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

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U ( B ), the double-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 group B * as the ‘negative root space’ of U ( B ). The choice B = U q ( n + ) recovers Lusztig's construction of U q ( g ); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct 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 projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.