to that of the HN which act on K/k. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras.

Item Type: Article
Additional Information: This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Taylor and Francis at https://www.tandfonline.com/toc/lagb20/current. Please refer to any applicable terms of use of the publisher.
Uncontrolled Keywords: Hopf-Galois extension, Greither-Pareigis theory, regular subgroup, MSC 16T05,20B35,11S20
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Natural Sciences > School of Computing and Maths
Depositing User: Symplectic
Date Deposited: 20 Aug 2018 13:37
Last Modified: 20 Aug 2018 14:08
URI: http://eprints.keele.ac.uk/id/eprint/5228

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