A sequence to compute the Brauer group of certain quasi-triangular Hopf algebras

Abstract

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 (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 BM (C; B)contains a subgroup isomorphic to Br(C) × H 2 (C; B, I), where H 2(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 BM(K, B × H, R) 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 2 (HM; B, K) is a subgroup of BM(K, B × H, R), confirming the suspicion that a certain cohomology group of B × 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.