Some isomorphisms for the Brauer groups of a Hopf algebra

Using equivalences of categories we provide isomorphisms between the Brauer groups of different Hopf algebras. As an example, we show that when k is a field of characteristic different from 2 the Brauer groups BC(k,H_4,r_t) for every dual quasitriangular structure r_t on Sweedler's Hopf algebra H_4 are all isomorphic to the direct sum of (k,+) and the Brauer-Wall group of k. We provide an isomorphism between the Brauer group of a Hopf algebra H and the Brauer group of the dual Hopf algebra H^* generalizing a result of Tilborghs. Finally we relate the Brauer groups of H and of its opposite and co-opposite Hopf algebras.