Summary: Representations \(\pi_{\Lambda\mu}\) of the principal degenerate series of the quantum algebra \(U_ q(u_{r,s})\) are introduced. The structure of these representations is studied. Several classes of unitary irreducible representations of the algebra \(U_ q(u_{r,s})\) are separated in the set of irreducible representations \(\pi_{\Lambda\mu}\) and irreducible constituents of reducible representations \(\pi_{\Lambda\mu}\). The formulas of action of operators of irreducible representations of the quantum algebra \(U_ q(u_{r+s})\) in the basis corresponding to restriction of representations onto the subalgebra \(U_ q(u_ r+u_ s)\) are given.