Quantum orthogonal planes:ISOq,r(N) and SOq,r(N) - bicovariant calculi and differential geometry on quantum Minkowski space

We construct differential calculi on multiparametric quantum orthogonal planes in any dimension N. These calculi are bicovariant under the action of the full inhomogeneous (multiparametric) quantum group , and do contain dilatations. If we require bicovariance only under the quantum orthogonal group , the calculus on the q-plane can be expressed in terms of its coordinates , differentials and partial derivatives without the need of dilatations, thus generalizing known results to the multiparametric case. Using real forms that lead to the signature with n, n + 1, we find and bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space is treated in detail. The conjugated partial derivatives can be expressed as linear combinations of the . This allows a deformation of the phase-space where no additional operators (besides and ) are needed.