Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equation

TL;DR

This paper extends the Ambrose-Singer theorem to diffeological bundles, reformulates regular Lie groups, and applies differential geometry to analyze the complete integrability of the KP hierarchy using $q$-deformed pseudo-differential operators.

Contribution

It generalizes the Ambrose-Singer theorem to diffeological bundles and develops a differential geometric framework for the KP hierarchy involving $q$-deformed operators.

Findings

Extended Ambrose-Singer theorem for diffeological bundles.

Reformulated regular Lie groups and holonomy in this context.

Achieved a geometric integration of the KP hierarchy using $q$-deformed operators.

Abstract

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Fr\"olicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, 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 Fr\"olicher (Lie) algebras such as formal $q-$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 in the sense of Frobenius. 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 $q-$deformed operators. The integration obtained by Mulase \cite{M1}, and the key tools he developped, 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 Fr\"olicher Lie group. The case of a (classical) Lie group modelled on a complete locally convex topological vector space is also recovered and the work developped in \cite{Ma,Ma2} is completed.