2008/09/14 by Juan Cuadra, Cuadra, Juan, Bojana Femić +1
Mathematics · Physics and Astronomy · #16W30 #18D10 #18D35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0809.2517
openalex publication_date 2008/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) × \Hc(\C;B,I), where \Hc(\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) × \Hc(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.