Centrally Extended W_{1+\infty} and the KP Hierarchy

TL;DR

This paper demonstrates that the centrally extended W_{1+ abla} algebra introduces a new Poisson structure for the KP hierarchy, expanding the understanding of its Hamiltonian formulations.

Contribution

It explicitly constructs a new Poisson structure for the KP hierarchy using the centrally extended W_{1+ abla} algebra, revealing a novel Hamiltonian framework.

Findings

Centrally extended W_{1+ abla} algebra provides a new Poisson structure for KP.

Explicit construction of infinitely many new Hamiltonians in closed form.

The new structure parallels the known centerless case, enriching the integrable system's theory.

Abstract

It is well known that the centerless W_{1+\infty} algebra provides a hamiltonian structure for the KP hierarchy. In this letter we address the question whether the centerful version plays a similar r\^ole in any related integrable system. We find that, surprisingly enough, the centrally extended W_{1+\infty} algebra yields yet another Poisson structure for the same standard KP hierarchy. This is proven by explicit construction of the infinitely many new hamiltonians in closed form.