2 6 M ar 2 00 4 On the hermiticity of q - differential operators and forms on the quantum Euclidean spaces R Nq

We show that the complicated ⋆-structure characterizing for positive q the Uqso(N)-covariant differential calculus on R N q 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 Rq are made into Hilbert spaces by introducing nonconventional “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 antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. Preprint 03-55 Dip. Matematica e Applicazioni, Università di Napoli DSF/45-2003