A Sequence to Compute the Brauer Group of Certain Quasi-Triangular Hopf Algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category , and under certain assumptions on the braiding (fulfilled if is symmetric), we construct a sequence for the Brauer group of B-module algebras, generalizing Beattie's one. It allows one to prove that , where is the Brauer group of and the group of B-Galois objects. We also show that contains a subgroup isomorphic to where is the second Sweedler cohomology group of B with values in the unit object I of . These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct B x H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure is contained in H and B is a Hopf algebra in the category of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that is a subgroup of , confirming the suspicion that a certain cohomology group of B x H (second lazy cohomology group was conjectured) embeds into it. New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.