×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] DOI: 10.1007/BF02902200 · Zbl 0161.23606 · doi:10.1007/BF02902200
[2] Santilli R. M., Hadronic J. 1 pp 574– (1978)
[3] Santilli R. M., Hadronic J. 4 pp 1166– (1981)
[4] DOI: 10.1088/0305-4470/23/5/001 · Zbl 0715.17017 · doi:10.1088/0305-4470/23/5/001
[5] DOI: 10.1007/BF01077280 · Zbl 0606.17013 · doi:10.1007/BF01077280
[6] DOI: 10.1023/A:1021431709238 · Zbl 0942.17005 · doi:10.1023/A:1021431709238
[7] DOI: 10.1006/aima.1996.0066 · Zbl 0874.33011 · doi:10.1006/aima.1996.0066
[8] DOI: 10.1088/0305-4470/26/9/011 · Zbl 0780.17010 · doi:10.1088/0305-4470/26/9/011
[9] DOI: 10.1063/1.530440 · Zbl 0809.17010 · doi:10.1063/1.530440
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.