 

Lindsay N. Childs
Biskew braces and Hopf Galois structures
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Published: 
July 15, 2019. 
Keywords: 
skew brace, Hopf Galois extension, Galois extension of fields, semidirect product, radical algebra. 
Subject: 
12F10, 16T05. 


Abstract
A skew brace G is a set with two group operations, one defining a (not necessarily abelian) "additive group" on G and the other a "circle group" on G, so that G with the two operations satisfies a relation analogous to distributivity. If G is a skew brace, then G yields a Hopf Galois structure of type equal to the additive group of G on any Galois extension of fields with Galois group isomorphic to the circle group of G. A skew brace G is a biskew brace if it is also a skew brace with the roles of the circle and additive group reversed. In that event, then G also corresponds to a Hopf Galois structure of type equal to the circle group on a Galois extension of fields with Galois group isomorphic to the additive group. Many nontrivial examples exist. One source is radical rings A with A^{3} = 0, where one of the groups is abelian and the other need not be. We find that the left braces of degree p^{3} classified by Bachiller are biskew braces if and only they are radical rings. A different source of biskew braces is semidirect products of arbitrary finite groups, which yield many examples where both groups are nonabelian, and a skew brace proof of a result of Crespo, Rio and Vela that if G is a semidirect product of two finite groups H and J, then any Galois extension of fields with Galois group G has a Hopf Galois structure of type equal to the direct product of H and J. 

Acknowledgements
N/A.


Author information
Lindsay N. Childs:
Department of Mathematics and Statistics
University at Albany
Albany, NY 12222, USA
lchilds@albany.edu

