  The vector fields of the quantum Lie algebra are described for the quantum
groups $GL_q(N), SL_q(N)$ and $SO_q(N)$ as pseudodifferential operators on the
linear quantum spaces covariant under the corresponding quantum group. Their
expressions are simple and compact. It is pointed out that these vector fields
satisfy certain characteristic polynomial identities. The real forms $SU_q(N)$
and $SO_q(N,R)$ are discussed in detail.
