Pseudo-groups, moving frames, and differential invariants. (English) Zbl 1144.58013

Eastwood, Michael (ed.) et al., Symmetries and overdetermined systems of partial differential equations. Proceedings of the IMA summer program, Minneapolis, MN, USA, July 17–August 4, 2006. New York, NY: Springer (ISBN 978-0-387-73830-7/hbk). The IMA Volumes in Mathematics and its Applications 144, 127-149 (2008).
Lie pseudo-groups are the infinite-dimensional counterparts of local Lie groups of transformations. The first author has successfully reformulated the classical theory of moving frames in a general, algorithmic, and equivariant framework that can be readily applied to a wide range of finite-dimensional Lie group actions. The main goal in this paper is to survey the extension of the moving frame theory to general Lie pseudo-groups recently put forth by the authors with or without J. Chen [Sel. Math., New Ser. 11, No. 1, 99–126 (2005; Zbl 1091.58014), Symmetry and perturbation theory. Proceedings of the 5th international conference, Cala Gonone, Sardinia, Italy, May 30–June 6, 2004. Singapore: World Scientific, 244–254 (2005; Zbl 1136.58304), J. Math. Phys. 46, No. 2, 023504 (2005; Zbl 1076.37054)].
The paper consists of 6 sections. The first section is devoted to an introduction. In Section 2 the usual variational bicomplex structure on the diffeomorphism jets is employed to construct the Maurer-Cartan forms as invariant contact forms, and write out the complete system of structure equations. In Section 3 it is shown how the structure equations of a Lie pseudo-group are obtained by restricting the diffeomorphism structure equations to the solution space to the infinitesimal determining equations. In Section 4 the authors develop the moving frame constructions for the prolonged action on submanifold jets, and explain how to determine a complete system of differential invariants. In Section 5 they explicitly derive the recurrence formulae for the differentiated invariants, demonstrating, in particular, that the differential invariants of any transitive pseudo-group form a non-commutative rational differential algebra. In Section 6 they present a constructive version of the basis theorem that provides a finite system of generating differential invariants for a large class of pseudo-group actions and the generators of their differential syzygies.
For the entire collection see [Zbl 1126.35005].


58H05 Pseudogroups and differentiable groupoids
53A55 Differential invariants (local theory), geometric objects