Weakly nonassociative algebras, Riccati and KP hierarchies
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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.
16 pages, second version: LaTeX problem with L's in section 5 resolved, third version: example 2 in section 3 added, some minor corrections, forth version: a few small changes and corrections. Proceedings of the workshop Algebra, Geometry, and Mathematical Physics, Lund, October, 2006
16 pages, second version: LaTeX problem with L's in section 5 resolved, third version: example 2 in section 3 added, some minor corrections, forth version: a few small changes and corrections. Proceedings of the workshop Algebra, Geometry, and Mathematical Physics, Lund, October, 2006