Quantum orthogonal planes: ISO_q,r(N) and SO_q,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 ISO_q,r(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SO_q,r(N), the calculus on the q-plane can be expressed in terms of its coordinates cursive Greek chi^a, differentials dcursive Greek chi^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 ISO_q,r(n + 1, m) and SO_q,r(n + 1, m) bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space ISO_q,r(3, 1)/SO_q,r(3, 1) is treated in detail. The conjugated partial derivatives ∂*_a can be expressed as linear combinations of the ∂_a. This allows a deformation of the phase-space where no additional operators (besides cursive Greek chi^a and p_a) are needed.