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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Frobenius extensions of subalgebras
of Hopf algebras


Authors: D. Fischman, S. Montgomery and H.-J. Schneider
Journal: Trans. Amer. Math. Soc. 349 (1997), 4857-4895
MSC (1991): Primary 16W30; Secondary 17B35, 17B37
MathSciNet review: 1401518
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider when extensions $S\subset R$ of subalgebras of a Hopf algebra are $\beta $-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra $H$, we show that when $S\subset R$ are Hopf algebras in the Yetter-Drinfeld category for $H$, the extension is $\beta $-Frobenius provided $R$ is finite over $S$ and the extension of biproducts $S\star H\subset R\star H$ is cleft.

More generally we give conditions for an extension to be $\beta $-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.

We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.


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Additional Information

D. Fischman
Affiliation: Department of Mathematics, California State University, San Bernardino, California 92407
Email: fischman@math.csusb.edu

S. Montgomery
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
Email: smontgom@math.usc.edu

H.-J. Schneider
Affiliation: Mathematisches Institut, Universität München, Theresienstrasse 39, D-80333 Munich, Germany
Email: hanssch@rz.mathematik.uni-muenchen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01814-X
PII: S 0002-9947(97)01814-X
Received by editor(s): December 10, 1995
Article copyright: © Copyright 1997 American Mathematical Society