κ-deformed covariant quantum phase spaces as Hopf algebroids

We consider the general D=4D=4(10+10)(10+10)-dimensional κ  -deformed quantum phase space as given by Heisenberg double HH of D=4D=4κ  -deformed Poincaré–Hopf algebra HH. The standard (4+4)(4+4)-dimensional κ-deformed covariant quantum phase space spanned by κ  -deformed Minkowski coordinates and commuting momenta generators (xˆμ,pˆμ) is obtained as the subalgebra of HH. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicit Hopf algebroid structure of standard κ-deformed quantum covariant phase space in Majid–Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.