(untitled) (Communications in Mathematical Research, 2021)
Let _HL^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 _HL^B(F (π)) where the opposite group 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 _HL^B for a quasitriangular Turaev group-coalgebra H and a coquasitriangular Turaev group-algebra B.