Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously angular momentum is deformed to so(q)(3), it acts on the q-Euclidean space that becomes a so(q)(3)-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on C-infinity functions on R-3. On a factorspace of C-infinity (R-3) a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice.