Hopf Forms and Hopf-Galois Theory

2020/10/10 by Timothy Kohl, Kohl, Timothy, Robert G. Underwood +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2010.05067

Abstract

Let K be a finite field extension of \Q and let N be a finite group with automorphism group F=\Aut(N). R. Haggenmüller and B. Pareigis have shown that there is a bijection Θ: \mathcal Gal(K,F)→ \mathcal Hopf(K[N]) from the collection of F-Galois extensions of K to the collection of Hopf forms of the group ring K[N]. For N=C_n, n≥ 1, C_p^m, p prime, m≥ 1, and N=D₃,D₄,Q₈, we show that \Q[N] admits an absolutely semisimple Hopf form H and find L for which Θ(L)=H. Moreover, if H is the Hopf algebra given by a Hopf-Galois structure on a Galois extension E/K, we show how to construct the preimage of H under Θ assuming certain conditions.

Related

• Isomorphism problems for Hopf-Galois structures on separable field\n extensions
• Nilpotent and abelian Hopf-Galois structures on field extensions
• Automatic realization of Hopf Galois structures
• The Structure of Hopf Algebras Acting on Dihedral Extensions
• Minimal Hopf-Galois Structures on Separable Field Extensions