Towards a topological (dual of) quantum κ-Poincaré group

We argue that the κ-deformation is related to a factorization of a Lie group, therefore an appropriate version of κ-Poincaré does exist on the C*-algebraic level. The explicit form of this factorization is computed that leads to an “action with singularities” of the Lorentz group (with space reflections) considered in Amelino-Cameliaʹs Doubly Special Relativity theory. The orbit structure is found and “the momentum manifold” is extended in a way that removes singularities and results in a true action. It turns out that this manifold is nonsimply connected and nonorientable (in dimension 1+3). It can be equipped with a Lorentzian metric and the Lorentz group acts by isometries. In the introduction some aspects of κ-Poincaré algebra, among them of the symmetric and nonsymmetric coproduct, are discussed.