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

Koch, Alan; Truman, Paul J.

Abstract

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 structures admitted by a Galois extension of fields with Galois group isomorphic to G. 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 pq for p > q prime numbers.

Citation

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

• Journal Article Type: Article
• Acceptance Date: Jan 18, 2022
• Publication Date: Apr 15, 2022
• Publicly Available Date: Jan 12, 2022
• Journal: Journal of Algebra and Its Applications
• Print ISSN: 0219-4988
• Publisher: World Scientific Publishing
• Volume: 22
• Issue: 05
• Article Number: ARTN 2350118
• DOI: https://doi.org/10.1142/S0219498823501189
• Public URL: https://keele-repository.worktribe.com/output/422019
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S0219498823501189?journalCode=jaa