Representations of the q-deformed algebra U_q(iso₂)

An algebra homomorphism from the q-deformed algebra U_q(iso₂) with generating elements I, T₁, T₂ and defining relations [I,T₂]_q = T₁, [T₁,I]_q = T₂, [T₂,T₁]_q = 0 (where [A,B]_q = q^1/2AB-q^-1/2BA) to the extension Û_q(m₂) of the Hopf algebra U_q(m₂) is constructed. The algebra U_q(iso₂) at q = 1 leads to the Lie algebra iso₂~m₂ of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra U_q(m₂) (which is not isomorphic to U_q(iso₂)) is treated as a Hopf q-deformation of the universal enveloping algebra of iso₂ and is well known in the literature.

Not all irreducible representations of U_q(m₂) can be extended to representations of the extension Û_q(m₂). Composing the homomorphism with irreducible representations of Û_q(m₂) we obtain representations of U_q(iso₂). Not all of these representations of U_q(iso₂) are irreducible. The reducible representations of U_q(iso₂) are decomposed into irreducible components. In this way we obtain all irreducible representations of U_q(iso₂) when q is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra iso₂ when q1. Representations of the other part have no classical analogue.