Search results for " 18D10"
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Module categories of finite Hopf algebroids, and self-duality
2017
International audience; We characterize the module categories of suitably finite Hopf algebroids (more precisely, $X_R$-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.
The dual and the double of a Hopf algebroid are Hopf algebroids
2017
Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as well. If in addition the coopposite $\times$-bialgebra of $H$ is $\times$-Hopf, then the coopposite of the Drinfeld double of $H$ is $\times$-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.