Quantum group covariant (anti)symmetrizers, epsilon-tensors, vielbein,   Hodge map and Laplacian
  GL_q(N)- and SO_q(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 GL_q(N)- and SO_q(N)-covariant differential calculi on the corresponding quantum group covariant noncommutative spaces C_q^N, R_q^N, 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 SO_q(N)-covariant R_q^N-bilinear Hodge map acting on the bimodule of differential forms on R_q^N, 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. 
Gaetano Fiore