On the classification of irreducible finite-dimensional representations ofU-q(')(so(3)) algebra

In an earlier work [M. Havlicek , J. Math. Phys. 40, 2135 (1999)] we define d for any finite dimension five nonequivalent irreducible representations o f the nonstandard deformation U-q'(so(3)) of the Lie algebra so(3) where q is not a root of unity [for each dimension only one of them (called classic al) admits limit q --> 1]. In the first part of this paper we show that any finite-dimensional irreducible representation is equivalent to some of the se representations. In the case q(n) = 1 we derive new Casimir elements of U-q'(so(3)) and show that a dimension of any irreducible representation is not higher than n. These elements are Casimir elements of U-q'(so(m)) for a ll m and even of U-q'(iso(m+1)) due to Inonu-Wigner contraction. According to the spectrum of one of the generators, the representations are found to belong to two main disjoint sets. We give full classification and explicit formulas for all representations from the first set (we call them nonsingul ar representations). If n is odd, we have full classification also for the remaining singular case with the exception of a finite number of representa tions. (C) 2001 American Institute of Physics.