Vector fields on quantum groups

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left-invariant vector fields. We study the duality between vector fields and one-forms and generalize the construction to tensor fields. A Lie derivative along any (also non-left-invariant) vector field is proposed and a puzzling ambiguity in its definition discussed. These results hold for a generic Hopf algebra.