Module categories of finite Hopf algebroids, and self-duality

Schauenburg, Peter

doi:10.1090/tran6687

Module categories of finite Hopf algebroids, and self-duality
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Trans. Amer. Math. Soc. 369 (2017), 1127-1146 Request permission

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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