Representations of classical Lie algebras from their quantum deformations

We make use of a well-know deformation of the Poincaré Lie algebra in p + q + 1 dimensions (p + q > 0) to construct the Poincaré Lie algebra out of the Lie algebras of the de Sitter and anti de Sitter groups, the generators of the Poincaré Lie algebra appearing as certain irrational functions of the generators of the de Sitter groups. We have obtained generalizations of this "anti-deformation" for the SO(p + 2, q) and SO(p + 1, q + 1) cases with arbitrary p and q. Similar results have been established for q deformations U_q(so(p, q)) with small p and q values. Combining known results on representations of U_q(so(p, q)) (for q both generic and a root of unity) with our "anti-deformation" formulae, we get representations of classical Lie algebras which depend upon the deformation parameter q. Explicit results are given for the simplest example (of type A₁) i.e. that associated with U_q(so(2, 1)).