Realization of Vector fields for Quantum Groups as Pseudodifferential Operators on Quantum Spaces

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.