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 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) × 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 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 H M 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 ( H M; 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). New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.

The Hopf Automorphism Group and the Quantum Brauer Group in Braided Monoidal Categories

“…This leads us to conclude that (ι ⊗ Π)(BM(C; B) ⋊ Aut(C; B)) ∼ = BM(C; B) ⋊ k * is a subgroup of BQ(k; B ⋊ H), Proposition 6.2. Using the results we obtained in [8], in Corollary 6.3 we prove that there is an embedding…”

“…In [8] we studied the Brauer group BM(E; B) of the braided monoidal category B E of left modules over a Hopf algebra B in E, where E is a braided monoidal category. We computed this group for the family of Hopf algebras studied in [15,Section 4] and defined as follows.…”

“…Though, the Brauer-Long group and its subgroups have extensively been studied [1,9,3,4,22,6]. Furthermore, in [8] we gave a method to compute BM(R; H) -the Brauer group of module algebras over a Hopf algebra H -where H is a quasitriangular Radford biproduct of a certain kind. It unifies several computations of this group performed in the last years for certain cases of H. A first approximation of the quantum Brauer group itself was attained in [24].…”

“…We apply the sequence (1.2) to the family of Radford biproduct Hopf algebras B ⋊ H that we studied in [8]. Here H is quasitriangular and B is a module over H. Then B can be made into a left H-comodule so that it becomes a Yetter-Drinfel'd module.…”

Braided bi-Galois theory II: The cocommutative case

“…Nevertheless, such cocommutative Hopf algebras do occur and their Hopf-Galois theory was recently studied and applied by Cuadra and Femić [4].…”