Quasi-symmetric functions and the KP hierarchy

TL;DR

This paper explores the role of quasi-symmetric functions in solving the KP hierarchy, introducing a new nonassociative product that enhances their algebraic structure and yields identities related to the KP equations.

Contribution

It introduces a novel nonassociative product for quasi-symmetric functions and demonstrates its significance in forming an infinitesimal bialgebra linked to the KP hierarchy.

Findings

New nonassociative product for quasi-symmetric functions

Establishment of an infinitesimal bialgebra structure

Derivation of identities corresponding to KP hierarchy equations

Abstract

Quasi-symmetric functions show up in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.