A Note on Generalized Long Modules

Abstract

Let $_H\mathcal{L}^B$ be the category of generalized Long modules, that is, $H$-modules and $B$-comodules over Hopf algebras $B$ and $H$. We describe a new Turaev braided group category over generalized Long module $_H\mathcal{L}^B(\mathcal{F} (π))$ where the opposite group $\mathcal{F} (π)$ of the semidirect product of the opposite group $π^{op}$ of a group $π$ by $π$. As an application, we show that this is a Turaev braided group-category $_H\mathcal{L}^B$ for a quasitriangular Turaev group-coalgebra $H$ and a coquasitriangular Turaev group-algebra $B$.