It is well-known that the universal \(R\)-matrices for quantized semisimple finite dimensional Lie algebras and for quantized Kac-Moody algebras are uniquely determined (under some conditions) by quasi-cocommutativity and quasi-triangularity of the corresponding (quasi-triangular) Hopf algebra. The authors prove that the universal \(R\)-matrix is uniquely determined by quasi-cocommutativity, and quasi-triangularity (as well as the Yang-Baxter equation) for the \(R\)-matrix is a consequence of the linear equation for quasi-cocommutativity. To prove these statements, tensor products of Verma modules and the structure of extremal projectors (projectors onto highest weight vectors) for irreducible representations of quantum algebras are considered. The authors give explicit examples of applications of their theory.