CAYLEY-KLEIN LIE ALGEBRAS AND THEIR QUANTUM UNIVERSAL ENVELOPING ALGEBRAS 1 A. Ballesteros, F.J. Herranz, M.A. del Olmo and M. Santander Departamento de F´ ısica Te´orica, Universidad de Valladolid. E-47011, Valladolid. Spain. ABSTRACT. The N-dimensional Cayley-Klein scheme allows the simultaneous descrip- tion of 3 N geometries (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 deformation (Hopf algebra structure) is presented in a compact form by using a formalism developed for the case of (quasi)free Lie algebras. Their quasi- triangularity (i.e., the most usual way to study the associativity of their dual objects, the quantum groups) is also discussed. 1. Introduction We study here a certain type of Lie algebra deformations (so called “quantum” ones), that have recently appeared in the context of the Quantum Inverse Scattering Method. They are properly defined as deformations of the corresponding universal enveloping algebra U g and their dual objects (in certain restricted sense) generate the “quantum groups” –deformations of the algebra of functions on the group, in the spirit of non-commutative geometry [1,2]–. Their underlying algebraic structure (mainly Hopf algebra properties [3]) is rather rich and was soon described for the classical simple Lie algebras [4,5,6]. However, many physically interesting groups are not simple groups: for instance, the groups of inertial transformations of space–time such as Galilei or Poincar´ e ones. Some quantum deformations have been built for their associated Lie algebras [7,8] which also arises as symmetries of certain physical problems [9]. We present here an attempt towards their characterization based on a Cayley-Klein (CK) geometrical scheme that includes all these groups as well as their transformed by In¨on¨ u–Wigner contractions [10]. We also discuss the problems arising in the definition of the quantum groups as dual objects of these quantum algebras, mainly in connection with the way in which the associativity of the deformed algebra of functions on the group is guaranteed (the R–matrix problem). 2. The Cayley-Klein Lie algebras From a physical point of view, some interesting homogeneous symmetric spaces can be simultaneously described in the framework of CK geometries. To do this, we consider a N -dimensional (N -d) symmetric orthogonal geometry as a group G of dimension 1 2 N (N + 1) and a set of N commuting involutions S (i) in g, the Lie algebra of G. If we denote by h (i) the Lie subalgebras of elements invariant under 1 Communication presented in the “International Symposium On Non Associative Algebras and Applications”, Oviedo (Spain), July 1993. 1