Comodule Algebras and 2-Cocycles over the (Braided) Drinfeld Double

TL;DR

This paper explores how comodule algebras over dually paired bialgebras can be extended to their Drinfeld double, introducing new constructions and cohomology maps in braided monoidal categories.

Contribution

It introduces a method to construct comodule algebras over the Drinfeld double from those over paired bialgebras and develops a cohomology map for 2-cocycles in this context.

Findings

Provides a crossed product construction for comodule algebras over the Drinfeld double.

Generalizes constructions to braided monoidal categories.

Establishes a map between second Hopf algebra cohomology spaces.

Abstract

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.