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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 \(\psi\).

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Full Text: DOI arXiv
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