UBC Mathematics Department
http://www.math.ubc.ca
A pseudogroup of local diffeomorphisms of an analytic manifold is said to be a Lie pseudogroup if it is the set of solutions of an involutive exterior differential system. E. Cartan showed that any Lie pseudogroup can be characterized geometrically as the set of local automorphisms of a reduction of the frame bundle of a manifold associated to the given differential system. The Lie pseudogroups defined by exterior differential systems which are completely integrable in the sense of the Frobenius theorem are said to be of finite type, since their elements are parametrized locally by arbitrary constants. Classical examples are given by the local isometries of a Riemannian manifold, or the symmetries of a system of ordinary differential equations of order n\geq 2. Any Lie pseudogroup of finite type can be regarded as the representation by local diffeomorphisms of a finite-dimensional (local) Lie group. In contrast, we have the Lie pseudogroups of infinite type, whose elements are parametrized by arbitrary functions. In this case, the defining exterior differential system is involutive in the sense of the Cartan-Kahler Theorem. Simple examples are given by the local automorphisms of a symplectic or a contact manifold. There are basic questions about Lie pseudogroups of infinite type which are not yet fully understood. One would like for example to define an infinite-dimensional Lie group structure for these pseudogroups. This is of course not possible in general, because of the absence of a Frobenius theorem in most of the relevant infinite-dimensional spaces. We will review a number of classical results from the theory of Lie pseudogroups and sketch some recent developments concerning the latter problem.