Classical affine W-superalgebras via generalized Drinfeld-Sokolov reductions and related integrable systems

TL;DR

This paper introduces a new approach to constructing classical affine W-superalgebras using generalized Drinfeld-Sokolov reductions, revealing their structure and connection to integrable super-Hamiltonian systems.

Contribution

It provides a novel definition of W-superalgebras via D-S reduction, identifies their generators, and links them to integrable super-Hamiltonian systems.

Findings

New construction of W-superalgebras via D-S reduction

Explicit free generators of W-superalgebras identified

Connection established between W-superalgebras and integrable systems

Abstract

The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the Poisson vertex algebras theory. A W-superalgebra with certain properties can be understood as an underlying differential superalgebra of a series of integrable super-Hamiltonian systems.