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Book Chapter

Weakly nonassociative algebras, Riccati and KP hierarchies


Müller-Hoissen,  Folkert
Max Planck Institute for Dynamics and Self-Organization, Max Planck Society;

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Dimakis, A., & Müller-Hoissen, F. (2009). Weakly nonassociative algebras, Riccati and KP hierarchies. In S. Silvestrov, E. Paal, V. Abramov, & A. Stolin (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond (pp. 9-27). Berlin Heidelberg: Springer-Verlag.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0029-1335-C
It has recently been observed that certain nonassociative algebras (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. The resulting solutions of a matrix KP hierarchy then determine (under a rank 1 condition) solutions of the scalar KP hierarchy. We 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.