Braided Hopf algebras obtained from coquasitriangular Hopf algebras

Let $(H, \sigma)$ be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define $H_\sigma$, a sub-Hopf algebra of $H^0$, the finite dual of $H$. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we associate to $H$ a braided Hopf algebra structure in the category of Yetter-Drinfeld modules over $H_\sigma^{\rm cop}$. Specializing to $H={\rm SL}_q(N)$, we obtain explicit formulas which endow ${\rm SL}_q(N)$ with a braided Hopf algebra structure within the category of left Yetter-Drinfeld modules over $U_q^{\rm ext}({\rm sl}_N)^{\rm cop}$.