Geometrical Tools for Quantum Euclidean Spaces

TL;DR

This paper develops geometrical tools for quantum Euclidean spaces covariant under quantum groups, constructing covariant differential calculi, metrics, and connections with specific curvature properties, generalizing previous 3D results.

Contribution

It introduces a systematic construction of covariant differential calculi, metrics, and torsion-free covariant derivatives on quantum Euclidean spaces for arbitrary dimensions, extending prior 3D findings.

Findings

Constructed two covariant differential calculi on $R^N_q$.

Found metrics and covariant derivatives with vanishing linear curvature.

Discovered a homomorphism linking $R^N_q$ and $U_qso(N)$.

Abstract

We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-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 $R^3_q$. As in the case N=3, one has to slightly enlarge the algebra $R^N_q$; 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 $R^N_q$. 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 $R^N_q$ with $U_qso(N)$ into $R^N_q$, an interesting result in itself.