[Paper Review] The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions

Stuart Jonathon Taylor, Paul J. Truman|arXiv (Cornell University)|Sep 25, 2018

Algebraic structures and combinatorial models17 references5 citations

TL;DR

This paper classifies all Hopf-Galois structures on Galois extensions with Galois group isomorphic to the quaternion group $Q_8$, determining which of the associated Hopf algebras are isomorphic as Hopf algebras and as $F$-algebras when $F$ has characteristic zero. It computes explicit Wedderburn-Artin decompositions and shows that the $F$-algebra structure of the Hopf algebras depends on the isomorphism class of the associated quaternion algebra, with three distinct cases possible depending on whether the algebra is split or division.

ABSTRACT

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 algebras. In the case that $ F $ has characteristic zero we also determine which of these Hopf algebras are isomorphic as $ F $-algebras and explicitly compute their Wedderburn-Artin decompositions.

Motivation & Objective

• To classify all Hopf-Galois structures admitted by a Galois extension $L/F$ with Galois group isomorphic to the quaternion group $Q_8$.
• To determine which of the Hopf algebras arising from these structures are isomorphic as $F$-Hopf algebras.
• To compute the Wedderburn-Artin decompositions of these Hopf algebras when $F$ has characteristic zero.
• To identify which of these Hopf algebras are isomorphic as $F$-algebras, particularly through their decomposition into matrix algebras and quaternion algebras.

Proposed method

• Using the Greither-Pareigis theorem, the authors classify Hopf-Galois structures via regular subgroups $N$ of $\operatorname{Perm}(G)$ normalized by the left regular representation $\lambda(G)$.
• For each such regular subgroup $N$, the corresponding Hopf algebra is constructed as $L[N]^G$, the fixed subalgebra under the $G$-action on $L[N]$.
• The authors compute explicit $F$-bases for the Hopf algebras $L[D_{s, ho}]^G$, $L[D_{t, ho}]^G$, and $L[D_{st, ho}]^G$, identifying a 4-dimensional $F$-subalgebra isomorphic to a quaternion algebra.
• They apply the Wedderburn-Artin theorem to decompose each Hopf algebra as a product of matrix algebras and quaternion algebras over $F$, using the Brauer group to compare isomorphism types.
• The isomorphism classes of the resulting quaternion algebras are analyzed via the Brauer group, particularly comparing classes $[-1,a]$, $[-1,b]$, and $[-1,ab]$.
• Examples are constructed over $\mathbb{Q}$ to demonstrate all three possible cases: all three quaternion algebras split, exactly one splits, or all three are non-isomorphic division algebras.

Experimental results

Research questions

• RQ1 How many distinct Hopf-Galois structures exist on a Galois extension $L/F$ with Galois group $Q_8$?
• RQ2 Which of the Hopf algebras arising from these structures are isomorphic as $F$-Hopf algebras?
• RQ3 What is the Wedderburn-Artin decomposition of each Hopf algebra $L[N]^G$ when $F$ has characteristic zero?
• RQ4 When are the Hopf algebras isomorphic as $F$-algebras, and how does this depend on the isomorphism class of the associated quaternion algebra?
• RQ5 What are the possible isomorphism types of the $F$-algebra structures of these Hopf algebras, and how do they vary with the field $F$?

Key findings

• There are exactly five Hopf-Galois structures on a $Q_8$-extension $L/F$, corresponding to five regular subgroups $N$ of $\operatorname{Perm}(G)$ normalized by $\lambda(G)$.
• The classical Hopf-Galois structure corresponds to $N = \rho(G)$, giving the group algebra $F[G]$, while the canonical non-classical structure arises from $N = \lambda(G)$.
• The remaining three structures correspond to subgroups $D_{s,\rho}, D_{t,\rho}, D_{st,\rho}$, each giving a Hopf algebra $L[N]^G$ isomorphic to $F^4 \times (-1,a)_F$, $F^4 \times (-1,b)_F$, and $F^4 \times (-1,ab)_F$ respectively.
• When $F$ has characteristic zero, the $F$-algebra structure of each Hopf algebra is determined by its Wedderburn-Artin decomposition, with the 4-dimensional part being $F^4$ and the remaining part a quaternion algebra.
• The Hopf algebras are isomorphic as $F$-algebras if and only if their associated quaternion algebras are isomorphic, which occurs precisely when $(-1,x)_F \cong (-1,y)_F$ in the Brauer group.
• Three distinct isomorphism types of $F$-algebras can occur: all three quaternion algebras split (isomorphic to $M_2(F)$), exactly one splits, or all three are non-isomorphic division algebras, as demonstrated by examples over $\mathbb{Q}$.