Quantum Berezinian for quantum affine superalgebra \(\textrm{U}_q(\widehat{\mathfrak {gl}}_{M|N})\)

We introduce the quantum Berezinian for the quantum affine superalgebra \(\textrm{U}_q(\widehat{\mathfrak {gl}}_{M|N})\) and show that the coefficients of the quantum Berezinian belong to the center of \(\textrm{U}_q(\widehat{\mathfrak {gl}}_{M|N})\). We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of \(\textrm{U}_q(\widehat{\mathfrak {gl}}_{M|N})\).