MODULE CATEGORIES OF FINITE HOPF ALGEBROIDS, AND SELF-DUALITY

We characterize the module categories of suitably finite Hopf algebroids (more precisely, xR-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.