Fixed-point free endomorphisms and Hopf Galois structures

Author:
Lindsay N. Childs

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1255-1265

MSC (2000):
Primary 12F10

Published electronically:
September 21, 2012

MathSciNet review:
3008873

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Abstract | References | Similar Articles | Additional Information

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 Greither-Pareigis 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:
https://doi.org/10.1090/S0002-9939-2012-11418-2

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.