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 C, and under certain assumptions on the braiding (fulfilled if C is symmetric), we construct a sequence for the Brauer group BM(C;B) of B-module algebras, generalizing Beattie’s one. It allows one to prove that BM(C;B) ∼= Br(C) × Gal(C;B), where Br(C) is the Brauer group of C and Gal(C;B) the group of B-Galois objects. We also show that BM(C;B) contains a subgroup isomorphic to Br(C)×H2(C;B, I), where H2(C;B, I) is the second Sweedler cohomology group of B with values in the unit object I of C. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure R is contained in H and B is a Hopf algebra in the category HM of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that BM(K,H,R)×H(HM;B,K) is a subgroup of the Brauer group BM(K,B×H,R), confirming the suspicion that a certain cohomology group of B × H (second lazy cohomology group was conjectured) embeds into BM(K,B × H,R).