Quantum deformations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic o⋆(4)o⋆(4) symmetries in unified o(4;C)o(4;C) setting

We employ new calculational technique and present complete list of classical r  -matrices for D=4D=4 complex homogeneous orthogonal Lie algebra o(4;C)o(4;C), the rotational symmetry of four-dimensional complex space–time. Further applying reality conditions we obtain the classical r  -matrices for all possible real forms of o(4;C)o(4;C): Euclidean o(4)o(4), Lorentz o(3,1)o(3,1), Kleinian o(2,2)o(2,2) and quaternionic o⋆(4)o⋆(4) Lie algebras. For o(3,1)o(3,1) we get known four classical D=4D=4 Lorentz r-matrices, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide new results and mention some applications.