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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hopf algebroids and H-separable extensions


Author: Lars Kadison
Journal: Proc. Amer. Math. Soc. 131 (2003), 2993-3002
MSC (2000): Primary 13B02, 16H05, 16W30, 46L37, 81R15
Published electronically: December 30, 2002
MathSciNet review: 1993204
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Abstract: Since an H-separable extension $A \vert B$ is of depth two, we associate to it dual bialgebroids $S := \operatorname{End}{}_BA_B$ and $T := (A \otimes_B A)^B$over the centralizer $R$ as in Kadison-Szlachányi. We show that $S$ has an antipode $\tau$ and is a Hopf algebroid. $T^{\operatorname{op}}$ is also Hopf algebroid under the condition that the centralizer $R$ is an Azumaya algebra over the center $Z$ of $A$. For depth two extension $A \vert B$, we show that $\operatorname{End}{}_AA\otimes_B A \cong T \ltimes \operatorname{End}{}_BA$.


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Additional Information

Lars Kadison
Affiliation: Matematiska Institutionen, Göteborg University, S-412 96 Göteborg, Sweden
Address at time of publication: Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hamphsire 03824
Email: kadison@math.chalmers.se, kadison@math.unh.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06876-4
PII: S 0002-9939(02)06876-4
Received by editor(s): January 11, 2002
Received by editor(s) in revised form: April 22, 2002
Published electronically: December 30, 2002
Additional Notes: The author thanks Tomasz Brzezinski and U.W.S. for discussions and a hospitable visit to Swansea in the fall of 2001, as well as NORDAG in Bergen for partial support.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2002 American Mathematical Society