Representations of the q-deformed algebra U-q(iso(2))

An algebra homomorphism psi from the q-deformed algebra U-q(iso(2)) with ge nerating elements I, T-1, T-2 and defining relations [I, T-2](q) = T-1, [T- 1, I](q) = T-2, [T-2, T-1](q) = 0 (where [A, B](q) = q(1/2)AB-q(-1/2)BA) to the extension (U) over cap(q)(m(2)) of the Hopf algebra U-q(m(2)) is const ructed. The algebra U-q(iso(2)) at q = 1 leads to the Lie algebra iso(2) si milar to m(2) of the group ISO(2) of motions of the Euclidean plane. The Ho pf algebra U-q(m(2)) (which is not isomorphic to U-q(iso(2))) is treated as a Hopf q-deformation of the universal enveloping algebra of iso(2) and is well known in the literature. Not all irreducible representations of U-q(m(2)) can be extended to represe ntations of the extension (U) over cap(q)(m(2)). Composing the homomorphism psi with irreducible representations of (U) over cap(q)(m(2)) we obtain re presentations of U-q(iso(2)). Not all of these representations of U-q(iso(2 )) are irreducible. The reducible representations of U-q(iso(2)) are decomp osed into irreducible components. In this way we obtain all irreducible rep resentations of U-q(iso(2)) when q is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra i so(2) when q --> 1. Representations of the other part have no classical ana logue.