Fixedpoint free endomorphisms and Hopf Galois structures
Author:
Lindsay N. Childs
Journal:
Proc. Amer. Math. Soc. 141 (2013), 12551265
MSC (2000):
Primary 12F10
Published electronically:
September 21, 2012
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Abstract: Let be a Galois extension of fields with finite Galois group . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on and regular subgroups of normalized by , and Byott translated the problem into that of finding equivalence classes of embeddings of in the holomorph of groups of the same cardinality as . In 2007 we showed, using Byott's translation, that fixed point free endomorphisms of yield Hopf Galois structures on . Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the GreitherPareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are ``twistings'' of the Hopf Galois structure by , the Hopf algebra that arises from the left regular representation of in . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.
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Additional Information
Lindsay N. Childs
Affiliation:
Department of Mathematics and Statistics, University at Albany, Albany, New York 12222
Email:
childs@math.albany.edu
DOI:
http://dx.doi.org/10.1090/S000299392012114182
PII:
S 00029939(2012)114182
Received by editor(s):
November 18, 2009
Received by editor(s) in revised form:
November 14, 2010, July 21, 2011, and August 25, 2011
Published electronically:
September 21, 2012
Additional Notes:
The author thanks the mathematics department at Virginia Commonwealth University for its hospitality while this research was conducted and the referee for numerous helpful suggestions.
Communicated by:
Ted Chinburg
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
