ON THE HERMITICITY OF q-DIFFERENTIAL OPERATORS AND FORMS ON THE QUANTUM EUCLIDEAN SPACES $\mathbb{R}_q^N$

Abstract

We show that the complicated ⋆-structure characterizing for positive q the Uqso(N)-covariant differential calculus on the noncommutative manifold [Formula: see text] boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ṽ. Subspaces of the spaces of functions and of p-forms on [Formula: see text] 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 ṽ′±1 realizing the action of ṽ±1; this serves to make the partial q-derivatives anti-hermitean 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.