Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies

TL;DR

This paper demonstrates that all W-algebras W(gl_N,f) possess integrable Hamiltonian hierarchies, linking them to vector constrained KP hierarchies and their matrix generalizations through Dirac reduction, enriching the understanding of their algebraic structures.

Contribution

The paper introduces a new method to construct integrable Hamiltonian hierarchies for all W-algebras W(gl_N,f), establishing their connection to KP hierarchies via Dirac reduction.

Findings

All W(gl_N,f) algebras carry integrable Hamiltonian hierarchies.

Vector constrained KP hierarchies are derived from these hierarchies.

Provides bi-Poisson structures for the KP hierarchies.

Abstract

We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper, to show that all W-algebras W(gl_N,f) carry such a hierarchy. As an application, we show that all vector constrained KP hierarchies and their matrix generalizations are obtained from these hierarchies by Dirac reduction, which provides the former with a bi-Poisson structure.