  To construct a quantum group gauge theory one needs an algebra which is
invariant under gauge transformations. The existence of this invariant algebra
is closely related with the existence of a differential algebra $\delta _{{\cal
H}} G_{q}$ compatible with the Hopf algebra structure. It is shown that $\delta
_{{\cal H}} G_{q}$ exists only for the quantum group $U_{q}(N)$ and that the
quantum group $SU_q(N)$ as a quantum gauge group is not allowed. The
representations of the algebra $\delta _{{\cal H}} G_{q}$ are con- structed.
The operators corresponding to the differentials are realized via derivations
on the space of all irreducible *-representations of $U_q(2)$. With the help of
this construction infinitesimal gauge transformations in two-dimensional
classical space-time are described.
