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 H and quantum Poincare Hopf group Gˆ. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one H(10,10) describing dynamics on quantum group algebra Gˆ provides by the Heisenberg double algebra HD=H⋊Gˆ, and second, denoted by H˜(10,10), describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space xˆμ. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.