Weakly Nonassociative Algebras, Riccati and KP Hierarchies

It has recently been observed that certain nonassociative a lgebras (called “weakly nonassociative”, WNA) determine, via a universal hierarchy of ordinary differential equations, solutions of the KP hierarchy with dependent variable in an associative subalgebra (the middle nucleus). We recall central results and consider a class of WNA algebras for which the hierarchy of ODEs reduces to a matrix Riccati hierarchy, which can be easily solved in certain cases. The resulting solutions of a matrix KP hierarchy then determine (under a rank 1 condition) solutions of the scalar KP hierarchy. We also extend these results to the discrete KP hierarchy. Moreover, we build a bridge from the WNA framework to the Gelfand-Dickey formulation of the KP hierarchy.