Integrable systems on rectangular $\mathcal{W}$-superalgebras via super Adler-type operators

TL;DR

This paper introduces super Adler-type operators linked to Lie superalgebras, demonstrating their role in generating Poisson vertex superalgebras and constructing integrable hierarchies on rectangular $ ext{W}$-superalgebras.

Contribution

It establishes a novel connection between super Adler-type operators and rectangular $ ext{W}$-superalgebras, enabling the construction of integrable hierarchies.

Findings

Super Adler-type operators generate Poisson vertex superalgebras.

Isomorphism with classical $ ext{W}$-superalgebras established.

Construction of integrable hierarchies on these superalgebras.

Abstract

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