Two $theta_{mu u }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $theta_{mu u }$-deformed $D=4$ Poincare-Hopf algebra $mathbb{H}$. The first algebroid describes $theta_{mu u }$-deformed relativistic phase space with canonical NC space-time (constant $theta_{mu u }$ parameters) and the second one incorporates dual to $mathbb{H}$ quantum $theta_{mu u }$-deformed Poincare-Hopf group algebra $mathbb{G}$, which contains noncommutative space-time translations given by $Lambda $-dependent $Theta_{mu u }$ parameters ($% Lambda $ $equiv Lambda_{mu u }$ parametrize classical Lorentz group). The canonical $theta_{mu u }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.