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We construct new central elements in the quantum affine algebras of type $A$and prove Liouville-type formulas relating them to the quantum determinants. Weapply these formulas to calculate the eigenvalues of the quantum Gelfandinvariants as introduced by Reshetikhin, Takhtadzhyan and Faddeev (1989) actingin irreducible highest weight representations of the quantized envelopingalgebra for ${\mathfrak {gl}}_n$.

Liouville formulas for quantum affine algebras and eigenvalues of quantum Gelfand invariants