Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras

We show that if is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first-order differential calculus over a co-quasitriangular Hopf algebra (A,r), then a certain extension of it is a braided Lie algebra in the category of A-comodules. This is used to show that the Woronowicz quantum universal enveloping algebra is a bialgebra in the braided category of A-comodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standard differential calculi. We also find a quantum Lie functor for co-quasitriangular Hopf algebras, which has properties analogous to the classical one. This functor gives trivial results on standard quantum groups , but reasonable ones on examples closer to the classical case, such as the cotriangular Jordanian deformations. In addition, we show that split braided Lie algebras define ‘generalized-Lie algebras’ in a different sense of deforming the adjoint representation. We construct these and their enveloping algebras for , recovering the Witten algebra for n=2.