Universal T-matrices for quantum Poincaré groups: contractions and quantum reference frames

Universal T-matrices, or Hopf algebra dual forms, for quantum groups are revisited, and their contraction theory is developed. As a first illustrative example, the (1+1) timelike κ-Poincaré T-matrix is explicitly worked out. Afterwards, motivated by recent results on the role of the Hopf algebra dual form of a quantum (1+1) centrally extended Galilei group as the algebraic object underlying non-relativistic quantum reference frame transformations, a new quantum deformation of the (1+1) centrally extended Poincaré Lie algebra is obtained, and its universal T-matrix is presented. Finally, the Hopf algebra dual form contraction is applied to this Poincaré T-matrix, showing that its corresponding non-relativistic counterpart is precisely the Galilei T-matrix associated with quantum reference frames. In this way, the Poincaré Hopf algebra dual form introduced here stands as a natural candidate for describing the symmetry structure of relativistic quantum reference frame transformations. In the appropriate basis, the associated quantum Poincaré group is recognized, remarkably, as a non-trivial central extension of the (1+1) spacelike κ-Poincaré dual Hopf algebra.