Nonassociativity and Integrable Hierarchies
Abstract
Let A be a nonassociative algebra such that the associator (A,A^2,A) vanishes. If A is freely generated by an element f, there are commuting derivations delta_n, n=1,2,..., such that delta_n(f) is a nonlinear homogeneous polynomial in f of degree n+1. We prove that the expressions delta_{n_1} ... delta_{n_k}(f) satisfy identities which are in correspondence with the equations of the Kadomtsev-Petviashvili (KP) hierarchy. As a consequence, solutions of the `nonassociative hierarchy' partial_{t_n}(f) = delta_n(f), n=1,2,..., of ordinary differential equations lead to solutions of the KP hierarchy. The framework is extended by introducing the notion of an A-module and constructing, with the help of the derivations delta_n, zero curvature connections and linear systems.
Cite
@article{arxiv.nlin/0601001,
title = {Nonassociativity and Integrable Hierarchies},
author = {Aristophanes Dimakis and Folkert Muller-Hoissen},
journal= {arXiv preprint arXiv:nlin/0601001},
year = {2007}
}
Comments
36 pages, second version substantially revised and rewritten