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.

36 pages, second version substantially revised and rewritten

References in corpus (15)

• Integrable Hierarchies and Dispersionless Limit
• Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres
• Currents, Charges, and Canonical Structure of Pseudodual Chiral Models
• Vertex operator solutions to the discrete KP-hierarchy
• Gauge theory on nonassociative spaces
• Exact Noncommutative KP and KdV Multi-solitons
• Functional representation of the Ablowitz-Ladik hierarchy
• An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
• Functional representation of the Ablowitz-Ladik hierarchy. II
• Nonassociative geometry: Towards discrete structure of spacetime
• Darboux first integral conditions and integrability of the 3D Lotka-Volterra system
• Conservation laws for the nonlinear Schrodinger equation in Miwa variables
• The Paradigm of Pseudodual Chiral Models
• On a generalization of the Fay-Sato identity for KP Baker functions and its application to constrained hierarchies
• Algebraic identities associated with KP and AKNS hierarchies

Cited by in corpus (7)

• Noncommutative Ward's Conjecture and Integrable Systems
• With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices
• Burgers and KP hierarchies: A functional representation approach
• A new approach to deformation equations of noncommutative KP hierarchies
• Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics
• Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
• Hamilton's equations in a non-associative quantum theory