The Dual and the Double of a Hopf Algebroid are Hopf Algebroids
Résumé
Let $H$ be a x-bialgebra in the sense of Takeuchi. We show that if $H$ is x-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is x-Hopf as well. If in addition the coopposite x-bialgebra of $H$ is x-Hopf, then the coopposite of the Drinfeld double of $H$ is x-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.