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

TL;DR

This paper introduces the quantum Berezinian for the quantum affine superalgebra U_q(ˆgl_{M|N}) and establishes its centrality, along with related central elements and classical identities in the quantum setting.

Contribution

It defines the quantum Berezinian for U_q(ˆgl_{M|N}) and proves its coefficients are central, also deriving classical identities for the quantum generator matrices.

Findings

Quantum Berezinian coefficients are central in U_q(ˆgl_{M|N})

Constructed new central elements expressed via the Berezinian

Proved quantum analogues of classical identities like Jacobi and Sylvester

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