Embedding Euclidean Lie algebras into quantum structure

We show that it is possible to express the basis elements of the Lie algebra of the Euclidean group, E(2), as simple irrational functions of certain q deformed expressions involving the generators of the quantum algebra U-q(so(2,1)). We consider implications of these results for the representation theory of the Lie algebra of E(2). We briefly discuss analogous results for U-q(so(2,2)).