Quantum doubles in monoidal categories

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map tau is a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H) = H x(tau) H-*cop in C. In this paper, we first show that D(H) is' a quasitriangular Hopf algebra in C. Next, let A be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H, H) and B(H, D(H)) tin a rigid braided monoidal category) by Majid's method associated to the ordinary Hopf algebra maps id : H --> H and i(H) : H --> D(H), where D(H) is the Drinfel'd quantum double. We show that D(B(H, H)) and B(H, D(H)) are isomorphic Hopf algebras in the braided monoidal category.