Cayley-Klein Lie Algebras and Their Quantum Universal Enveloping Algebras

The N-dimensional Cayley-Klein scheme allows a simultaneous description of 3 N symmetric orthogonal homogeneous spaces by means of a set. of Lie algebras depending on N real parameters. We present here a quantum deformation of the Lie algebras generating the groups of motion of the two and three dimensional Cayley-Klein geometries. This Hopf algebra structure is presented in a compact form by using a formalism developed for the case of (quasi)free Lie algebras. Their quasitriangularity (i.e., the usual way to study the associativity of their dual objects, the quantum groups) is also discussed.