1 REPRESENTATIONS OF THE q-DEFORMED ALGEBRA Uq(isoq(2))

An algebra homomorphism ψ from the q-deformed algebra U q (iso 2) with generating 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 extensionˆU q (m 2) of the Hopf algebra U q (m 2) is constructed. The algebra U q (iso 2) at q = 1 leads to the Lie algebra iso 2 ∼ m 2 of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra U q (m 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 representations of the extensionˆU q (m 2). Composing the homomorphism ψ with irreducible representations ofˆU q (m 2) we obtain representations 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 decomposed into irreducible components. In this way we obtain all irreducible representations 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 iso 2 when q → 1. Representations of the other part have no classical analogue.