The first classification of all primitive Lie pseudo-groups of infinite type was obtained by E. Cartan . According to his theorem, every primitive Lie pseudo-group of transformations of infinite type, consisting of holomorphic local transformations, is locally isomorphic to one of the pseudo-groups of Examples a)–f).
E.g., the pseudo-group of conformal transformations of the plane is a Lie pseudo-group of transformations, determined by the Cauchy–Riemann equations (cf. Cauchy-Riemann equations ). The order of a Lie pseudo-group of transformations is the minimum order of its defining system of differential equations.
A pseudo-group of transformations of the form $ Gamma ( G) $ is called globalizable. E.g., a pseudo-group of local conformal transformations of the sphere $ S ^ {n} $ is globalizable for $ n > 2 $ and not globalizable for $ n = 2 $.
A pseudo-group $ Gamma $ of transformations of a manifold $ M $ is called transitive if $ M $ is its only orbit, and is called primitive if $ M $ does not admit non-trivial $ Gamma $-invariant foliations (otherwise the pseudo-group is called imprimitive).