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When weak Hopf algebras are Frobenius

Authors: Miodrag Cristian Iovanov and Lars Kadison
Journal: Proc. Amer. Math. Soc. 138 (2010), 837-845
MSC (2000): Primary 18D10; Secondary 16W30, 16S50, 16D90, 16L30
Published electronically: October 22, 2009
MathSciNet review: 2566549
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Abstract: We investigate when a weak Hopf algebra $ H$ is Frobenius. We show this is not always true, but it is true if the semisimple base algebra $ A$ has all its matrix blocks of the same dimension. However, if $ A$ is a semisimple algebra not having this property, there is a weak Hopf algebra $ H$ with base $ A$ which is not Frobenius (and consequently, it is not Frobenius ``over'' $ A$ either). Moreover, we give a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of a weak Hopf algebra.

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Additional Information

Miodrag Cristian Iovanov
Affiliation: Faculty of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109, Bucharest, Romania – and – State University of New York, Buffalo, 244 Mathematics Building, Buffalo, New York 14260-2900

Lars Kadison
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 S. 33rd Street, Philadelphia, Pennsylvania 19104
Address at time of publication: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, #0112, La Jolla, California 92093

Keywords: Weak Hopf algebra, quasi-Hopf algebra, Frobenius algebra, quasi-Frobenius algebra, tensor category, Tannakian reconstruction
Received by editor(s): November 20, 2008
Received by editor(s) in revised form: July 15, 2009
Published electronically: October 22, 2009
Additional Notes: The first author was partially supported by contract no. 24/28.09.07 with UEFISCU “Groups, quantum groups, corings and representation theory” of CNCIS, PN II (ID_1002).
Communicated by: Martin Lorenz
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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