Representations of the cyclically symmetric q-deformed algebra so(q)(3)

An algebra homomorphism psi from the nonstandard q-deformed (cyclically sym metric) algebra U-q (so(3)) to the extension (U) over cap(q) (sl(2)) of the Hopf algebra U-q (sl(2))is constructed. Not all irreducible representation s of U-q (sl(2)) can be extended to representations of (U) over cap(q) (sl( 2)). Composing the homomorphism psi with irreducible representations of (U) over cap(q) (sl(2)) we obtain representations of U-q (so(3)). Not all of t hese representations of (U) over cap(q) (so(3)) are irreducible. Reducible representations of U-q (so(3)) are decomposed into irreducible components. In this way we obtain all irreducible representations of U-q (so(3)) when q is not a root of unity. A part of these representations turns into irreduc ible representations of the Lie algebra so(3) when q --> 1. Representations of the other part have no classical analog. Using the homomorphism psi is shown how to construct tensor products of finite-dimensional representation s of U-q (so(3)). Irreducible representations of U-q (so(3)) when q is a ro ot of unity are constructed. Some of them are obtained from irreducible rep resentations of (U) over cap(q) (sl(2)) by means of the homomorphism psi. ( C) 1999 American Institute of Physics. [S0022-2488(99)01003-8].