DOUBLE-BOSONISATION OF BRAIDED GROUPS AND THE CONSTRUCTION OF Uq(g)

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonisation, associated to every braided group B in the category of Hmodules 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 = f recovers Lusztig’s construction of Uq(g), where f is Lusztig’s algebra associated to a Cartan datum; 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 Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane Aq . 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.