In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Prölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions. Bull. Sd. Math. 128 (2004) 513-529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal g-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov-Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of ç-deformed operators. The integration obtained by Muíase [Complete integrability of the Kadomtsev-Petviashvili equation Adv. Math. 54 (1984) 57-66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose-Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibre d'Holonomie en dimension infinie. Math. Rep. Canadian Roy. Math. Soc. 28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions. Bull. Sd. Math. 128 (2004) 513-529] is completed.

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of Mulase's deep algebraic factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients which are series in formal parameters, we describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and we discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-I equation.

Using weighted traces which are linear functionals of the type A → tr Q (A) := tr(AQ −z ) − z −1 tr(AQ −z ) z=0 defined on the whole algebra of (classical) pseudo-differential operators (P.D.O.s) and where Q is some positive invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces. RésuméA l'aide de traces pondérées qui sont des fonctionnelles linéaires du type:définies sur toute l'algèbre des opérateurs pseudo-différentiels classiques, Qétant un opérateur elliptique inversible, onétudie la géométrie de l'espace des lacetsà la lumière de la cohomologie des opérateurs pseudo-différentiels. On met en place un cadre géométrique afin d'étudier une classe de variétés de dimension infinie, cadre dans lequel on retrouve, toujoursà l'aide des traces pondérées, des résultats concernant la géométrie des lacets. Ces traces pondérées nous permettent aussi d'étendre la notion de cocycle de Radul et de Schwinger et d'enétudier certaines propriétés.