Quantum deformations of nonsemisimple algebras: The example of D=4 inhomogeneous rotations

A general class of deformations of the complexified D=4 Poincare algebra O(3,1;C)⊇T4(C) is considered with a classical (undeformed) subalgebra O(3;C)⊇T4(C) and deformed relations preserving the O(3;C) tensor structure. We distinguish the class of quantum deformations—the complex noncocommutative Hopf algebras—which depend on one complex mass parameter κ. Further, we consider the real Hopf algebras, obtained by imposing the reality conditions. For any choice of real metric [O(4), O(3,1), or O(2,2)] the parameter κ becomes real. All (e.g., Minkowski as well as Euclidean) real quantum algebras with standard reality condition contain as nonlinearities the hyperbolic functions of the energy operator and can be interpreted as introducing an imaginary time lattice. The symmetries of the models with real time lattice are described by a real quantum algebra with nonstandard reality conditions and trigonometric nonlinearities.