Degenerate Series Representations of the $q$-Deformed Algebra ${\rm so}'_q(r,s)$

Abstract

The q-deformed algebra ${\rm so}'_q(r,s)$ is a real form of the q-deformed algebra $U'_q({\rm so}(n,\mathbb{C}))$, $n=r+s$, which differs from the quantum algebra $U_q({\rm so}(n,\mathbb{C}))$ of Drinfeld and Jimbo. We study representations of the most degenerate series of the algebra ${\rm so}'_q(r,s)$. The formulas of action of operators of these representations upon the basis corresponding to restriction of representations onto the subalgebra ${\rm so}'_q(r)\times {\rm so}'_q(s)$ are given. Most of these representations are irreducible. Reducible representations appear under some conditions for the parameters determining the representations. All irreducible constituents which appear in reducible representations of the degenerate series are found. All $*$-representations of ${\rm so}'_q(r,s)$ are separated in the set of irreducible representations obtained in the paper.