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

M. Havlíček; A. U. Klimyk; S. Pošta

doi:10.1063/1.532856

paper

Representations of the cyclically symmetric q-deformed algebra $s o_{q} (3)$

arXiv:math/9805048 · doi:10.1063/1.532856

Abstract

An algebra homomorphism $ψ$ from the nonstandard q-deformed (cyclically symmetric) algebra $U_{q} (s o_{3})$ to the extension $U^_{q} (s l_{2})$ of the Hopf algebra $U_{q} (s l_{2})$ is constructed. Not all irreducible representations of $U_{q} (s l_{2})$ can be extended to representations of $U^_{q} (s l_{2})$ . Composing the homomorphism $ψ$ with irreducible representations of $U^_{q} (s l_{2})$ we obtain representations of $U_{q} (s o_{3})$ . Not all of these representations of $U_{q} (s o_{3})$ are irreducible. Reducible representations of $U_{q} (s o_{3})$ are decomposed into irreducible components. In this way we obtain all irreducible representations of $U_{q} (s o_{3})$ 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 analogue. Using the homomorphism $ψ$ it is shown how to construct tensor products of finite dimensional representations of $U_{q} (s o_{3})$ . Irreducible representations of $U_{q} (s o_{3})$ when $q$ is a root of unity are constructed. Part of them are obtained from irreducible representations of $U^_{q} (s l_{2})$ by means of the homomorphism $ψ$ .

28 pages, LaTeX

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

Abstract

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Representations of the cyclically symmetric q-deformed algebra $s o_{q} (3)$