Quantum group covariant (anti)symmetrizers, ε-tensors, vielbein, Hodge map and Laplacian

GLq(N)and SOq(N)-covariant deformations of the completely symmetric/antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices. The precise relation between the completely antisymmetric projectors and the completely antisymmetric tensor is determined. Adopting the GLq(N)and SOq(N)-covariant differential calculi on the corresponding quantum group covariant noncommutative spaces Cq , R N q , we introduce a generalized notion of vielbein basis (or “frame”), based on differential-operator-valued 1-forms. We then give a thorough definition of a SOq(N)-covariant R N q bilinear Hodge map acting on the bimodule of differential forms on Rq , introduce the exterior coderivative and show that the Laplacian acts on differential forms exactly as in the undeformed case, namely it acts on each component as it does on functions. Preprint 04-5 Dip. Matematica e Applicazioni, Università di Napoli DSF/ 08-2004 ∗Work partially supported by the European Commission RTN Programme HPRN-CT-200000131 and by MIUR