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On semisimple Hopf algebras of dimension $pq$

Authors: Shlomo Gelaki and Sara Westreich
Journal: Proc. Amer. Math. Soc. 128 (2000), 39-47
MSC (1991): Primary 16W30
Published electronically: June 24, 1999
Erratum: Proc. Amer. Math. Soc. 128 (2000), 2829-2831.
MathSciNet review: 1618670
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of the classification of semisimple Hopf algebras $A$ of dimension $pq$ where $p<q$ are two prime numbers. First we prove that the order of the group of grouplike elements of $A$ is not $q$, and that if it is $p$, then $q=1$ $(\operatorname{mod}\,p)$. We use it to prove that if $A$ and its dual Hopf algebra $A^*$ are of Frobenius type, then $A$ is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension $3p$, and a partial classification in dimensions $5p$ and $7p$.

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Additional Information

Shlomo Gelaki
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, University of Southern California, Los Angeles, California 90089

Sara Westreich
Affiliation: Interdisciplinary Department of the Social Science, Bar-Ilan University, Ramat-Gan, Israel

Received by editor(s): August 1, 1997
Received by editor(s) in revised form: March 17, 1998
Published electronically: June 24, 1999
Additional Notes: The second author’s research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities
Communicated by: Lance W. Small
Article copyright: © Copyright 1999 American Mathematical Society