an:01711138
Zbl 1032.17022
Havl????ek, M.; Po??ta, S.
On the classification of irreducible finite-dimensional representations of \(U_q'(\text{so}_3)\) algebra
EN
J. Math. Phys. 42, No. 1, 472-500 (2001).
00083500
2001
j
17B37 81R50
nonstandard deformation; Lie algebra; finite-dimensional irreducible representation; Casimir elements; In??n??-Wigner contraction
Summary: In an earlier work [\textit{M. Havl????ek}, \textit{A. U. Klimyk} and \textit{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 In??n??-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.
Zbl 0959.17015