VECTOR-FIELDS ON QUANTUM GROUPS

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