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

AbstractWe show that if gΓ 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 U(gΓ) 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 Oq(G), 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 Oq(SL(n)), recovering the Witten algebra for n=2