Weakly nonassociative algebras, Riccati and KP hierarchies

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.Comment: 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, 200

Extension of Moyal-deformed hierarchies of soliton equations

Moyal-deformed hierarchies of soliton equations can be extended to larger hierarchies by including additional evolution equations with respect to the deformation parameters. A general framework is presented in which the extension is universally determined and which applies to several deformed hierarchies, including the noncommutative KP, modified KP, and Toda lattice hierarchy. We prove a Birkhoff factorization relation for the extended ncKP and ncmKP hierarchies. Also reductions of the latter hierarchies are briefly discussed. Furthermore, some results concerning the extended ncKP hierarchy are recalled from previous work.Comment: 15 pages, proceedings XI-th International Conference Symmetry Methods in Physics (Prague, June 2004

Differential Calculi on Quantum Spaces determined by Automorphisms

If the bimodule of 1-forms of a differential calculus over an associative algebra is the direct sum of 1-dimensional bimodules, a relation with automorphisms of the algebra shows up. This happens for some familiar quantum space calculi.Comment: 7 pages, Proceedings of XIIIth International Colloquium Integrable Systems and Quantum Group

Algebraic identities associated with KP and AKNS hierarchies

Explicit KP and AKNS hierarchy equations can be constructed from a certain set of algebraic identities involving a quasi-shuffle product.Comment: 6 pages, proceedings of Integrable Systems 2005, Pragu

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions show up in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.Comment: 16 page