QUANTUM DEFORMATIONS OF NONSEMISIMPLE ALGEBRAS - THE EXAMPLE OF D = 4INHOMOGENEOUS ROTATIONS

A general class of deformations of the complexified D = 4 Poincare alg ebra O(3,1;C) +superset-of BT4(C) is considered with a classical (unde formed) subalgebra O(3;C) +superset-of BT4(C) and deformed relations p reserving the O(3;C) tensor structure. We distinguish the class of qua ntum deformations-the complex noncocommutative Hopf algebras-which dep end on one complex mass parameter kappa. Further, we consider the real Hopf algebras, obtained by imposing the reality conditions. For any c hoice of real metric [O(4), O(3,1), or O(2,2)] the parameter kappa bec omes real. All (e.g., Minkowski as well as Euclidean) real quantum alg ebras with standard reality condition contain as nonlinearities the hy perbolic functions of the energy operator and can be interpreted as in troducing an imaginary time lattice. The symmetries of the models with real time lattice are described by a real quantum algebra with nonsta ndard reality conditions and trigonometric nonlinearities.