On the structure of Lie pseudo-groups. (English) Zbl 1241.58008

Olver, Peter J.; Pohjanpelto, Juha; Valiquette, Francis

SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 077, 14 p. (2009).

The authors compare and contrast two approaches to the structure equations for Lie pseudo-groups, the first due to Cartan, and the second due to the first two authors. The latter relies on the contact structure of the infinite diffeomorphism jet bundles, whereas Cartan’s is based on the prolongation of exterior differential systems. The authors construct the Maurer-Cartan forms associated to a Lie pseudo-group \(\mathcal G\) by pulling back the canonical Maurer-Cartan forms of the bundle of infinite jets of all diffeomorphisms of the underlying manifold to the bundle of infinite jets of the transformations of \(\mathcal G\). Then they deduce that both approaches are equivalent for transitive Lie pseudo-groups. Two examples illustrate what can happen in the nontransitive case. In the referee’s opinion, the contact form approach is so conceptual that we are entitled to expect it will lead to many further applications.

Reviewer: Ivan Kolář (Brno)

MSC:

58H05	Pseudogroups and differentiable groupoids
58A15	Exterior differential systems (Cartan theory)

Keywords:

Lie pseudo-group; jet; contact form; Maurer-Cartan form; structure equations; essential invariant