Lie Pseudogroups à la Cartan from a Modern Perspective

Abstract

In 1904-05, the mathematician Élie Cartan published two pioneer papers in which he introduced a structure theory for Lie pseudogroups. Lie pseudogroups are mathematical objects that appear in both differential geometry and in the theory of differential equations as local symmetries of geometric structures and of systems of (partial) differential equations. In this thesis, we present a modern formulation of Cartan's theory. We introduce the notions of a Cartan algebroid and its realizations. These are global structures which, locally, correspond to Cartan’s notions of structure equations and their associated infinitesimal data, the structure functions, the main objects in Cartan´s theory. A Cartan algebroid generalizes the notion of a Lie algebra, while a realization of a Cartan algebroid generalizes the notion of a Lie algebra-valued Maurer-Cartan form. The starting point of Cartan’s theory are the three fundamental theorems. In the first and second fundamental theorems, Cartan proves that one can associate structure equations, and hence structure functions, with any Lie pseudogroup, and that these fully encode the Lie pseudogroup. In the third fundamental theorem, Cartan solves the so called “realization problem”; namely, that under certain conditions a given set of structure functions can be “integrated” to a set of structure equations, and hence to a Lie pseudogroup. Using our modern framework, we present coordinate-free proofs of the first and second fundamental theorems, with the main ingredients being jet spaces and Lie groupoids and algebroids. Turning to the third fundamental theorem, we present a new method for tackling this integrability problem and apply the method to prove two special cases of the theorem. We also indicate how one may use this method in order to address the general case, which, to date, has only been proven in the analytic case and locally (first by Cartan himself). Slightly deviating from the main storyline, we apply our method to obtain a new proof for the existence of local symplectic realizations of Poisson structures, one which sheds light on the role of the Poisson equation in the integration process. In the final part of the thesis, we turn to two key notions in Cartan’s structure theory. The first is the notion of a prolongation. We discuss prolongations in both the global and the infinitesimal levels, identify obstructions to their existence in the form of Spencer cohomology classes and prove two formal integrability theorems. The second is the notion of the systatic system and reduction, a type of algorithm introduced by Cartan which allows one to reduce a given Lie pseudogroup to one which is “smaller” but “equivalent”. In our modern formulation, the systatic system takes on a rather elegant form, namely that of a Lie algebroid contained within every Cartan algebroid together with a canonical action on all realizations of the Cartan algebroid. Inspired by Cartan’s reduction algorithm, we prove a reduction theorem which goes beyond Cartan’s reduction and leads us naturally to the realm of generalized pseudogroups, i.e. pseudogroups of bisections of Lie groupoids.