On the classification of irreducible finite-dimensional representations of Uq′(so3) algebra

Abstract

In an earlier work ͓M. Havlíček et al., J. Math. Phys. 40, 2135 ͑1999͔͒ we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation U q Ј(so 3 ) of the Lie algebra so 3 where q is not a root of unity ͓for each dimension only one of them ͑called classical͒ admits limit q→1͔. 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 ϭ1 we derive new Casimir elements of U q Ј(so 3 ) and show that a dimension of any irreducible representation is not higher than n. These elements are Casimir elements of U q Ј(so m ) for all m and even of U q Ј(iso mϩ1 ) due to Inönu-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 nonsingular representations͒. If n is odd, we have full classification also for the remaining singular case with the exception of a finite number of representations.

Classification theorem on irreducible representations of the q‐deformed algebra U′q(son)

“…This assumption is true for the algebras U q (so 3 ) and U q (so 4 ) (see [10,11]). As we know from the previous section, irreducible finite-dimensional representations T of U q (so n ) are divided into two classes-irreducible representations of the classical type and irreducible representations of the nonclassical type.…”

“…This proposition is true for the algebra U q (so 3 ). It follows from complete reducibility of finite-dimensional representations of U q (so 3 ) (see [12]) and from the fact that representations of the classical and of the nonclassical types exhaust all irreducible representations of U q (so 3 ) (see [11] 3 ) and U q (so 4 ), this assumption is true (see [10,11,12] Proof. The restriction of T to the subalgebra U q (so n−1 ) is completely reducible due to the assumption.…”

Finite-Dimensional Simple Poisson Modules

“…There is an alternative 4-homogeneous version of U q (sl 2 ), appearing, for example, in [15,16,20,36] and obtained by replacing the relation…”

Center of quantum algebra $U_q^{\prime }({\rm so}_3)$Uq′( so 3)

“…There are three new Casimir elements which take general form (cf., Havlíček, Klimyk and Pošta (2000) and Havlíček and Pošta (2001), formula (5))…”

“…In the paper, Havlíček and Pošta (2001) classification of finite dimensional representations and their explicit form were derived. However, the structure of the center was not studied in detail, which is the purpose of this paper.…”