Integrable Systems on Rectangular \(\mathcal {W}\)-Superalgebras via Super Adler-Type Operators

In this paper, we introduce a class of super Adler-type operators associated with the Lie superalgebra \(\mathfrak {gl}(m|n)\) $\mathfrak{gl}(m|n)$ . We show that these operators generate Poisson vertex superalgebras which are isomorphic to the classical \(\mathcal {W}\) $\mathcal{W}$ -superalgebras associated with \(\mathfrak {gl}(m|n)\) $\mathfrak{gl}(m|n)$ and some rectangular nilpotent elements. We use this isomorphism to construct integrable hierarchies on these rectangular \(\mathcal {W}\) $\mathcal{W}$ -superalgebras.