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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
Email: peter.schauenburg@u-bourgogne.fr

DOI: https://doi.org/10.1090/tran6687
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