Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures

We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $\mathbb{H}$ and quantum Poincare Hopf group $\widehat{\mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $$ describing dynamics on quantum group algebra $$ provided by the Heisenberg double algebra \mathcal{HD=% }\mathbb{H}\rtimes \widehat{\mathbb{G}}, and second, denoted by \mathcal{% \tilde{H}}^{(10,10)}, describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space $\hat{x}_{\mu }$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.