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

TL;DR

This paper explores the subgroup structure of the Brauer group associated with the Sweedler Hopf algebra, introducing new algebraic families and exact sequences to understand their intersections and relations.

Contribution

It introduces a parametric family of algebras in the Brauer group of the Sweedler Hopf algebra and constructs an exact sequence linking a new subgroup to the Brauer group of a Nichols Hopf algebra.

Findings

Defined a family of algebras representing elements in BQ(k,H_4)

Described intersections of subgroups from quasitriangular structures

Established an exact sequence relating subgroups to Nichols 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.