paper

Geometrical Tools for Quantum Euclidean Spaces

arXiv:math/0002007 · doi:10.1007/PL00005553

Abstract

We apply one of the formalisms of noncommutative geometry to $R_{q}^{N}$ , the quantum space covariant under the quantum group $S O_{q} (N)$ . Over $R_{q}^{N}$ there are two $S O_{q} (N)$ -covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of $R_{q}^{3}$ . As in the case N=3, one has to slightly enlarge the algebra $R_{q}^{N}$ ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over $R_{q}^{N}$ . While in our previous article the frame was found `by hand', here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of $R_{q}^{N}$ with $U_{q} so (N)$ into $R_{q}^{N}$ , an interesting result in itself.

latex, 38 pages, typos corrected