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 ISOq,r(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SOq,r(N), the calculus on the q-plane can be expressed in terms of its coordinates x a , differentials dx a and partial derivatives ∂a without the need of dilatations, thus generalizing known results to the multiparametric case. Using real forms that lead to the signature (n + 1, m) with m = n − 1, n, n+ 1, we find ISOq,r(n+ 1, m) and SOq,r(n+ 1, m) bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space ISOq,r(3,1)/SOq,r(3,1) is treated in detail. The conjugated partial derivatives ∂ ∗ can be expressed as linear combinations of the ∂a. This allows a deformation of the phase-space where no additional operators (besides x a and pa) are needed.