Double-bosonization of braided groups and the construction of U-q(g)

We introduce a quasitriangular Hopf algebra or 'quantum group' U(B), the do uble-bosonization, associated to every braided group B in the category of H -modules over a quasitriangular Hopf algebra H, such that B appears as the 'positive root space', H as the Cartan subalgebra' and the dual braided gro up B* as the 'negative root space of U(B). The choice B = U-q(n(+)) recover s Lusztig's construction of U-q(g); other choices give more novel quantum g roups. As an application, our construction provides a canonical way of buil ding up quantum groups from smaller ones by repeatedly extending their posi tive and negative root spaces by linear braided groups; we explicitly const ruct U-q(sl(3)) from U-q(sl(2)) by this method, extending it by the quantum -braided plane. We provide a fundamental representation of U(B) in B. A pro jection from the quantum double, a theory of double biproducts and a Tannak a-Krein reconstruction point of view are also provided.