A new approach to deformation equations of noncommutative KP hierarchies

TL;DR

This paper introduces a novel approach to deformation equations of noncommutative KP hierarchies, utilizing a matrix Riccati hierarchy and weakly nonassociative algebra structures to extend the KP framework.

Contribution

It develops a new matrix Riccati hierarchy linked to noncommutative KP hierarchies using WNA algebra, and formulates deformation flow equations incorporating star product deformations.

Findings

Matrix Riccati hierarchy derived from linear ODEs

WNA algebra structure underlying the hierarchy

Deformation parameters lead to extended KP hierarchies

Abstract

Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP) hierarchy, we start with a quite general hierarchy of linear ordinary differential equations in a space of matrices and derive from it a matrix Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly nonassociative' (WNA) algebra structure, from which we can conclude, refering to previous work, that any solution of the Riccati system also solves the potential KP hierarchy (in the corresponding matrix algebra). We then turn to the case where the components of the matrices are multiplied using a (generalized) star product. Associated with the deformation parameters, there are additional symmetries (flow equations) which enlarge the respective KP hierarchy. They have a compact formulation in terms of the WNA structure. We also present a formulation of the KP hierarchy equations themselves as deformation flow equations.