A Note on the Quantum Berezinian for the Double Yangian of the Lie Superalgebra \(\mathfrak {gl}_{m|n}\)

In this note, we generalize the notion of quantum Berezinian to the double Yangian \(\textrm{DY}(\mathfrak {gl}_{m|n})\) $\text{DY}({\mathfrak{gl}}_{m|n})$ of the Lie superalgebra \( \mathfrak {gl}_{m|n} \) ${\mathfrak{gl}}_{m|n}$ . We show that its coefficients form a family of algebraically independent topological generators of the center of \(\textrm{DY}(\mathfrak {gl}_{m|n})\) $\text{DY}({\mathfrak{gl}}_{m|n})$ .