Recently, Olver and Pohjanpelto have successfully extended the theory of equivariant moving frames to infinite-dimensional Lie pseudo-groups. Based on its finite-dimensional counterpart, this new theory promises to be a source of interesting new results and applications. In this thesis, we look at two applications of this new theory.
By combining the powerful theories of Lie groupoids and variational bicomplexes, Olver and Pohjanpelto have developed a practical algorithm for determining the Maurer--Cartan structure equations of Lie pseudo-groups. The structure equations obtained with this new theory are compared with those derived by Cartan. It is shown that for transitive Lie pseudo-groups the two structure theories are isomorphic while for intransitive Lie pseudo-groups the two sets of structure equations do not agree. To make the two structure theory isomorphic we argue that Cartan's structure equations need to be slightly modified. The effect of this modification on Cartan's definition of essential invariants is analyzed.
In 1965, Singer and Sternberg gave an infinitesimal interpretation of Cartan's structure equations for transitive Lie pseudo-groups. This interpretation is extended to intransitive Lie pseudo-groups and the result is used to state a symmetry-based linearization theorem for systems of nonlinear partial differential equations which does not require the integration of the infinitesimal determining equations of the symmetry group.
The theory of equivariant moving frames is a powerful tool for determining a generating set of the differential invariant algebra of Lie pseudo-groups. After reviewing this theory, the method is illustrated with three applications. In the first two applications, generating sets of differential invariant algebra for the symmetry groups of the Infeld--Rowlands equation and the Davey--Stewartson equations are determined. Then we show that for two and three dimensional Riemannian manifolds the sectional curvatures generate the differential invariant algebra of the pseudo-group of locally invertible changes of variables.