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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Module categories of finite Hopf algebroids, and self-duality


Author: Peter Schauenburg
Journal: Trans. Amer. Math. Soc. 369 (2017), 1127-1146
MSC (2010): Primary 16T99, 18D10
DOI: https://doi.org/10.1090/tran6687
Published electronically: June 20, 2016
MathSciNet review: 3572267
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Abstract:

We characterize the module categories of suitably finite Hopf algebroids (more precisely, $\times _R$-bialgebras in the sense of Takeuchi (1977) that are Hopf and finite in the sense of a work by the author (2000)) as those $k$-linear abelian monoidal categories that are module categories of some algebra, and admit dual objects for “sufficiently many” of their objects.

Then we proceed to show that in many situations the Hopf algebroid can be chosen to be self-dual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them.


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Additional Information

Peter Schauenburg
Affiliation: Institut de Mathématiques de Bourgogne — UMR 5584 du CNRS, Université de Bourgogne, BP 47870, 21078 Dijon Cedex, France
MR Author ID: 346687
Email: peter.schauenburg@u-bourgogne.fr

Keywords: Finite tensor category, fusion category, Hopf algebroid, weak Hopf algebra, self-duality
Received by editor(s): August 13, 2014
Received by editor(s) in revised form: February 11, 2015
Published electronically: June 20, 2016
Additional Notes: This research was partially supported through a FABER Grant by the Conseil régional de Bourgogne
Article copyright: © Copyright 2016 American Mathematical Society