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Representations of the cyclically symmetric $$q$$-deformed algebra $$\text{so}_q(3)$$. (English) Zbl 0959.17015
Summary: An algebra homomorphism $$\psi$$ from the nonstandard $$q$$-deformed (cyclically symmetric) algebra $$U_q(so_3)$$ to the extension $$\widehat{U}_q(sl_2)$$ of the Hopf algebra $$U_q(sl_2)$$ is constructed. Not all irreducible representations of $$U_q(sl_2)$$ can be extended to representations of $$\widehat{U}_q(sl_2)$$. Composing the homomorphism $$\psi$$ with irreducible representations of $$\widehat{U}_q(sl_2)$$ we obtain representations of $$U_q(so_3)$$. Not all of these representations of $$U_q(so_3)$$ 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 irreducible representations of the Lie algebra $$so_3$$ when $$q\to 1$$. Representations of the other part have no classical analog. Using the homomorphism $$\psi$$ it is shown how to construct tensor products of finite-dimensional representations of $$U_q(so_3)$$. Irreducible representations of $$U_q(so_3)$$ when $$q$$ is a root of unity are constructed. Some of them are obtained from irreducible representations of $$\widehat{U_q} (sl_2)$$ by means of the homomorphism $$\psi$$.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
##### Keywords:
quantum algebra; irreducible representations
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##### References:
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