After refreshing the well-known works by Mulase, Reyman and Semenov-Tian-Shanski on the algebraic integration of the Kadomtsev-Petviashvili (KP) hierarchy via r-matrices, we show in this talk:
- firstly how the classical algebra of pseudo-differential operators over $S^1$ present in the formulation of the KP hierarchy, which coincide with the formal part of the so-called odd class (non formal) operators defined by Kontsevich and Vishik, can be extended to the class of formal classical pseudo-differential operators. For this, we describe the link between these two algebras which stands in an almost complex integrable structure. This part is based on a joint work with V. Rubtsov.
- secondly how the category of Frölicher spaces, which is a subcategory of diffeological spaces, can apply here to state rigorously smooth dependence of the solution of the KP hierarchy on the initial condition. This part is mostly based on a joint work with Enrique G. Reyes on the "classical" KP hierarchy, which applies also to KP hierarchy in an "extended" class of operators of previously exposed.