Representations of Classical Lie Algebras from their Quantum Deformations

Abstract

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 $A_1$) i.e. that associated with $U_q(so(2,1))$.