Double Quantum Groups,

The construction of the Drinfeld double D(H) of a finite dimensional Hopf algebra H was one of the first examples of a quasitriangular Hopf algebra whose category of modules D(H) is braided. The braided category of Yetter–Drinfeld modules HH is the analogue for infinite dimensional Hopf algebras. It uses a strong dependence between the H-module and the H-comodule structures. We generalize this construction to the category CA(ψ) of entwined modules, that is, A-modules and C-comodules over Hopf algebras A and C where the structures are only related by an entwining map ψ: C A → A C. We show that CA(ψ) is braided iff there is an r-map r: C C → A A satisfying suitable axioms that generalize the axioms of an R-matrix. For finite dimensional C there is a quasitriangular Hopf algebra structure on Hom(C, A), the quantum group double, generalizing the construction of the Drinfeld double.