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On the classification of irreducible finite-dimensional representations of $$U_q'(\text{so}_3)$$ algebra. (English) Zbl 1032.17022
Summary: In an earlier work [M. Havlíček, A. U. Klimyk and S. Posta, J. Math. Phys. 40, 2135-2161 (1999; Zbl 0959.17015)] we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation $$U_q'(\text{so}_3)$$ of the Lie algebra $$\text{so}_2$$ where $$q$$ is not a root of unity [for each dimension only one of them (called classical) admits limit $$q\to 1]$$. In the first part of this paper we show that any finite-dimensional irreducible representation is equivalent to some of these representations. In the case $$q^n=1$$ we derive new Casimir elements of $$U_q'(\text{so}_3)$$ and show that the dimension of any irreducible representation is not higher than $$n$$. These elements are Casimir elements of $$U_q'(\text{so}_m)$$ for all $$m$$ and even of $$U_q(\text{iso}_{m+1})$$ due to Inönü-Wigner contraction. According to the spectrum of one of the generators, the representations are found to belong to two main disjoint sets. We give a full classification and explicit formulas for all representations from the first set (we call them nonsingular representations). If $$n$$ is odd, we have a full classification also for the remaining singular case with the exception of a finite number of representations.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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##### References:
 [1] DOI: 10.1007/BF00420371 · Zbl 0735.17020 · doi:10.1007/BF00420371 [2] Nelson J. E., Commun. Math. Phys. 141 pp 211– (1991) · Zbl 0746.53062 · doi:10.1007/BF02100010 [3] Havlı\'ček M., J. Math. Phys. 40 pp 2135– (1999) · Zbl 0959.17015 · doi:10.1063/1.532856 [4] Havlı\'ček M., Czech. J. Phys. 50 pp 79– (2000) · Zbl 0977.22005 · doi:10.1023/A:1022825031633 [5] Samoilenko Y., Quantum Groups and Quantum Spaces 40 pp 21– (1997)
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