Recent Research:  Hopf Galois structures

A  major direction of my research since 1989 has been the study of Hopf Galois extensions of fields.  
This area of mathematics arose as a natural generalization of the classical concept of a Galois extension 
of fields, first studied by Galois (1831).  S. Chase and M. Sweedler developed a Hopf algebra 
generalization of the concept in 1969, a generalization that, in the language of Grothendieck's 
version of algebraic geometry,  translates into the concept of a principal homogeneous space for 
a finite group scheme.  The idea is that for a classical Galois extension L/K with Galois group G, 
L becomes a KG-module, and KG is a K-Hopf algebra.  

To generalize, suppose H is a K-Hopf algebra that acts on L as an H-module algebra.  Then L/K is 
an H-Galois extension if certain properties of a classical Galois extension hold, for example, 
that  the obvious map from L \times H to the ring  End_K(L) of K-module endomorphisms of L 
is bijective. (For L = KG, this is a consequence of "linear independence of characters" in Galois 
theory.)   

The concept lay mostly fallow until the publication of a 1987 paper of Greither and Pareigis.  
For L/K a finite separable extension of fields with normal closure E, with Gal(E/K) = G, Gal(E/L)= G', 
they showed that there is a bijection between Hopf Galois structures on L/K and regular subgroups 
of Perm(G/G') mormalized by the left regular representation of G in Perm(G/G').  This result 
translated the problem of finding Hopf Galois structures on L/K into a problem in permutation groups.

In 1989 I published [31], in which I observed that the left regular representation of G normalizes 
a regular subgroup N of Perm(G/G') if and only if there is a subgroup of the holomorph Hol(N) of N 
that is isomorphic to G.   This result, already implicit in Greither-Pareigis, translated the study 
of Hopf Galois structures on a Galois extension with Galois group to a study of regular subgroups 
of the holomorph of a finite group.  

An immediate consequence in [31] is that if L/K is a separable extension of prime degree, 
then L/K has a Hopf Galois structure if and only if the Galois group of E/K is solvable.

The holomorph result was turned by Byott (1996) into a way to count the number of Hopf Galois 
structures on L/K of type N  (that is, where E \otimes H is isomorphic to the group ring EN) by 
counting the number of equivalence classes of regular embeddings of G into  Hol(N).  
One consequence of his result was that if E=L, that is, if L/K is a classical Galois extension of 
fields, then L/K has a non-classical Hopf Galois structure unless the degree n = [L:K} is coprime 
to phi(n) (Euler's phi-function).

Beginning in the late 1990's I started exploring, often with students, ways of counting Hopf 
Galois structures on finite Galois extensions of fields, beginning with Dan Pragel, who wrote an 
undergraduate thesis counting Hopf Galois structures on Galois extensions with dihedral Galois groups, 
and Scott Carnahan, who did the same for Galois extensions with Galois group a non-abelian simple 
group or a symmetric group [42].  Tim Kohl (1997) and Steve Featherstonhaugh (2003) wrote doctoral theses 
studying Hopf Galois structures on Galois extensions with Galois group cyclic of prime power degree, 
and an elementary abelian p-group, respectively, and Jesse Corradino obtained results 
on Hopf Galois structures on Galois extensions where the Galois group is a semidirect product of 
cyclic groups [50].  Papers [45, 46, 49, 53 and 55] also count Hopf Galois structures.  Others 
publishing on the topic include Tim Kohl, N. Byott and his students, and T. Crespo, A. Rio 
and M Vela.

When Steve Featherstonhaugh's thesis was written up and submitted for publication, it came 
to the attention of Andrea Caranti, who with Della Volta and Sala had obtained (in 2006) 
a bijection between regular abelian subgroups of the holomorph of an abelian group N and 
abelian nilpotent ring structures on the additive group N.  In [51] he contributed a proof of the 
main result in Featherstonhaugh's thesis that utilized that translation.  

Meanwhile, motivated by a question of Drin'feld (1990) on solutions of the Yang-Baxter equation, W. 
Rump (2007) defined a left brace.  A left brace is a generalization of a finite nilpotent ring.   
In turn, the concept of left brace has recently (2016) been generalized to the concept 
of a skew brace, which is a set with an additive group structure A and a multiplicative group 
structure G where the two operations satisfy a certain compatibility condition.  A skew brace is 
a brace if the additive group is abelian.  It turns out that given a finite group A which 
is the additive group of a skew brace with multiplicative group G, then G may 
be viewed as a regular subgroup of the holomorph Hol(A) contained in Perm(A).  The Caranti-della 
Volta-Sala bijection between abelian regular subgroups G of Hol(N) for N a finite abelian group and 
commutative nilpotent ring structures on N generalizes to the skew brace setting, where neither N 
nor G need be abelian.  So all of the work on describing skew braces and their special cases 
translates into results on Hopf Galois structures.  An appendix by Byott and Vendramin in a 2017  
paper by Smoktunowicz and Vendramin, arxiv:1705.06958 describes this translation in detail.

The main lemma in [51] that yields Featherstonehaugh's Theorem has been 
generalized by D. Bachiller [2016] to skew left braces.

I've used the Caranti-Della Volta-Sala result: in [57] to estimate the number of Hopf Galois 
structures of elementary abelian type on an elementary abelian Hopf Galois extension of degree p^n 
for n large, in [58] to study Hopf Galois structures on such extensions where the corresponding 
nilpotent algebra structure A satisfies A^3 = 0, in [59] to show that for an abelian Hopf Galois 
structure H on an abelian Galois extension of p-power order, the image of the Galois correspondence 
from Hopf subalgebras to subfields corresponds to the ideals of the corresponding nilpotent algebra 
structure, and most recently, in [60] with Cornelius Greither, to quantify the failure of 
surjectivity of the Galois correspondence for Hopf Galois structures on a Galois extension L/K  
with Galois group an elementary abelian p-group. (As a referee observed, 
"the surjectivity of the Galois correspondence fails spectacularly".)

Most recently in [61] I extended the main result of [59] from Hopf Galois extensions corresponding 
to commutative nilpotent algebra structures on finite  elementary abelian p-groups to the 
very general setting of Hopf Galois extensions that correspond to a left skew brace.  The image of 
the Galois correspondence for the Hopf Galois structure of type A is in bijective correspondence 
with certain subgroups of the circle group G of the left skew brace corresponging to the Hopf Galois
structure. I am currently exploring some of the implications of this result for various known 
constructions of Hopf Galois structures.

April 1, 2018