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

Abstract

We introduce a three-parameter family of two-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 define a new subgroup of BQ(k,H _4) and construct an exact sequence relating it to the Brauer group of Nichols 8-dimensional Hopf algebra with respect to the quasitriangular structure attached to the 2 x 2-matrix with 1 in the (1, 2)-entry and zero elsewhere

The Group of Bi-Galois Objects Over the Coordinate Algebra of the Frobenius–lusztig Kernel of Sl(2)

“…The Brauer-Picard group of C also coincides with the Brauer group of the Drinfeld center of C as defined in [29], in terms of Azumaya algebras, see [11]. If C = mod(H) is the category of finite-dimensional representations of a finite-dimensional Hopf algebra H, then the Brauer-Picard group is known as the strong Brauer of H [8], which is also notoriously known to be difficult to compute: see [9] for the latest developments (for the case of the Sweedler algebra) before the new technology from [13,11], and see [7,19] for recent computations.…”

Group Gradings and Actions of Pointed Hopf Algebras

“…Taft pairs and algebras. Now let H contain a group-like element g and a (1, g)-primitive element x such that (7) g n = 1, x n = 0, gx = ωxg, where ω is a primitive nth root of 1.…”

“…We also use heavily the well-known fact (see [14] and also [15,Chapter 6]) that any action of a Hopf algebra on a central simple algebra is inner. One motivation for this work is to try to extend the work of [19] and [7] on the Hopf Brauer group to Hopf algebras which are not quasitriangular".…”