  For q generic or a primitive l-th root of unity, q-Witt algebras are
described by means of q-divided power algebras. The structure of the universal
q-central extension of the q-Witt algebra, the q-Virasoro algebra, is also
determined. q-Lie algebras are investigated and the q-PBW theorem for the
universal enveloping algebras of q-Lie algebras is proved. A realization of a
class of representations of the q-Witt algebras is given. Based on it, the
q-holomorph structure for the q-Witt algebras is constructed, which interprets
the realization in the context of representation theory.
