Integrability of the Frobenius algebra-valued KP hierarchy

TL;DR

This paper introduces a Frobenius algebra-valued KP hierarchy, constructs its tau-function and Hamiltonian structures, and explores related bi-Hamiltonian structures and generalizations of classical W-algebras.

Contribution

It presents the first formulation of a Frobenius algebra-valued KP hierarchy, along with its tau-function and Hamiltonian structures, extending classical integrable systems.

Findings

Existence of Frobenius algebra-valued tau-function.

Construction of Hamiltonian structures using Adler-Dickey-Gelfand method.

Identification of multiple local bi-Hamiltonian structures and Frobenius algebra-valued W-algebra generalizations.

Abstract

We introduce a Frobenius algebra-valued KP hierarchy and show the existence of Frobenius algebra-valued $\tau$-function for this hierarchy. In addition we construct its Hamiltonian structures by using the Adler-Dickey-Gelfand method. As a byproduct of these constructions, we show that the coupled KP hierarchy defined by P.Casati and G.Ortenzi in \cite{CO2006} has at least $n$-``basic" different local bi-Hamiltonian structures. Finally, via the construction of the second Hamiltonian structures, we obtain some local matrix, or Frobenius algebra-valued, generalizations of classical $W$-algebras.