On the subgroup structure of the full Brauer group of Sweedler Hopf algebra

Abstract

We introduce a family of three parameters 2-dimensional algebras representing elements in the Brauer group BQ(k,H_4) of Sweedler Hopf algebra H_4 over a field k. They allow us to describe the mutual intersection of the subgroups arising from a quasitriangular or coquasitriangular structure. We also introduce a new subgroup of BQ(k,H_4) whose elements are represented by algebras for which the two natural Z_2-gradings coincide. We construct an exact sequence relating this subgroup to the Brauer group of Nichols 8-dimensional Hopf algebra E(2) with respect to the quasitriangular structure attached to the 2x2-matrix N with 1 in the (1,2)-entry and zero elsewhere.