Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$

Abstract: We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}q(\widehat{\mathfrak{gl}}{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}q(\widehat{\gl}{M|N})$. 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 $\mathrm{U}q(\widehat{\gl}{M|N})$.