On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N

Abstract

We show that the complicated *-structure characterizing for positive q the U_qso(N)-covariant differential calculus on the non-commutative manifold R_q^N boils down to similarity transformations involving the ribbon element of a central extension of U_qso(N) and its formal square root v. Subspaces of the spaces of functions and of p-forms on R_q^N are made into Hilbert spaces by introducing non-conventional ``weights'' in the integrals defining the corresponding scalar products, namely suitable positive-definite q-pseudodifferential operators realizing the action of v^{\pm 1}; this serves to make the partial q-derivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the `radial coordinate' r. Unless we choose a constant m, then the square-integrables functions/forms must fulfill an additional condition, namely their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are q-special functions with `quantized' free parameters.