We establish duality between real forms of the quantum deformation of the 4-dimensional orthogonal group studied by Fioresi et al. and the classification work made by Borowiec et al.. Classically these real forms are the isometry groups of $\mathbb{ R}^4$ equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named $q$-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.
Categories: hep-th math-ph math.MP math.QA