FRAME FORMALISM FOR THE N-DIMENSIONAL QUANTUM EUCLIDEAN SPACES

Abstract

We sketch our application1of a non-commutative version of the Cartan "moving-frame" formalism to the quantum Euclidean space [Formula: see text]the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over [Formula: see text] based on the R-matrix formalism, we summarize our construction of a frame, the dual inner derivations, a metric and two torsion-free almost metric compatible covariant derivatives with a vanishing curvature. To obtain these results we have developed a technique which fully exploits the quantum group covariance of [Formula: see text]. We first find a frame in the larger algebra [Formula: see text]. Then we define homomorphisms from [Formula: see text] to [Formula: see text] which we use to project this frame in [Formula: see text].