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 $\mathrm{U}_q(\widehat{\gl}_{M|N})$.

20pp. Revised version

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