Two θμν-deformed covariant relativistic quantum phase spaces as Poincaré-Hopf algebroids

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known θμν-deformed D=4 Poincaré-Hopf algebra H. The first algebroid describes θμν-deformed relativistic phase space with canonical NC space-time (constant θμν parameters) and the second one incorporates dual to H quantum θμν-deformed Poincaré-Hopf group algebra G, which contains noncommutative space-time translations given by Λ-dependent Θμν parameters (Λ≡Λμν parametrize classical Lorentz group). The canonical θμν-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincaré transformations described by G. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.