Nonassociativity and Integrable Hierarchies

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 δn, n = 1,2, . . ., such that δn(f) is a nonlinear homogeneous polynomial in f of degree n + 1. We prove that the expressions δn1 � � � δnk (f) satisfy identities which are in correspondence with the equations of the Kadomtsev-Petviashvili (KP) hierarchy. As a consequence, solutions of the ‘nonassociative hierarchy’ ∂tn (f) = δn(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 δn, zero curvature connections and linear systems.