Double-bosonization of braided groups and the construction of Uq(g)
We introduce a quasitriangular Hopf algebra or âquantum groupâ 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 âpositive root spaceâ, H as the âCartan subalgebraâ and the dual braided group B* as the ânegative root spaceâ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(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 Uq(sl3) from Uq(sl2) 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 TannakaâKrein reconstruction point of view are also provided.